TopoTuner: Topological Finetuning of Large Language Models

# TopoTuner: Topological Finetuning of Large Language Models
Source: [https://arxiv.org/html/2607.16637](https://arxiv.org/html/2607.16637)
Abdulkadir Erol Department of Computer Science University of Central Florida Orlando, USA abdulkadir\.erol@ucf\.edu&Yash Mahajan Department of Computer Science University of Central Florida Orlando, USA yash\.mahajan@ucf\.edu&Vepaul Hariprashad Department of Computer Science University of Central Florida Orlando, USA vepaul\.hariprashad@ucf\.edu&Baha Rababah Department of Computer Science University of Manitoba Canada rababahb@myumanitoba\.ca&Santu Karmaker Department of Computer Science University of Central Florida Orlando, USA santu\.karmaker@ucf\.edu&Cuneyt G\. Akcora Department of Finance University of Central Florida Orlando, USA cuneyt\.akcora@ucf\.edu&Mubarak Shah Department of Computer Science University of Central Florida Orlando, USA mubarak\.shah@ucf\.edu

###### Abstract

Full fine\-tuning remains a strong way to adapt pretrained LLMs, but it updates all weights and can be expensive\. LoRA reduces the number of trainable parameters, but it does not directly answer which pretrained components should be trained and which can be frozen during adaptation\. We introduceTopoTuner, a topology\-guided fine\-tuning framework for selective freezing of attention projection matrices\.TopoTunertreats each projection matrix as a row cloud and uses Wasserstein distances between persistence diagrams to measure how its topology changes during fine\-tuning\.

TopoTunerlearns a reusable freezing profile from a source dataset and transfers it to efficiently fine\-tune models on out\-of\-domain datasets, evaluating whether task\-specific topological drift generalizes across question answering and sentiment analysis tasks\.

Across LLaMA\-3\.1\-8B, Mistral\-7B\-v0\.3, and Qwen3\-8B\-Base,TopoTuneris competitive with full fine\-tuning while training only 1\-2% of the model parameters, and outperforms LoRA in 7 out of 9 model\-dataset settings, which can change up to 39\.57% of the projection parameters\. Along with minimized updates,TopoTunerreduces training time by 20\.4% relative to full fine\-tuning and 5\.5% relative to LoRA on average\.TopoTuneropens a new direction for reusable freezing profiles, where fine\-tuning behavior learned on one dataset can be shared across multiple tasks\.

## 1Introduction

Fine\-tuning\[[18](https://arxiv.org/html/2607.16637#bib.bib33),[37](https://arxiv.org/html/2607.16637#bib.bib21)\]is the standard mechanism for adapting pretrained large language models \(LLMs\) to downstream tasks, such as question answering\[[40](https://arxiv.org/html/2607.16637#bib.bib20)\]\. Full fine\-tuning updates all model weights and remains a strong baseline, but it is expensive and can damage capabilities acquired during pretraining with risks of catastrophic forgetting\[[20](https://arxiv.org/html/2607.16637#bib.bib28)\], overfitting to narrow task distributions\[[15](https://arxiv.org/html/2607.16637#bib.bib22)\], and instability in alignment behavior\[[19](https://arxiv.org/html/2607.16637#bib.bib27)\]\.

Parameter\-efficient methods \(PEFT\)\[[37](https://arxiv.org/html/2607.16637#bib.bib21)\], such as LoRA\[[13](https://arxiv.org/html/2607.16637#bib.bib32)\], reduce the number of trainable parameters and often recover strong task performance\. However, matching downstream accuracy does not answer a basic structural question: which parts of the pretrained model actually need to change during adaptation? Furthermore, although LoRA uses low\-rank updates, it can still apply updates to many, or even all, weight matrices without explicit control over pretrained components\.

The question matters because fine\-tuning is rarely single\-task in practice\[[14](https://arxiv.org/html/2607.16637#bib.bib31)\]\. Models are adapted across domains and often reused across datasets\. A method that only controls the number of trainable parameters does not show which parts of the model change, whether the same parts change across related tasks, or at what epoch the main structural change has already occurred\. Standard summaries such as spectra, norms, and sparsity measure the size or rank structure of an update, but they do not directly describe how the structure of a weight matrix changes\[[29](https://arxiv.org/html/2607.16637#bib.bib30)\]\.

Topology\[[3](https://arxiv.org/html/2607.16637#bib.bib46)\]offers a principled way to study how model parameters reorganize during fine\-tuning\. We can treat each attention projection matrix as a point cloud formed by its rows and use topological data analysis\[[5](https://arxiv.org/html/2607.16637#bib.bib43)\]to obtain stable multiscale summaries of this geometry\. The distance between the persistence diagrams of pretrained and fine\-tuned matrices then gives a matrix\-level measure of structural change, which can be used to rank matrices for freezing\. This view also suggests a sufficient condition for reusing such freezing profiles across tasks: the fine\-tuning trajectories need not be identical; it suffices that the induced gradient fields remain close to the pretrained model\.

This insight allows us to introduceTopoTuner, a topology\-guided framework for measuring and controlling LLM fine\-tuning\. The method tracks topological drift, the structural change of each weight matrix relative to its pretrained state, and uses this signal to build reusable freezing profiles and to stop training once the main structural change has occurred\. This reframes fine\-tuning from a parameter\-count problem into a structural\-control problem: instead of asking only how many parameters to update,TopoTunerasks which pretrained components should be allowed to reorganize\.

We evaluateTopoTuneron three language models across six task families: question answering, sentiment analysis, information retrieval, summarization, instruction following, and code generation\. Our results show that drift profiles are model\-dependent, ruling out a fixed projection\-level rule across architectures\. Yet, within a fixed model, profiles learned from one dataset can transfer to other tasks, reducing full fine\-tuning time by20\.4%20\.4\\%and LoRA fine\-tuning time by5\.5%5\.5\\%on average while preserving competitive accuracy\.

Topological drift also reveals how freezing changes the adaptation path\. Freezing low\-drift matrices can leave the main route open and spread updates across many layers, while freezing high\-drift matrices, as shown in Figure[1](https://arxiv.org/html/2607.16637#S1.F1), can block that route and force the remaining trainable matrices to absorb updates in a more concentrated pattern, reducing overfitting\. This redistribution explains why drift is useful for matrix selection, task comparison, and early stopping\.

![Refer to caption](https://arxiv.org/html/2607.16637v1/x1.png)Figure 1:LLaMAOO\-projection change under high\-drift Wasserstein freezing \(33to1515layers\)\. As more matrices are frozen during training, the remaining trainable matrices absorb updates in a more concentrated manner; selective freezing redistributes adaptation across layers\.Our contributions are as follows\.

- •We introduceTopoTuner, a topology\-guided framework that measures how attention projection matrices reorganize during fine\-tuning and uses this signal to observe and control which matrices remain trainable\.To our knowledge, this is the first work that uses topological summaries of LLM weights to construct reusable freezing profiles across tasks\.
- •We define matrix\-level topological drift through Wasserstein distances between persistence diagrams of pretrained and fine\-tuned weights\. We show that this drift captures structural change beyond entrywise magnitude, and use it for selective matrix freezing and topology\-based early stopping\.
- •We compareTopoTuneragainst PEFT and full\-fine\-tuning baselines\. Across transferred\-knowledge experiments,TopoTuneroutperforms LoRA in77out of99model\-dataset settings while reducing wall\-clock training time by20\.4%20\.4\\%on average and using88\.36%88\.36\\%fewer parameter updates than LoRA\.TopoTunerstops fine\-tuning after two to three epochs in most runs, reducing the six\-epoch budget by50%50\\%–66\.7%66\.7\\%while retaining near\-final validation accuracy\.

![Refer to caption](https://arxiv.org/html/2607.16637v1/x2.png)Figure 2:Overview ofTopoTuner\.TopoTuneruses a prior fine\-tuning task to compute the topological drift of attention projection matrices\. The computed topological drift is used to \(A\) build reusable freezing profiles for selective fine\-tuning on new datasets and tasks, and to \(B\) track structural change across epochs for early stopping\. Freezing profiles are provided in Appendix[B](https://arxiv.org/html/2607.16637#A2)\.
## 2Related Work

Topological methods\.Persistent homology \(PH\) has emerged as a powerful tool for analyzing neural network representations\. Pérez\-Fernández et al\. treat a neural network as an abstract simplicial complex and extract its topological fingerprint via persistent homology\[[24](https://arxiv.org/html/2607.16637#bib.bib40)\]\. The authors define a PH\-based descriptor that can serve as a similarity measure between trained models\. Moreover, persistent homology can reveal layer\-wise organizational differences\. Purvine et al\. analyze hidden representations of CNNs by treating each layer’s activations as a high\-dimensional point cloud\[[25](https://arxiv.org/html/2607.16637#bib.bib41)\]\. The authors compute persistence\-based distances between layers’ activation spaces and show that these distances capture meaningful differences in layer behavior\. These works establish that persistent homology, via persistence diagrams and their distances, provides a rigorous framework to quantify structural similarities in deep networks\. However, to the best of our knowledge, no prior work has used topological signals to guide LLM fine\-tuning via selective freezing\.

Fine\-tuning strategies\.Fine\-tuning large transformer models can be approached at the granularity of individual weight matrices or entire layers\. A variety of matrix\-level adaptation techniques, often termed parameter\-efficient fine\-tuning \(PEFT\[[37](https://arxiv.org/html/2607.16637#bib.bib21)\]\), restrict which parameters are updated to reduce computational cost\. A prominent example is LoRA\[[13](https://arxiv.org/html/2607.16637#bib.bib32)\], which inserts low\-rank update matrices into weight layers\. Building on this idea, “Deep LoRA”\[[39](https://arxiv.org/html/2607.16637#bib.bib42)\]applies a low\-rank decomposition to multiple transformer layers simultaneously, to confine the fine\-tuning updates to a low\-dimensional subspace at each layer\. This strategy mitigates overfitting with limited data\. On the other end of the spectrum, layer\-level strategies decide which whole layers to train or freeze\. Zheng et al\.\[[42](https://arxiv.org/html/2607.16637#bib.bib44)\]freeze the bottom25%25\\%of transformer layers, and achieve better downstream performance in continual learning scenarios\. In addition to static freezing, recent methods dynamically adjust training at the layer level\. A representative approach is DropBP, which randomly omits gradient backpropagation through certain layers at each update, guided by a sensitivity metric that estimates each layer’s impact on the training loss\[[35](https://arxiv.org/html/2607.16637#bib.bib17)\]\. Layers with lower sensitivity receive a higher drop rate, effectively being temporarily frozen during some training steps\. This adaptive schedule trains only the most“critical”layers at full frequency and reduces the time costs\. In contrast, we use topological drift to measure how each attention projection matrix reorganizes relative to the pretrained model, and use this signal to build reusable freezing profiles and to decide when fine\-tuning has stabilized\.

## 3Preliminaries

Letfθf\_\{\\theta\}denote a pretrained large language model, parameterized by a collection of weight matricesθ=\{Wi\}i=1m\\theta=\\\{W\_\{i\}\\\}\_\{i=1\}^\{m\}, whereWi∈ℝdi×di′W\_\{i\}\\in\\mathbb\{R\}^\{d\_\{i\}\\times d\_\{i\}^\{\\prime\}\}\. We useℐ⊆\{1,…,m\}\\mathcal\{I\}\\subseteq\\\{1,\\dots,m\\\}to denote the index set of matrices\.

Task and fine\-tuning\.Given a downstream task𝒯\\mathcal\{T\}with training data𝒟𝒯=\{\(xr,yr\)\}r=1N\\mathcal\{D\}\_\{\\mathcal\{T\}\}=\\\{\(x\_\{r\},y\_\{r\}\)\\\}\_\{r=1\}^\{N\}, fine\-tuning solvesθ~=arg⁡minϑ⁡1N​∑r=1Nℓ​\(yr,fϑ​\(xr\)\),\\tilde\{\\theta\}=\\arg\\min\_\{\\vartheta\}\\frac\{1\}\{N\}\\sum\_\{r=1\}^\{N\}\\ell\\big\(y\_\{r\},\\ f\_\{\\vartheta\}\(x\_\{r\}\)\\big\),whereℓ\\ellis a supervised loss function\. The resulting model is denoted byfθ~f\_\{\\tilde\{\\theta\}\}with parametersθ~=\{W~i\}i=1m\\tilde\{\\theta\}=\\\{\\tilde\{W\}\_\{i\}\\\}\_\{i=1\}^\{m\}\.

Topological tools\.Persistent homology is a tool from topological data analysis that summarizes how topology evolves across a filtration\[[5](https://arxiv.org/html/2607.16637#bib.bib43),[12](https://arxiv.org/html/2607.16637#bib.bib45)\]\. Let𝒳=x1,…,xn⊂ℝd\\mathcal\{X\}=\{x\_\{1\},\\dots,x\_\{n\}\}\\subset\\mathbb\{R\}^\{d\}be a point cloud equipped with a distance functionΔ\\Delta\. We construct the Vietoris–Rips filtration\{VRε​\(𝒳\)\}ε≥0\\\{\\mathrm\{VR\}\_\{\\varepsilon\}\(\\mathcal\{X\}\)\\\}\_\{\\varepsilon\\geq 0\}, where simplices are added according to pairwise distance at scaleε\\varepsilon\[[5](https://arxiv.org/html/2607.16637#bib.bib43)\]\. For simplicity, we write this filtration as𝒦1⊂𝒦2⊂⋯⊂𝒦n\\mathcal\{K\}\_\{1\}\\subset\\mathcal\{K\}\_\{2\}\\subset\\dots\\subset\\mathcal\{K\}\_\{n\}\.

Topological features and persistence diagrams\.For each complex𝒦i\\mathcal\{K\}\_\{i\}in the filtration, letHk​\(𝒦i\)H\_\{k\}\(\\mathcal\{K\}\_\{i\}\)denote itskkth homology group, whose nontrivial classes representkk\-dimensional topological features such as connected components fork=0k=0, loops fork=1k=1, and voids fork=2k=2\. As the filtration grows, features appear and disappear;qj=\(bj,dj\)∈PD​\(𝒳\)q\_\{j\}=\(\{\\color\[rgb\]\{0,\.5,\.5\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,\.5,\.5\}b\_\{j\}\},\{\\color\[rgb\]\{1,\.5,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,\.5,0\}d\_\{j\}\}\)\\in\\rm\{PD\}\(\\mathcal\{X\}\)represents thebirthanddeathtimes of a topological feature in𝒳\\mathcal\{X\}within a Persistence Diagram \(PD\\rm\{PD\}\)\. Thekkth persistence diagram, denoted byPDk​\(𝒳\)\\mathrm\{PD\}\_\{k\}\(\\mathcal\{X\}\), is the multiset ofbirth\-deathpairs for all such features\. Features with larger persistencedj−bjd\_\{j\}\-b\_\{j\}are often interpreted as more salient\.

pp\-Wasserstein distance\.Given two persistence diagramsPDk​\(𝒳\)\\rm\{PD\}\_\{k\}\(\\mathcal\{X\}\)andPDk​\(𝒳′\)\\rm\{PD\}\_\{k\}\(\\mathcal\{X\}^\{\\prime\}\)in the same homological dimensionkk, theirpp\-Wasserstein distance is𝒲assp\(PDk\(𝒳\),PDk\(𝒳′\)\)=minϕ\(∑j∥qj−ϕ\(qj\)∥∞p\)1/p,\\mathcal\{W\}\{ass\}\_\{p\}\(\\rm\{PD\}\_\{k\}\(\\mathcal\{X\}\),\\rm\{PD\}\_\{k\}\(\\mathcal\{X\}^\{\\prime\}\)\)=\\min\_\{\\phi\}\\left\(\\sum\_\{j\}\\\|q\_\{j\}\-\\phi\(q\_\{j\}\)\\\|\_\{\\infty\}^\{p\}\\right\)^\{1/p\},whereϕ\\phiranges over partial matchings with diagonal matching allowed\. This distance measures how much the underlying topological summaries are different between two point clouds\.

## 4Topological Characteristics of LLM Fine\-tuning

Topological data analysis is based on the principle that data has shape and shape has meaning\[[3](https://arxiv.org/html/2607.16637#bib.bib46)\]\. Accordingly, we treat each projection \(i\.e\., weight matrix\) in an LLM as a geometric object and study how its internal structure evolves during fine\-tuning across epochs\. Our topology\-based hypothesis is that persistence diagrams of a projection provide a topological summary of this organization, and that the distance between pre\- and post\-fine\-tuning diagrams yields a useful projection\-level measure of structural change\. Our goal is to use this measure to make fine\-tuning more controlled: identify which projections can be frozen, reuse those freezing profiles across datasets, and stop training once the remaining trainable projections have stabilized\.

Letfθf\_\{\\theta\}be a pretrained transformer with layer setℒ=\{1,…,L\}\\mathcal\{L\}=\\\{1,\\dots,L\\\}\. For each layerl∈ℒl\\in\\mathcal\{L\}, letWl\(q\)W\_\{l\}^\{\(q\)\},Wl\(k\)W\_\{l\}^\{\(k\)\},Wl\(v\)W\_\{l\}^\{\(v\)\}, andWl\(o\)W\_\{l\}^\{\(o\)\}denote the query, key, value, and output projections\. Letℐ\\mathcal\{I\}denote the collection of matrices considered for ranking and freezing\.

Topological sensitivity pipeline\.For each matrixWi∈ℝdi×di′W\_\{i\}\\in\\mathbb\{R\}^\{d\_\{i\}\\times d\_\{i\}^\{\\prime\}\}, we form the row cloudXi=\{x1\(i\),…,xdi\(i\)\}⊂ℝdi′X\_\{i\}=\\\{x\_\{1\}^\{\(i\)\},\\dots,x\_\{d\_\{i\}\}^\{\(i\)\}\\\}\\subset\\mathbb\{R\}^\{d\_\{i\}^\{\\prime\}\}, wherexa\(i\)x\_\{a\}^\{\(i\)\}is theaath row ofWiW\_\{i\}\. We equipXiX\_\{i\}with a distance metricδi\\delta\_\{i\}and, for each thresholdε≥0\\varepsilon\\geq 0, build the graph𝒢ε\(i\)\\mathcal\{G\}\_\{\\varepsilon\}^\{\(i\)\}with vertex setXiX\_\{i\}and edges between two data points\(a,b\)\(a,b\)whenever their distanceδi​\(a,b\)≤ε\\delta\_\{i\}\(a,b\)\\leq\\varepsilon\. Promoting every clique in𝒢ε\(i\)\\mathcal\{G\}\_\{\\varepsilon\}^\{\(i\)\}to a simplex gives the Vietoris–Rips complexVRε\(i\)\\mathrm\{VR\}\_\{\\varepsilon\}^\{\(i\)\}\[[12](https://arxiv.org/html/2607.16637#bib.bib45)\], and the nested family\{VRε\(i\)\}ε≥0\\\{\\mathrm\{VR\}\_\{\\varepsilon\}^\{\(i\)\}\\\}\_\{\\varepsilon\\geq 0\}defines a filtration\. From this filtration we compute persistence diagramsP​Dk​\(Xi\)PD\_\{k\}\(X\_\{i\}\)in homological dimensionsk∈\{0,1,…\}k\\in\\\{0,1,\\ldots\\\}\.

###### Definition 4\.1\(Topological distance to pretrained model\)\.

LetWi\(0\)W^\{\(0\)\}\_\{i\}be the pretrained version of matrixi∈ℐi\\in\\mathcal\{I\}, and letW~i\(t\)\\tilde\{W\}\_\{i\}^\{\(t\)\}denote its version at the fine\-tuning epocht≥1t\\geq 1\. For a fixed homological dimensionkk, we define the topological distance of matrixiiatttbyDi\(t\):=𝒲​a​s​sp​\(PDk​\(Xi\(0\)\),PDk​\(Xi\(t\)\)\)D\_\{i\}^\{\(t\)\}:=\\mathcal\{W\}\{ass\}\_\{p\}\(\\mathrm\{PD\}\_\{k\}\(X^\{\(0\)\}\_\{i\}\),\\mathrm\{PD\}\_\{k\}\(X\_\{i\}^\{\(t\)\}\)\)\.

Distances are always computed with respect to the pretrained model \(denoted as epocht=0t=0\)\. We next define drift, which uses the distance from the pretrained model to the first epoch as a reference\.

###### Definition 4\.2\(Topological drift\)\.

LetDi\(t\)D\_\{i\}^\{\(t\)\}denote the fine\-tuning induced topological distance for matrixiiat the fine\-tuning epochtt\. We define the topological drift att≥2t\\geq 2byΔi\(t\):=Di\(t\)−Di\(t−1\)Di\(1\)\\Delta\_\{i\}^\{\(t\)\}:=\\frac\{D\_\{i\}^\{\(t\)\}\-D\_\{i\}^\{\(t\-1\)\}\}\{D\_\{i\}^\{\(1\)\}\}\.

The denominator \(first epoch as a reference\) provides a per\-layer, epoch\-to\-epoch scaling of change\.

Stability\.The usefulness ofΔi\\Delta\_\{i\}rests on the stability principle of PH: small perturbations in the underlying metric induce small perturbations in the resulting diagrams\. In our setting, this means that if fine\-tuning changes pairwise distances between rows only slightly, then the corresponding topological drift must also remain small\. The next proposition makes this statement explicit\.

###### Proposition 4\.3\(Stability of topological sensitivity\)\.

LetX=\{x1,…,xd\}X=\\\{x\_\{1\},\\dots,x\_\{d\}\\\}andX~=\{x~1,…,x~d\}\\tilde\{X\}=\\\{\\tilde\{x\}\_\{1\},\\dots,\\tilde\{x\}\_\{d\}\\\}be the row clouds ofWiW\_\{i\}andW~i\\tilde\{W\}\_\{i\}, indexed consistently\. Letddandd~\\tilde\{d\}be the metrics used to construct the Vietoris–Rips filtrations ofXXandX~\\tilde\{X\}\. If‖d−d~‖∞:=maxa,b⁡\|d​\(a,b\)−d~​\(a,b\)\|≤η\\\|d\-\\tilde\{d\}\\\|\_\{\\infty\}:=\\max\_\{a,b\}\|d\(a,b\)\-\\tilde\{d\}\(a,b\)\|\\leq\\eta, then for everyk≥0k\\geq 0we havedB​\(P​Dk​\(X\),P​Dk​\(X~\)\)≤ηd\_\{B\}\(PD\_\{k\}\(X\),PD\_\{k\}\(\\tilde\{X\}\)\)\\leq\\eta, and thereforeWp​\(P​Dk​\(X\),P​Dk​\(X~\)\)≤Nk1/p​ηW\_\{p\}\(PD\_\{k\}\(X\),PD\_\{k\}\(\\tilde\{X\}\)\)\\leq N\_\{k\}^\{1/p\}\\eta, wheredBd\_\{B\}is the bottleneck distance andNkN\_\{k\}is the maximum number of off\-diagonal points in the two diagrams\. Due to space limitations, the proof is given in Appendix[Q](https://arxiv.org/html/2607.16637#A17)\.

## 5Freezing Strategies Based on Topological Drift

We use topological drift to guide efficient adaptation and to study how matrix topology reorganizes during fine\-tuning\. Our framework is visualized in Figure[2](https://arxiv.org/html/2607.16637#S1.F2)\. We consider two settings: prior\-based freezing and dynamic stopping\. The first uses a prior fine\-tuning task to construct a fine\-tuning profile, which is then reused for fast selective fine\-tuning on new tasks and datasets\. The second uses topological drift observed during the ongoing fine\-tuning run to track structural change across epochs and terminate fine\-tuning as early as possible\. An algorithm for the freezing procedure is provided in Appendix[H](https://arxiv.org/html/2607.16637#A8)\.

### Scenario A: Fine\-tuning with Freezing Priors

We posit a subtle but important insight: reusable freezing profiles do not require identical fine\-tuning trajectories across tasks; it suffices that the induced gradient fields remain close to the pretrained model, limiting drift \(i\.e\.,model change\) that could otherwise cause catastrophic forgetting\.

Letθ\(t\)\\theta^\{\(t\)\}denote the model obtained by fine\-tuning the pretrained networkfθf\_\{\\theta\}on a single prior task𝒯\\mathcal\{T\}fort=\{1,…,tm​a​x\}t=\\\{1,\\ldots,t\_\{max\}\\\}epochs\. For each matrixWi∈ℐW\_\{i\}\\in\\mathcal\{I\}, we first compute its prior\-task topological distanceDi:=𝒲​a​s​sp​\(PDk​\(Xi\(0\)\),PDk​\(Xi\(tm​a​x\)\)\),D\_\{i\}:=\\mathcal\{W\}\{ass\}\_\{p\}\(\\mathrm\{PD\}\_\{k\}\(X^\{\(0\)\}\_\{i\}\),\\mathrm\{PD\}\_\{k\}\(X\_\{i\}^\{\(t\_\{max\}\)\}\)\),whereXi\(tm​a​x\)X\_\{i\}^\{\(t\_\{max\}\)\}is the point cloud of the post\-fine\-tuning version ofWiW\_\{i\}\. Next, we also measure the magnitude change ofWiW\_\{i\}byMi:=1\|Wi\|​∑a,b\|Wi,a​b\(tm​a​x\)−Wi,a​b\(0\)\|Wi,a​b\(0\)\|\|M\_\{i\}:=\\frac\{1\}\{\|W\_\{i\}\|\}\\sum\_\{a,b\}\\left\|\\frac\{W\_\{i,ab\}^\{\(t\_\{max\}\)\}\-W\_\{i,ab\}^\{\(0\)\}\}\{\|W\_\{i,ab\}^\{\(0\)\}\|\}\\right\|element\-wise\. We use this magnitude\-based change as a baseline to discuss the usefulness of a TDA\-guided strategy\. Using the prior\-task model, we build distance\-based matrix rankings that are later reused on a new task𝒯∗\\mathcal\{T\}^\{\\ast\}\.

We now rank matrices from the prior task and reuse the resulting freezing set on the target task\. For a budgetbb, letℱpriorTDA​\-​H​\(b\)\\mathcal\{F\}\_\{\\mathrm\{prior\}\}^\{\\mathrm\{TDA\\text\{\-\}H\}\}\(b\)be thebbmatrices with the largest values ofDiD\_\{i\}, and letℱpriorTDA​\-​L​\(b\)\\mathcal\{F\}\_\{\\mathrm\{prior\}\}^\{\\mathrm\{TDA\\text\{\-\}L\}\}\(b\)be thebbmatrices with the smallest values\. The magnitude baseline uses the same ranking procedure withMiM\_\{i\}\. The selected matrices are frozen from the start of fine\-tuning on𝒯∗\\mathcal\{T\}^\{\\ast\}\. The theorem below gives a sufficient condition under which the prior\-task ranking is preserved on𝒯∗\\mathcal\{T\}^\{\\ast\}, making the freezing profile reusable\.

###### Theorem 5\.1\(Stability of reusable freezing profiles\)\.

Fix a pretrained transformerfθ\(0\)f\_\{\\theta^\{\(0\)\}\}and a monitored matrix setℐ\\mathcal\{I\}\. Let𝒯\\mathcal\{T\}be a prior task and let𝒯∗\\mathcal\{T\}^\{\\ast\}be a target task\. For each task, letθ𝒯\(t\)=\{Wi,𝒯\(t\)\}i=1m\\theta\_\{\\mathcal\{T\}\}^\{\(t\)\}=\\\{W\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\\\}\_\{i=1\}^\{m\}andθ𝒯∗\(t\)=\{Wi,𝒯∗\(t\)\}i=1m\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}=\\\{W\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\\\}\_\{i=1\}^\{m\}denote the fine\-tuned parameters afterttoptimization steps, both initialized atθ\(0\)\\theta^\{\(0\)\}\. LetL𝒯​\(θ\)L\_\{\\mathcal\{T\}\}\(\\theta\)andL𝒯∗​\(θ\)L\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\)be the corresponding population fine\-tuning objectives, and writeG𝒯​\(θ\)=∇θL𝒯​\(θ\)G\_\{\\mathcal\{T\}\}\(\\theta\)=\\nabla\_\{\\theta\}L\_\{\\mathcal\{T\}\}\(\\theta\)andG𝒯∗​\(θ\)=∇θL𝒯∗​\(θ\)G\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\)=\\nabla\_\{\\theta\}L\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\)\. Assume both trajectories remain in a ballB​\(θ\(0\),R\)B\(\\theta^\{\(0\)\},R\), both gradient fields areLL\-Lipschitz on this ball, andsupθ∈B​\(θ\(0\),R\)‖G𝒯​\(θ\)−G𝒯∗​\(θ\)‖2≤ϵ\\sup\_\{\\theta\\in B\(\\theta^\{\(0\)\},R\)\}\\\|G\_\{\\mathcal\{T\}\}\(\\theta\)\-G\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\)\\\|\_\{2\}\\leq\\epsilon\. If both tasks are fine\-tuned with the same gradient descent update and step sizeα\\alpha, then for everyt≤tmaxt\\leq t\_\{\\max\}we have‖θ𝒯\(t\)−θ𝒯∗\(t\)‖2≤Rt\\\|\\theta\_\{\\mathcal\{T\}\}^\{\(t\)\}\-\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\\\|\_\{2\}\\leq R\_\{t\}, whereRt:=α​ϵ​∑r=0t−1\(1\+α​L\)rR\_\{t\}:=\\alpha\\epsilon\\sum\_\{r=0\}^\{t\-1\}\(1\+\\alpha L\)^\{r\}\.

Assume that the row\-cloud metricδi\\delta\_\{i\}is fixed across tasks and stable under row perturbations\. For eachi∈ℐi\\in\\mathcal\{I\}and fixed homological dimensionkk, defineDi,𝒯\(t\):=𝒲​a​s​sp​\(PDk​\(Xi\(0\)\),PDk​\(Xi,𝒯\(t\)\)\)D\_\{i,\\mathcal\{T\}\}^\{\(t\)\}:=\\mathcal\{W\}\{ass\}\_\{p\}\(\\mathrm\{PD\}\_\{k\}\(X\_\{i\}^\{\(0\)\}\),\\mathrm\{PD\}\_\{k\}\(X\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\)\)andDi,𝒯∗\(t\):=𝒲​a​s​sp​\(PDk​\(Xi\(0\)\),PDk​\(Xi,𝒯∗\(t\)\)\)D\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}:=\\mathcal\{W\}\{ass\}\_\{p\}\(\\mathrm\{PD\}\_\{k\}\(X\_\{i\}^\{\(0\)\}\),\\mathrm\{PD\}\_\{k\}\(X\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\)\)\. Then\|Di,𝒯\(t\)−Di,𝒯∗\(t\)\|≤2​Ni,k1/p​Rt\|D\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\-D\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\|\\leq 2N\_\{i,k\}^\{1/p\}R\_\{t\}, whereNi,kN\_\{i,k\}is the maximum number of off\-diagonal points in the two PDs for matrixiiin dimensionkk\.

Takek=0k=0and define the prior and target ranking scores bySiprior:=Di,𝒯\(tmax\)S\_\{i\}^\{\\mathrm\{prior\}\}:=D\_\{i,\\mathcal\{T\}\}^\{\(t\_\{\\max\}\)\}andSi∗:=Di,𝒯∗\(tmax\)S\_\{i\}^\{\\ast\}:=D\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\_\{\\max\}\)\}\. LetS\(1\)prior≥⋯≥S\(\|ℐ\|\)priorS\_\{\(1\)\}^\{\\mathrm\{prior\}\}\\geq\\cdots\\geq S\_\{\(\|\\mathcal\{I\}\|\)\}^\{\\mathrm\{prior\}\}be the sorted prior scores, and letgH​\(b\):=S\(b\)prior−S\(b\+1\)priorg\_\{H\}\(b\):=S\_\{\(b\)\}^\{\\mathrm\{prior\}\}\-S\_\{\(b\+1\)\}^\{\\mathrm\{prior\}\}be the high\-drift margin for freezing budgetbb\. Defineℱ∗TDA​\-​H​\(b\)\\mathcal\{F\}\_\{\\ast\}^\{\\mathrm\{TDA\\text\{\-\}H\}\}\(b\)andℱ∗TDA​\-​L​\(b\)\\mathcal\{F\}\_\{\\ast\}^\{\\mathrm\{TDA\\text\{\-\}L\}\}\(b\)analogously from the target scoresSi∗S\_\{i\}^\{\\ast\}\. Letδmax:=2​maxi∈ℐ⁡Ni,01/p​Rtmax\\delta\_\{\\max\}:=2\\max\_\{i\\in\\mathcal\{I\}\}N\_\{i,0\}^\{1/p\}R\_\{t\_\{\\max\}\}\. IfgH​\(b\)\>2​δmaxg\_\{H\}\(b\)\>2\\delta\_\{\\max\}, thenFpriorTDA−H​\(b\)=F∗TDA−H​\(b\)F\_\{\\mathrm\{prior\}\}^\{\\mathrm\{TDA\-H\}\}\(b\)=F\_\{\\ast\}^\{\\mathrm\{TDA\-H\}\}\(b\)\. The same statement holds forFpriorTDA−L​\(b\)F\_\{\\mathrm\{prior\}\}^\{\\mathrm\{TDA\-L\}\}\(b\)after sorting the scores in increasing order and using the corresponding low\-drift margin\. Due to space limitations, the proof is given in Appendix[Q](https://arxiv.org/html/2607.16637#A17)\.

### Scenario B: Fine\-tuning with Dynamic Stopping

We now consider the second setting, where no prior fine\-tuning profile is available\. Let\{θ\(t\)\}t=0tmax\\\{\\theta^\{\(t\)\}\\\}\_\{t=0\}^\{t\_\{\\max\}\}denote the checkpoint sequence of a single full fine\-tuning run\. In this setting, our goal is to stop fine\-tuning once the monitored matrices have stabilized topologically\. Prior work offers evidence that“a few epochs of fine\-tuning are typically sufficient to achieve strong performance”\[[31](https://arxiv.org/html/2607.16637#bib.bib34)\], with later epochs often providing limited additional gains\.

Using the topological distanceDi\(t\)D\_\{i\}^\{\(t\)\}from Definition[4\.1](https://arxiv.org/html/2607.16637#S4.Thmtheorem1), we compute the normalized epoch\-to\-epoch driftΔi\(t\)\\Delta\_\{i\}^\{\(t\)\}from Definition[4\.2](https://arxiv.org/html/2607.16637#S4.Thmtheorem2)\. For a monitored set of matrices𝒲⊆ℐ\\mathcal\{W\}\\subseteq\\mathcal\{I\}, we average this drift into a model\-level stabilization score as follows\.

###### Definition 5\.2\(Topology\-based stopping score\)\.

Let𝒲⊆ℐ\\mathcal\{W\}\\subseteq\\mathcal\{I\}be the monitored set of matrices\. Fort≥2t\\geq 2, we definer\(t\):=1\|𝒲\|​∑i∈𝒲\|Δi\(t\)\|r^\{\(t\)\}:=\\frac\{1\}\{\|\\mathcal\{W\}\|\}\\sum\_\{i\\in\\mathcal\{W\}\}\|\\Delta\_\{i\}^\{\(t\)\}\|\. This score measures the average normalized topological drift of the monitored matrices between consecutive checkpoints\.

Given a thresholdη∈\(0,1\)\\eta\\in\(0,1\), we choose the stopping epocht∗:=min⁡\{t≥2:r\(t\)≤η\}t^\{\\ast\}:=\\min\\\{t\\geq 2:r^\{\(t\)\}\\leq\\eta\\\}\. If no such epoch is found within the training budget, we stop at the final epoch\. The rule stops only when the average structural drift of the monitored matrices has become small; if substantial drift continues at a later epoch, thenr\(t\)r^\{\(t\)\}remains large and training continues\.

Computational complexity\.TopoTuneris computationally efficient\. For each monitored matrixWi∈ℝdi×di′W\_\{i\}\\in\\mathbb\{R\}^\{d\_\{i\}\\times d^\{\\prime\}\_\{i\}\}, computing pairwise row distances costsO​\(di2​di′\)O\(d\_\{i\}^\{2\}d^\{\\prime\}\_\{i\}\)\. Constructing the Vietoris–Rips filtration and computing persistence has worst\-case exponential complexity indid\_\{i\}, since the number of simplices can grow exponentially\. In practice, we compute only low\-dimensional persistence,i\.e\.,H0H\_\{0\}and limit the analysis to selected attention projection matrices\. Acrossttcheckpoints and monitored matrices𝒲\\mathcal\{W\}, the post\-hoc topological analysis costsO​\(t​∑i∈𝒲\(di2​di′\+PH​\(di\)\)\)O\\\!\\left\(t\\sum\_\{i\\in\\mathcal\{W\}\}\(d\_\{i\}^\{2\}d^\{\\prime\}\_\{i\}\+\\mathrm\{PH\}\(d\_\{i\}\)\)\\right\), wherePH​\(di\)\\mathrm\{PH\}\(d\_\{i\}\)denotes the persistence computation cost for the row cloud of matrixii\.

## 6Experiments

Hardware and implementation\.All experiments were conducted on NVIDIA H200 GPUs and AMD EPYC 9555 64\-Core Processor CPUs with our PyTorch\-based code\. We compute persistence diagrams with Ripser\[[34](https://arxiv.org/html/2607.16637#bib.bib36)\]and evaluate diagram distances using Gudhi’s Wasserstein implementation\[[22](https://arxiv.org/html/2607.16637#bib.bib35)\]\. Our implementation is available as open\-source code on GitHub \(https://anonymous\.4open\.science/r/TopoTuner/\)

Table 1:Datasets and evaluation metrics used in our experiments\.QA: question answering,SA: sentiment analysis, IR: information retrieval,SU: summarization,IF: instruction following,CG: code generation\.TaskDatasetTrain sizeEval sizeEvaluation metricQAGSM8K\[[7](https://arxiv.org/html/2607.16637#bib.bib4)\]7,4731,319AccuracyQAMMLU\[[11](https://arxiv.org/html/2607.16637#bib.bib5),[10](https://arxiv.org/html/2607.16637#bib.bib6)\]99,84214,042AccuracySAIMDB\[[21](https://arxiv.org/html/2607.16637#bib.bib7)\]25,00025,000AccuracySASST\-2\[[30](https://arxiv.org/html/2607.16637#bib.bib8)\]67,349872AccuracyForgettingIRHotpotQA\[[38](https://arxiv.org/html/2607.16637#bib.bib9)\]90,4477,405EM and F1IRSQuAD v1\.1\[[27](https://arxiv.org/html/2607.16637#bib.bib10)\]87,59910,570EM and F1SUXSum\[[23](https://arxiv.org/html/2607.16637#bib.bib11)\]204,04511,334ROUGE\-1, ROUGE\-2, ROUGE\-LSUCNN/DailyMail\[[28](https://arxiv.org/html/2607.16637#bib.bib12)\]287,11311,490ROUGE\-1, ROUGE\-2, ROUGE\-LIFDatabricks\-dolly\-15k\[[8](https://arxiv.org/html/2607.16637#bib.bib13)\]15,01115,011ROUGE\-1/2/L, BLEU, METEORIFAlpaca\[[32](https://arxiv.org/html/2607.16637#bib.bib14)\]52,00252,002ROUGE\-1/2/L, BLEU, METEORCGHumanEval\[[6](https://arxiv.org/html/2607.16637#bib.bib15)\]164164Pass@1CGMBPP\[[2](https://arxiv.org/html/2607.16637#bib.bib16)\]257257Pass@1

Datasets\.We evaluateTopoTuneron a diverse set of downstream tasks spanning question answering, sentiment analysis, information retrieval, summarization, instruction following, and code generation\. Table[1](https://arxiv.org/html/2607.16637#S6.T1)summarizes the datasets, splits, and evaluation metrics\. We useQAandSAdatasets for fine\-tuning and evaluation, and all twelve datasets for catastrophic forgetting after fine\-tuning\. For large datasets, training is capped at 20,000 examples\.

Models and training procedure\.We evaluate three open\-source pretrained language models:LLaMA\-3\.1\-8B\[[9](https://arxiv.org/html/2607.16637#bib.bib2)\]\(LLaMA\),Mistral\-7B\-v0\.3\[[17](https://arxiv.org/html/2607.16637#bib.bib3)\]\(Mistral\), andQwen3\-8B\-Base\[[33](https://arxiv.org/html/2607.16637#bib.bib1)\]\(Qwen\)\. For each model, we run both full fine\-tuning and LoRA\-based fine\-tuning on each downstream task\. We also tune the prominent models Spectrum\[[26](https://arxiv.org/html/2607.16637#bib.bib18)\]and DropBP\[[35](https://arxiv.org/html/2607.16637#bib.bib17)\], as explained in Appendix[L](https://arxiv.org/html/2607.16637#A12)\. We use cross\-entropy loss and a fixed epoch budget of six for all runs to ensure consistency across models and tasks\. Persistent homology uses the cosine distance andH0H\_\{0\}\. The experimental setup and hyperparameters are provided in the Appendix[C](https://arxiv.org/html/2607.16637#A3)\.

Evaluation metrics\.Our goal is to achieve the highest accuracy with the fewest parameter changes and the shortest fine\-tuning time\. We evaluate each method along three axes: downstream task performance, trainable\-parameter efficiency, and training time\. Downstream performance is measured on held\-out evaluation sets using the task\-specific metrics listed in Table[1](https://arxiv.org/html/2607.16637#S6.T1)\. Efficiency is measured by i\) the percentage of trainable model weights \(Train\. %\) and ii\) the percentage of updated model weights \(Upd\. %\)\. Computational cost is measured by wall\-clock training time in seconds on fixed hardware\. Accuracy \(Acc\.\) is computed on the full held\-out evaluation split, while Std is estimated from three fixed, non\-overlapping random subsets of the same evaluation split\.

### 6\.1Fine\-tuning with Freezing Priors

We use the question\-answering datasetQA:GSM8Kas the prior because it provides a standard reasoning task with clean, exact\-answer evaluation\. However, as Figure[3](https://arxiv.org/html/2607.16637#S6.F3)shows, for a fixed LLM, we find that the topological change patterns observed onQA:GSM8Kremain consistent across the other datasets in our benchmark \(topological drifts of all models on datasets in Appendix[I](https://arxiv.org/html/2607.16637#A9)\)\.

![Refer to caption](https://arxiv.org/html/2607.16637v1/x3.png)Figure 3:Epoch\-6 WassersteinH0H\_\{0\}distances for theVVprojection of LLaMA across datasets\. Full Wasserstein distance profiles across models are reported in Appendix[G](https://arxiv.org/html/2607.16637#A7)\.We compare full fine\-tuning, LoRA\[[13](https://arxiv.org/html/2607.16637#bib.bib32)\], Spectrum\[[26](https://arxiv.org/html/2607.16637#bib.bib18)\], DropBP\[[35](https://arxiv.org/html/2607.16637#bib.bib17)\], the magnitude\-based selector, and ourTopoTunerand report our metrics\.

We evaluate low\-drift and high\-drift freezing over budgetsb∈\{3,6,9,12,15\}b\\in\\\{3,6,9,12,15\\\}\. The low\-drift strategy tests the hypothesis that stable matrices are less necessary for adaptation\. The high\-drift strategy tests the hypothesis that blocking the dominant adaptation path can constrain task\-specific movement, reduce overfitting, and force the remaining trainable matrices to absorb the update\. In our experiments,the high\-drift strategy consistently yields stronger performance\. Accordingly, we omit the low\-drift results from the main text for clarity and report them in Appendix[K](https://arxiv.org/html/2607.16637#A11)\.

We freeze MLP weights and selectively freeze only the attention projectionsK,Q,V,OK,Q,V,O\. This keeps the trainable budget comparable to PEFT methods, since MLP weights form most of the model parameters \(e\.g\.,62% for LLaMA\)\. Our projection\-level ranks in Appendix Table[5](https://arxiv.org/html/2607.16637#A4.T5)show that the dominant adaptation route is model\-dependent: LLaMA and Mistral concentrate drift inV/OV/O, while Qwen concentrates drift inK/QK/Q; therefore, the candidate set includes all four attention projections\.

We run bothperfect\-knowledgeandtransferred\-knowledgeexperiments\. The perfect\-knowledge setting selects frozen projections from the sameQA:GSM8Kfull\-fine\-tuning profile used for evaluation; this setup is mainly for diagnostics\. As shown in Appendix[J](https://arxiv.org/html/2607.16637#A10), TDA\-High3 outperforms LoRA by2\.432\.43points on LLaMA and3\.593\.59points on Mistral while changing only2\.64%2\.64\\%–2\.82%2\.82\\%of model parameters\.

Table 2:Transferred knowledge\.Fine\-tuning results acrossSA:SST\-2,SA:IMDB, andQA:MMLU\.TopoTunerselective tuning freezes projections learned fromQA:GSM8K\.\*: LoRA trains only 0\.17–0\.19% extra parameters, but after merging can affect up to 100% of the targeted projection matrices\.DatasetMethodQwen3\-8B\-BaseLLaMA\-3\.1\-8BMistral\-7B\-v0\.3Train\.%Upd\. %Acc\.±\\pmStd\. \(↑\\uparrow\)Time \(min\)Train\.%Upd\. %Acc\.±\\pmStd\. \(↑\\uparrow\)Time \(min\)Train\.%Upd\. %Acc\.±\\pmStd\. \(↑\\uparrow\)Time \(min\)SST\-2Full Fine\-tuning10013\.0195\.76±1\.0095\.76\\pm 1\.0032\.710021\.8896\.10±0\.5396\.10\\pm 0\.5330\.310025\.0096\.10±0\.8796\.10\\pm 0\.8732\.9LoRA100\*11\.0295\.75±1\.7895\.75\\pm 1\.7824\.8100\*16\.6295\.64±2\.2295\.64\\pm 2\.2221\.5100\*18\.4196\.10±1\.4496\.10\\pm 1\.4424\.7DropBP100\*18\.1985\.66±1\.1285\.66\\pm 1\.1234\.0100\*16\.6195\.53±0\.3495\.53\\pm 0\.3426\.4100\*18\.4296\.10±0\.2196\.10\\pm 0\.2134\.2Spectrum17\.550\.4792\.78±0\.6892\.78\\pm 0\.6830\.315\.801\.2495\.76±0\.5395\.76\\pm 0\.5326\.96\.711\.2395\.53±0\.9295\.53\\pm 0\.9230\.9TopoTuner8\.451\.1595\.99±0\.41\\mathbf\{95\.99\\pm 0\.41\}23\.97\.572\.0295\.98±1\.44\\mathbf\{95\.98\\pm 1\.44\}21\.28\.392\.4496\.21±0\.70\\mathbf\{96\.21\\pm 0\.70\}24\.0IMDBFull Fine\-tuning1008\.3482\.83±0\.3182\.83\\pm 0\.31195\.110024\.9590\.66±0\.2190\.66\\pm 0\.21172\.410028\.6783\.10±0\.1883\.10\\pm 0\.18164\.8LoRA100\*18\.3882\.89±0\.3582\.89\\pm 0\.35177\.2100\*16\.6783\.36±0\.3083\.36\\pm 0\.30153\.2100\*18\.4982\.72±0\.1682\.72\\pm 0\.16147\.9DropBP100\*18\.2970\.79±0\.3070\.79\\pm 0\.30214\.6100\*16\.6395\.30±0\.20\\mathbf\{95\.30\\pm 0\.20\}158\.4100\*18\.4093\.64±0\.2293\.64\\pm 0\.22164\.2Spectrum17\.550\.8281\.62±0\.2881\.62\\pm 0\.28172\.015\.802\.5694\.98±0\.2294\.98\\pm 0\.22150\.96\.711\.8294\.82±0\.20\\mathbf\{94\.82\\pm 0\.20\}143\.6TopoTuner8\.450\.7882\.97±0\.34\\mathbf\{82\.97\\pm 0\.34\}163\.67\.572\.3683\.55±0\.2683\.55\\pm 0\.26143\.08\.392\.9279\.76±0\.3079\.76\\pm 0\.30137\.7MMLUFull Fine\-tuning1009\.3973\.22±2\.1573\.22\\pm 2\.15157\.610031\.2960\.03±0\.0560\.03\\pm 0\.05139\.610035\.3861\.85±2\.6261\.85\\pm 2\.62132\.2LoRA100\*18\.4174\.13±2\.96\\mathbf\{74\.13\\pm 2\.96\}141\.9100\*16\.6761\.86±1\.6761\.86\\pm 1\.67122\.9100\*18\.4956\.82±2\.8556\.82\\pm 2\.85118\.4DropBP100\*18\.1937\.54±0\.3637\.54\\pm 0\.36165\.1100\*16\.6154\.19±0\.9154\.19\\pm 0\.91122\.3100\*18\.4250\.01±0\.5150\.01\\pm 0\.51126\.1Spectrum17\.550\.6666\.88±0\.6266\.88\\pm 0\.62143\.315\.802\.8154\.79±0\.9454\.79\\pm 0\.94126\.26\.712\.0052\.31±0\.6352\.31\\pm 0\.63119\.0TopoTuner8\.450\.6872\.83±1\.9772\.83\\pm 1\.97131\.77\.572\.4963\.75±1\.79\\mathbf\{63\.75\\pm 1\.79\}115\.48\.392\.9860\.68±3\.06\\mathbf\{60\.68\\pm 3\.06\}110\.3

In the second and main setting, Table[2](https://arxiv.org/html/2607.16637#S6.T2)evaluates whether a freezing profile learned fromQA:GSM8Ktransfers to new datasets \(competitors are not transferred but are fine\-tuned on the same dataset\)\. In this setting, we report TDA\-High3 asTopoTunerbecause it is the smallest high\-drift budget, already improves over LoRA in77out of99settings, reduces training time, shows a smaller train\-test generalization gap, indicating reduced overfitting \(see Appendix[P](https://arxiv.org/html/2607.16637#A16)\) and keeps parameter change below3%3\\%; larger budgets are reported in Appendix Tables[9](https://arxiv.org/html/2607.16637#A10.T9),[10](https://arxiv.org/html/2607.16637#A10.T10)and[11](https://arxiv.org/html/2607.16637#A10.T11)\. The gains are clearest for LLaMA, whereTopoTuneroutperforms LoRA across all three datasets, with the largest improvement onQA:MMLU\. For Mistral,TopoTunerimproves over LoRA onSA:SST\-2andQA:MMLU\. For Qwen,TopoTuneris strongest onSA:SST\-2andSA:IMDB\. Across all settings,TopoTuneris also faster than competitors and trains on \(Train\. %\) substantially fewer parameters, indicating that the transferred topological profile can reduce training time and parameter updates while preserving or improving accuracy in most cases\.

The forgetting results in Appendix Tables[15](https://arxiv.org/html/2607.16637#A15.T15),[16](https://arxiv.org/html/2607.16637#A15.T16), and[17](https://arxiv.org/html/2607.16637#A15.T17)show the same pattern:TopoTuneroften preserves stronger out\-of\-domain performance onQA,SA, andIRevaluations, with more mixed effects on generation\-style tasks\.

### 6\.2Fine\-tuning with Dynamic Stopping

We next test whether topological drift can decide when fine\-tuning should stop\. For each fine\-tuning epoch, we compute the topological drift of the monitored projection matrices from initialization and aggregate the displacement into the stopping score from Definition[5\.2](https://arxiv.org/html/2607.16637#S5.Thmtheorem2)\. Training stops once the normalized epoch\-to\-epoch change falls below a threshold based on the model’s stabilization\.

![Refer to caption](https://arxiv.org/html/2607.16637v1/x4.png)Figure 4:Evolution of WassersteinH0H\_\{0\}distancesDi\(t\)D^\{\(t\)\}\_\{i\}over layers for theVVprojection of LLaMA fully fine\-tuned onQA:MMLU\. Each line is a distance to the pretrained model\.Figure[4](https://arxiv.org/html/2607.16637#S6.F4)shows the Wasserstein distanceDi\(t\)D^\{\(t\)\}\_\{i\}across epochs for LLaMA fully fine\-tuned onQA:MMLU\. The first epoch produces the largest change in early layers, while subsequent epochs contribute little additional change there\. Figure[5](https://arxiv.org/html/2607.16637#S6.F5)shows the corresponding epoch\-to\-epoch drift bars for LLaMA onQA:MMLU,SA:SST\-2, andSA:IMDB\(see other models in the Appendix[I](https://arxiv.org/html/2607.16637#A9)\)\. Drift values in Figure[5](https://arxiv.org/html/2607.16637#S6.F5)show that fine\-tuning onSA:SST\-2, andSA:IMDBcan stop after the second epoch\. The accuracy curves in Appendix Figure[15](https://arxiv.org/html/2607.16637#A11.F15)yield the same conclusion regarding performance\. Accuracy rises early and then moves within a narrow range, while the topological drift has already decayed\. Training beyond this point mostly involves small residual updates\.

![Refer to caption](https://arxiv.org/html/2607.16637v1/x5.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x6.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x7.png)QA:MMLUSA:SST\-2SA:IMDBFigure 5:Epoch\-to\-epoch topological drift bars for theVVprojections of LLaMA underH0H\_\{0\}during full fine\-tuning\. Most layers stabilize by epoch22, with subsequent epochs showing only small residual changes\. See all datasets and models in Appendix[I](https://arxiv.org/html/2607.16637#A9)\.Figure[6](https://arxiv.org/html/2607.16637#S6.F6)shows how the selected stopping epoch changes withη\\etaonSA:SST\-2\. The rule first stops at epoch 3 for LLaMA and Qwen atη=0\.05\\eta=0\.05, and reaches epoch\-2 stopping for Qwen atη=0\.15\\eta=0\.15\. Appendix[M](https://arxiv.org/html/2607.16637#A13)provides the stopping\-epoch curves forSA:IMDBandQA:MMLUacross the three model families\.

![Refer to caption](https://arxiv.org/html/2607.16637v1/x8.png)Figure 6:Topology\-based stopping epochs under full fine\-tuning\. The plots show selected stopping epochs acrossη\\etathresholds forSA:SST\-2across the three model families\.Time efficiency\.OnQA:GSM8K, persistence computation takes approximately 2\.1 min for LLaMA, 2\.4 min for Qwen, and 2\.2 min for Mistral per epoch; the subsequent Wasserstein computation is shorter, taking approximately 1\.8 min, 2\.1 min, and 2\.0 min\. These costs are CPU\-side, highly parallelizable, and can be amortized because the resulting profiles are reused for freezing and transfer experiments\. This cost is paid offline after checkpoints are saved and does not change the asymptotic cost of gradient\-based fine\-tuning; the online savings come from freezing selected projection matrices and stopping training early\. As a result, Table[2](https://arxiv.org/html/2607.16637#S6.T2)shows thatTopoTuneris more time\-efficient than the competitors in transferred fine\-tuning, and the time gains are significant on large datasets such asQA:MMLU\.

### 6\.3Reusable Fine\-Tuning

TDA\-High3\-LLaMAModelLLaMA\-3\.1\-8BPrior task[QA:GSM8K](https://huggingface.co/datasets/openai/gsm8k)Freezing strategyCosine\-WassersteinH0H\_\{0\}High\-33Frozen IDsVV:\{V8,V2,V0\}\\\{V\_\{8\},V\_\{2\},V\_\{0\}\\\}OO:\{O11,O10,O13\}\\\{O\_\{11\},O\_\{10\},O\_\{13\}\\\}Transferred to[SA:SST\-2](https://huggingface.co/datasets/stanfordnlp/sst2)Acc±\{\\pm\}Std95\.98±1\.4495\.98\{\\pm\}1\.44ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)21\.221\.2minUpd\.%2\.02%2\.02\\%[SA:IMDB](https://huggingface.co/datasets/stanfordnlp/imdb)Acc±\{\\pm\}Std83\.55±0\.2683\.55\{\\pm\}0\.26ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)143\.0143\.0minUpd\.%2\.36%2\.36\\%[QA:MMLU](https://huggingface.co/datasets/cais/mmlu)Acc±\{\\pm\}Std63\.75±1\.7963\.75\{\\pm\}1\.79ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)115\.4115\.4minUpd\.%2\.49%2\.49\\%Forgetting✓[MMLU](https://huggingface.co/datasets/cais/mmlu),✓[SST\-2](https://huggingface.co/datasets/stanfordnlp/sst2),✓[IMDB](https://huggingface.co/datasets/stanfordnlp/imdb),✓[SQuAD](https://huggingface.co/datasets/rajpurkar/squad),✓[HotpotQA](https://huggingface.co/datasets/hotpotqa/hotpot_qa),✓[XSum](https://huggingface.co/datasets/EdinburghNLP/xsum),✓[Dolly\-15k](https://huggingface.co/datasets/databricks/databricks-dolly-15k),✗[CNN/DM](https://huggingface.co/datasets/abisee/cnn_dailymail),✗[Alpaca](https://huggingface.co/datasets/tatsu-lab/alpaca),✗[HumanEval](https://huggingface.co/datasets/openai/openai_humaneval),✗[MBPP](https://huggingface.co/datasets/Muennighoff/mbpp)Stoppingη\\etaSA:SST\-2η\\eta=0\.25, SA:IMDBη\\eta=0\.20, QA:MMLUη\\eta=0\.25Figure 7:Reusable freezing profile for LLaMA\-3\.1\-8B learned fromQA:GSM8K\.Our results and descriptive analysis in the Appendix Section[E](https://arxiv.org/html/2607.16637#A5)suggest that fine\-tuning changes LLMs in a structured but highly constrained manner\. Entrywise weight movements are small, and most parameters remain close to their pretrained values\. This makes purely magnitude\-based descriptions incomplete: the important question is not only how much a matrix moves but also whether the topology changes coherently\. Wasserstein distances on persistence diagrams provide such a signal\.

A first result is that projection\-level drift is architecture\-dependent\. As Appendix Tables[5](https://arxiv.org/html/2607.16637#A4.T5)and[6](https://arxiv.org/html/2607.16637#A4.T6)show, LLaMA and Mistral usually place the largest changes inVVandOO, while Qwen often shows the opposite pattern, withKKandQQreceiving stronger drift\. This rules out a universal projection rule such as always tuningV/OV/Oor always freezingK/QK/Q\.

This model dependence is paired with a second \(useful and surprising\) observation: for a fixed model, the drift profile is reusable across tasks\. This does not imply that all tasks induce the same geometry \(we analyze layer change similarity across models for tasks in Appendix[N](https://arxiv.org/html/2607.16637#A14)\)\. However, each pretrained model appears to have a stable set of adaptation channels\.

We propose a reusable profile card that summarizes which attention projections consistently absorb topological change during a prior fine\-tuning run\. The card records the source task, the topological ranking strategy, the selected frozen projection IDs, the stopping threshold, transfer performance on downstream tasks, parameter update rates, and forgetting behavior across held\-out evaluations\. Figure[7](https://arxiv.org/html/2607.16637#S6.F7)shows an example profile learned fromQA:GSM8Kfor LLaMA\-3\.1\-8B using a cosine distance\-WassersteinH0H\_\{0\}High\-3 strategy\. The same frozen projection set transfers to sentiment analysis, question answering, and summarization tasks without recomputing a new freezing search\. This suggests that pretrained models contain reusable adaptation routes that persist across downstream fine\-tuning runs\. The profile card therefore acts as a compact transferable prior over how a model changes during adaptation\.

Limitations\.TopoTunerrequires access to checkpoints and attention projection weights, so closed\-weight API models cannot be studied without provider\-side access to weights and freezing controls\. Future work should test larger open\-weight models, broader task families, and profile reuse across longer fine\-tuning sequences\.

## 7Conclusion

We have proposedTopoTuner, a topology\-guided approach for controlling LLM fine\-tuning through matrix\-level structural drift\.TopoTunermeasures how attention projection matrices reorganize relative to the pretrained model and uses this signal for selective freezing and early stopping\. Across our experiments,TopoTuneroutperforms LoRA in77out of99transferred\-knowledge settings, reduces training time by20\.4%20\.4\\%on average, and improves over LoRA by2\.43%2\.43\\%on LLaMA and3\.59%3\.59\\%on Mistral\. Compared with baselines,TopoTuneris faster and changes fewer projection matrices, only about 2\.64%–2\.82%, which gives a more constrained adaptation process with lower risk of unintended changes\. More importantly,TopoTunerproduces reusable freezing profiles that can instruct future users which projection matrices to train when fine\-tuning the same model on new datasets\.

## References

- \[1\]\(2021\)Intrinsic dimensionality explains the effectiveness of language model fine\-tuning\.InProceedings of the 59th annual meeting of the association for computational linguistics and the 11th international joint conference on natural language processing \(volume 1: long papers\),pp\. 7319–7328\.Cited by:[Appendix Q](https://arxiv.org/html/2607.16637#A17.p5.1)\.
- \[2\]J\. Austin, A\. Odena, M\. Nye, M\. Bosma, H\. Michalewski, D\. Dohan, E\. Jiang, C\. Cai, M\. Terry, Q\. Le,et al\.\(2021\)Program synthesis with large language models\.arXiv preprint arXiv:2108\.07732\.Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.14.2)\.
- \[3\]G\. Carlsson\(2009\)Topology and data\.Bulletin of the American Mathematical Society46\(2\),pp\. 255–308\.Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p4.1),[§4](https://arxiv.org/html/2607.16637#S4.p1.1)\.
- \[4\]F\. Chazal, V\. de Silva, M\. Glisse, and S\. Oudot\(2016\)The structure and stability of persistence modules\.Springer\.Cited by:[Appendix Q](https://arxiv.org/html/2607.16637#A17.1.p1.19),[Appendix Q](https://arxiv.org/html/2607.16637#A17.p5.1)\.
- \[5\]F\. Chazal and B\. Michel\(2021\)An introduction to topological data analysis: fundamental and practical aspects for data scientists\.Frontiers in Artificial Intelligence4\.Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p4.1),[§3](https://arxiv.org/html/2607.16637#S3.p3.5)\.
- \[6\]M\. Chen, J\. Tworek, H\. Jun, Q\. Yuan, H\. P\. de Oliveira Pinto, J\. Kaplan, H\. Edwards, Y\. Burda, N\. Joseph, G\. Brockman, A\. Ray, R\. Puri, G\. Krueger, M\. Petrov, H\. Khlaaf, G\. Sastry, P\. Mishkin, B\. Chan, S\. Gray, N\. Ryder, M\. Pavlov, A\. Power, L\. Kaiser, M\. Bavarian, C\. Winter, P\. Tillet, F\. P\. Such, D\. Cummings, M\. Plappert, F\. Chantzis, E\. Barnes, A\. Herbert\-Voss, W\. H\. Guss, A\. Nichol, A\. Paino, N\. Tezak, J\. Tang, I\. Babuschkin, S\. Balaji, S\. Jain, W\. Saunders, C\. Hesse, A\. N\. Carr, J\. Leike, J\. Achiam, V\. Misra, E\. Morikawa, A\. Radford, M\. Knight, M\. Brundage, M\. Murati, K\. Mayer, P\. Welinder, B\. McGrew, D\. Amodei, S\. McCandlish, I\. Sutskever, and W\. Zaremba\(2021\)Evaluating large language models trained on code\.External Links:2107\.03374Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.13.2)\.
- \[7\]K\. Cobbe, V\. Kosaraju, M\. Bavarian, M\. Chen, H\. Jun, L\. Kaiser, M\. Plappert, J\. Tworek, J\. Hilton, R\. Nakano, C\. Hesse, and J\. Schulman\(2021\)Training verifiers to solve math word problems\.arXiv preprint arXiv:2110\.14168\.Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.2.2)\.
- \[8\]M\. Conover, M\. Hayes, A\. Mathur, J\. Xie, J\. Wan, S\. Shah, A\. Ghodsi, P\. Wendell, M\. Zaharia, and R\. Xin\(2023\)Free dolly: introducing the world’s first truly open instruction\-tuned llm\.External Links:[Link](https://www.databricks.com/blog/2023/04/12/dolly-first-open-commercially-viable-instruction-tuned-llm)Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.11.2)\.
- \[9\]A\. Grattafiori, A\. Dubey, A\. Jauhri, A\. Pandey, A\. Kadian, A\. Al\-Dahle, A\. Letman, A\. Mathur, A\. Schelten, A\. Vaughan,et al\.\(2024\)The llama 3 herd of models\.arXiv preprint arXiv:2407\.21783\.Cited by:[§6](https://arxiv.org/html/2607.16637#S6.p3.1)\.
- \[10\]D\. Hendrycks, C\. Burns, S\. Basart, A\. Critch, J\. Li, D\. Song, and J\. Steinhardt\(2021\)Aligning ai with shared human values\.Proceedings of the International Conference on Learning Representations \(ICLR\)\.Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.3.2)\.
- \[11\]D\. Hendrycks, C\. Burns, S\. Basart, A\. Zou, M\. Mazeika, D\. Song, and J\. Steinhardt\(2021\)Measuring massive multitask language understanding\.Proceedings of the International Conference on Learning Representations \(ICLR\)\.Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.3.2)\.
- \[12\]F\. Hensel, M\. Moor, and B\. Rieck\(2021\)A survey of topological machine learning methods\.Frontiers in Artificial Intelligence4,pp\. 52\.Cited by:[§3](https://arxiv.org/html/2607.16637#S3.p3.5),[§4](https://arxiv.org/html/2607.16637#S4.p3.17)\.
- \[13\]E\. J\. Hu, Y\. Shen, P\. Wallis, Z\. Allen\-Zhu, Y\. Li, S\. Wang, L\. Wang, W\. Chen,et al\.\(2022\)Lora: low\-rank adaptation of large language models\.\.Iclr1\(2\),pp\. 3\.Cited by:[Appendix Q](https://arxiv.org/html/2607.16637#A17.p5.1),[§1](https://arxiv.org/html/2607.16637#S1.p2.1),[§2](https://arxiv.org/html/2607.16637#S2.p2.1),[§6\.1](https://arxiv.org/html/2607.16637#S6.SS1.p2.1)\.
- \[14\]Y\. Hu, S\. Kang, H\. Yang, H\. Xu, and M\. Zhang\(2025\)Beyond single\-task: robust multi\-task length generalization for llms\.arXiv preprint arXiv:2502\.11525\.Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p3.1)\.
- \[15\]W\. Huang, J\. Liang, G\. Wan, D\. Zhu, H\. Li, J\. Shao, M\. Ye, B\. Du, and D\. Tao\(2025\)Be confident: uncovering overfitting in mllm multi\-task tuning\.InForty\-second International Conference on Machine Learning,Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p1.1)\.
- \[16\]G\. Ilharco, M\. T\. Ribeiro, M\. Wortsman, L\. Schmidt, H\. Hajishirzi, and A\. Farhadi\(2023\)Editing models with task arithmetic\.InThe Eleventh International Conference on Learning Representations,Cited by:[Appendix Q](https://arxiv.org/html/2607.16637#A17.p5.1)\.
- \[17\]A\. Q\. Jiang, A\. Sablayrolles, A\. Mensch, C\. Bamford, D\. S\. Chaplot, D\. de las Casas, F\. Bressand, G\. Lengyel, G\. Lample, L\. Saulnier, L\. R\. Lavaud, M\. Lachaux, P\. Stock, T\. Le Scao, T\. Lavril, T\. Wang, T\. Lacroix, and W\. El Sayed\(2023\)Mistral 7b\.arXiv preprint arXiv:2310\.06825\.External Links:[Link](https://arxiv.org/abs/2310.06825)Cited by:[§6](https://arxiv.org/html/2607.16637#S6.p3.1)\.
- \[18\]K\. Kumar, T\. Ashraf, O\. Thawakar, R\. M\. Anwer, H\. Cholakkal, M\. Shah, M\. Yang, P\. H\. Torr, F\. S\. Khan, and S\. Khan\(2025\)Llm post\-training: a deep dive into reasoning large language models\.arXiv preprint arXiv:2502\.21321\.Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p1.1)\.
- \[19\]G\. Liu, W\. Lin, Q\. Mu, T\. Huang, R\. Mo, Y\. Tao, and L\. Shen\(2025\)Targeted vaccine: safety alignment for large language models against harmful fine\-tuning via layer\-wise perturbation\.IEEE Transactions on Information Forensics and Security\.Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p1.1)\.
- \[20\]Y\. Luo, Z\. Yang, F\. Meng, Y\. Li, J\. Zhou, and Y\. Zhang\(2025\)An empirical study of catastrophic forgetting in large language models during continual fine\-tuning\.IEEE Transactions on Audio, Speech and Language Processing\.Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p1.1)\.
- \[21\]A\. L\. Maas, R\. E\. Daly, P\. T\. Pham, D\. Huang, A\. Y\. Ng, and C\. Potts\(2011\-06\)Learning word vectors for sentiment analysis\.InProceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies,Portland, Oregon, USA,pp\. 142–150\.External Links:[Link](http://www.aclweb.org/anthology/P11-1015)Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.4.2)\.
- \[22\]C\. Maria, J\. Boissonnat, M\. Glisse, and M\. Yvinec\(2014\)The gudhi library: simplicial complexes and persistent homology\.InInternational congress on mathematical software,pp\. 167–174\.Cited by:[§6](https://arxiv.org/html/2607.16637#S6.p1.1)\.
- \[23\]S\. Narayan, S\. B\. Cohen, and M\. Lapata\(2018\)Don’t give me the details, just the summary\! topic\-aware convolutional neural networks for extreme summarization\.ArXivabs/1808\.08745\.Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.9.2)\.
- \[24\]D\. Pérez\-Fernández, A\. Gutiérrez\-Fandiño, J\. Armengol\-Estapé, and M\. Villegas\(2021\)Characterizing and measuring the similarity of neural networks with persistent homology\.arXiv preprint arXiv:2101\.07752\.Cited by:[Appendix Q](https://arxiv.org/html/2607.16637#A17.p5.1),[§2](https://arxiv.org/html/2607.16637#S2.p1.1)\.
- \[25\]E\. Purvine, D\. Brown, B\. Jefferson, C\. Joslyn, B\. Praggastis, A\. Rathore, M\. Shapiro, B\. Wang, and Y\. Zhou\(2023\)Experimental observations of the topology of convolutional neural network activations\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.37,pp\. 9470–9479\.Cited by:[Appendix Q](https://arxiv.org/html/2607.16637#A17.p5.1),[§2](https://arxiv.org/html/2607.16637#S2.p1.1)\.
- \[26\]QuixiAI\(2024\)Spectrum: targeted training on signal to noise ratio\.Note:[https://github\.com/QuixiAI/spectrum](https://github.com/QuixiAI/spectrum)GitHub repositoryCited by:[Appendix L](https://arxiv.org/html/2607.16637#A12.SS0.SSS0.Px1.p1.3),[§6\.1](https://arxiv.org/html/2607.16637#S6.SS1.p2.1),[§6](https://arxiv.org/html/2607.16637#S6.p3.1)\.
- \[27\]P\. Rajpurkar, J\. Zhang, K\. Lopyrev, and P\. Liang\(2016\-11\)SQuAD: 100,000\+ questions for machine comprehension of text\.InProceedings of the 2016 Conference on Empirical Methods in Natural Language Processing,J\. Su, K\. Duh, and X\. Carreras \(Eds\.\),Austin, Texas,pp\. 2383–2392\.External Links:[Link](https://aclanthology.org/D16-1264),[Document](https://dx.doi.org/10.18653/v1/D16-1264),1606\.05250Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.8.2)\.
- \[28\]A\. See, P\. J\. Liu, and C\. D\. Manning\(2017\-07\)Get to the point: summarization with pointer\-generator networks\.InProceedings of the 55th Annual Meeting of the Association for Computational Linguistics \(Volume 1: Long Papers\),Vancouver, Canada,pp\. 1073–1083\.External Links:[Link](https://www.aclweb.org/anthology/P17-1099),[Document](https://dx.doi.org/10.18653/v1/P17-1099)Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.10.2)\.
- \[29\]R\. S\. Shuttleworth, J\. Andreas, A\. Torralba, and P\. Sharma\(2025\)LoRA vs full fine\-tuning: an illusion of equivalence\.InProceedings of the Thirty\-ninth Annual Conference on Neural Information Processing Systems,Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p3.1)\.
- \[30\]R\. Socher, A\. Perelygin, J\. Wu, J\. Chuang, C\. D\. Manning, A\. Ng, and C\. Potts\(2013\-10\)Recursive deep models for semantic compositionality over a sentiment treebank\.InProceedings of the 2013 Conference on Empirical Methods in Natural Language Processing,Seattle, Washington, USA,pp\. 1631–1642\.External Links:[Link](https://www.aclweb.org/anthology/D13-1170)Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.5.2)\.
- \[31\]Z\. Sun and G\. Raskutti\(2026\)A theoretical framework for llm fine\-tuning using early stopping for non\-random initialization\.arXiv preprint arXiv:2602\.13942\.Cited by:[§5](https://arxiv.org/html/2607.16637#S5.SSx2.p1.1)\.
- \[32\]R\. Taori, I\. Gulrajani, T\. Zhang, Y\. Dubois, X\. Li, C\. Guestrin, P\. Liang, and T\. B\. Hashimoto\(2023\)Stanford alpaca: an instruction\-following llama model\.GitHub\.Note:[https://github\.com/tatsu\-lab/stanford\_alpaca](https://github.com/tatsu-lab/stanford_alpaca)Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.12.2)\.
- \[33\]Q\. Team\(2025\)Qwen3 technical report\.External Links:2505\.09388,[Link](https://arxiv.org/abs/2505.09388)Cited by:[§6](https://arxiv.org/html/2607.16637#S6.p3.1)\.
- \[34\]C\. Tralie, N\. Saul, and R\. Bar\-On\(2018\)Ripser\. py: a lean persistent homology library for python\.Journal of Open Source Software3\(29\),pp\. 925\.Cited by:[§6](https://arxiv.org/html/2607.16637#S6.p1.1)\.
- \[35\]S\. Woo, B\. Park, B\. Kim, M\. Jo, S\. J\. Kwon, D\. Jeon, and D\. Lee\(2024\)Dropbp: accelerating fine\-tuning of large language models by dropping backward propagation\.Advances in Neural Information Processing Systems37,pp\. 20170–20197\.Cited by:[Appendix L](https://arxiv.org/html/2607.16637#A12.SS0.SSS0.Px2.p1.1),[§2](https://arxiv.org/html/2607.16637#S2.p2.1),[§6\.1](https://arxiv.org/html/2607.16637#S6.SS1.p2.1),[§6](https://arxiv.org/html/2607.16637#S6.p3.1)\.
- \[36\]M\. Wortsman, G\. Ilharco, S\. Y\. Gadre, R\. Roelofs, R\. Gontijo\-Lopes, A\. S\. Morcos, H\. Namkoong, A\. Farhadi, Y\. Carmon, S\. Kornblith,et al\.\(2022\)Model soups: averaging weights of multiple fine\-tuned models improves accuracy without increasing inference time\.InInternational conference on machine learning,pp\. 23965–23998\.Cited by:[Appendix Q](https://arxiv.org/html/2607.16637#A17.p5.1)\.
- \[37\]L\. Xu, H\. Xie, S\. J\. Qin, X\. Tao, and F\. L\. Wang\(2026\)Parameter\-efficient fine\-tuning methods for pretrained language models: a critical review and assessment\.IEEE Transactions on Pattern Analysis and Machine Intelligence\.Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p1.1),[§1](https://arxiv.org/html/2607.16637#S1.p2.1),[§2](https://arxiv.org/html/2607.16637#S2.p2.1)\.
- \[38\]Z\. Yang, P\. Qi, S\. Zhang, Y\. Bengio, W\. W\. Cohen, R\. Salakhutdinov, and C\. D\. Manning\(2018\)HotpotQA: a dataset for diverse, explainable multi\-hop question answering\.InConference on Empirical Methods in Natural Language Processing \(EMNLP\),Cited by:[Table 1](https://arxiv.org/html/2607.16637#S6.T1.9.1.7.2)\.
- \[39\]C\. Yaras, P\. Wang, L\. Balzano, and Q\. Qu\(2026\)Compressible dynamics in deep overparameterized low\-rank learning & adaptation\.InForty\-first International Conference on Machine Learning,Cited by:[§2](https://arxiv.org/html/2607.16637#S2.p2.1)\.
- \[40\]M\. Yue\(2025\)A survey of large language model agents for question answering\.arXiv preprint arXiv:2503\.19213\.Cited by:[§1](https://arxiv.org/html/2607.16637#S1.p1.1)\.
- \[41\]E\. B\. Zaken, Y\. Goldberg, and S\. Ravfogel\(2022\)Bitfit: simple parameter\-efficient fine\-tuning for transformer\-based masked language\-models\.InProceedings of the 60th Annual Meeting of the Association for Computational Linguistics \(Volume 2: Short Papers\),pp\. 1–9\.Cited by:[Appendix Q](https://arxiv.org/html/2607.16637#A17.p5.1)\.
- \[42\]J\. Zheng, X\. Cai, S\. Qiu, and Q\. Ma\(2025\)Spurious forgetting in continual learning of language models\.InThe Thirteenth International Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2607.16637#S2.p2.1)\.

## Appendix

## Appendix ABroader Impacts

This work improves the efficiency and control of fine\-tuning for large language models by identifying which components need to change and when training can stop\. A primary benefit is reduced computational cost\. By training a small fraction of parameters and avoiding unnecessary epochs, the method lowers energy consumption and shortens development cycles\. This makes fine\-tuning more accessible to smaller research groups and organizations with limited compute resources\.

The method also contributes to better understanding of model adaptation\. By measuring structural change at the level of individual weight matrices, it provides insight into how different tasks affect pretrained models\. This can support more reliable deployment by helping practitioners identify stable and unstable components during adaptation\. In settings where consistency is important, such as finance or data\-driven decision systems, controlled updates can reduce unintended deviations from pretrained behavior\.

Another positive aspect is reusability\. Freezing profiles learned on one task can be applied to new tasks, which reduces repeated exploration and simplifies the fine\-tuning process\. This can accelerate experimentation and enable more systematic development of downstream applications without requiring full retraining\.

## Appendix BReusable Freezing Profiles

The following cards summarize the reusable freezing profiles produced byTopoTuner\. Each profile is learned from a prior fine\-tuning task, ranks attention projection matrices by prior\-task topological drift, and freezes the selected drift matrices during target\-task fine\-tuning\.

Profile DefinitionProfile IDIdentifier of the reusable freezing profile, combining the freezing strategy and model\.ModelPretrained modelfθf\_\{\\theta\}for which the freezing profile is learned\.Prior taskPrior task𝒯\\mathcal\{T\}and prior dataset𝒟𝒯\\mathcal\{D\}\_\{\\mathcal\{T\}\}used to compute the matrix\-level ranking scores\.Freezing strategyRanking score and selection rule used to choose frozen matrices from the candidate set, e\.g\., Cosine\-Wasserstein for ranking score and High\-33for selection rule\.Frozen IDsSelected matrices or layers frozen from the start of target\-task fine\-tuning\.Transferred toAcc±\{\\pm\}Std, ETA \(with GPU Used\), Upd\.%, andΔf\\Delta\_\{f\}, denoting accuracy with standard deviation, total training time, updated\-parameter percentage, and forgetting datasets\.✓denotes our profile card is forgetting less than full fine\-tuning and✗denotes the opposite\.Stoppingη\\etaTopology\-based stopping threshold\. Fine\-tuning stops at the first epoch where the normalized epoch\-to\-epoch topological drift falls belowη\\eta\.

TDA\-High3\-LLaMAModelLLaMA\-3\.1\-8BPrior task[QA:GSM8K](https://huggingface.co/datasets/openai/gsm8k)Freezing strategyCosine\-WassersteinH0H\_\{0\}High\-33Frozen IDsVV:\{V8,V2,V0\}\\\{V\_\{8\},V\_\{2\},V\_\{0\}\\\}OO:\{O11,O10,O13\}\\\{O\_\{11\},O\_\{10\},O\_\{13\}\\\}Transferred to[SA:SST\-2](https://huggingface.co/datasets/stanfordnlp/sst2)Acc±\{\\pm\}Std95\.98±1\.4495\.98\{\\pm\}1\.44ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)21\.221\.2minUpd\.%2\.02%2\.02\\%[SA:IMDB](https://huggingface.co/datasets/stanfordnlp/imdb)Acc±\{\\pm\}Std83\.55±0\.2683\.55\{\\pm\}0\.26ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)143\.0143\.0minUpd\.%2\.36%2\.36\\%[QA:MMLU](https://huggingface.co/datasets/cais/mmlu)Acc±\{\\pm\}Std63\.75±1\.7963\.75\{\\pm\}1\.79ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)115\.4115\.4minUpd\.%2\.49%2\.49\\%Forgetting✓[MMLU](https://huggingface.co/datasets/cais/mmlu),✓[SST\-2](https://huggingface.co/datasets/stanfordnlp/sst2),✓[IMDB](https://huggingface.co/datasets/stanfordnlp/imdb),✓[SQuAD](https://huggingface.co/datasets/rajpurkar/squad),✓[HotpotQA](https://huggingface.co/datasets/hotpotqa/hotpot_qa),✓[XSum](https://huggingface.co/datasets/EdinburghNLP/xsum),✓[Dolly\-15k](https://huggingface.co/datasets/databricks/databricks-dolly-15k),✗[CNN/DM](https://huggingface.co/datasets/abisee/cnn_dailymail),✗[Alpaca](https://huggingface.co/datasets/tatsu-lab/alpaca),✗[HumanEval](https://huggingface.co/datasets/openai/openai_humaneval),✗[MBPP](https://huggingface.co/datasets/Muennighoff/mbpp)Stoppingη\\etaSA:SST\-2η\\eta=0\.25, SA:IMDBη\\eta=0\.20, QA:MMLUη\\eta=0\.25TDA\-High3\-QwenModelQwen3\-8B\-BasePrior task[QA:GSM8K](https://huggingface.co/datasets/openai/gsm8k)Freezing strategyCosine\-WassersteinH0H\_\{0\}High\-33Frozen IDsKK:\{K29,K34,K33\}\\\{K\_\{29\},K\_\{34\},K\_\{33\}\\\}QQ:\{Q21,Q32,Q33\}\\\{Q\_\{21\},Q\_\{32\},Q\_\{33\}\\\}Transferred to[SA:SST\-2](https://huggingface.co/datasets/stanfordnlp/sst2)Acc±\{\\pm\}Std95\.99±0\.4195\.99\{\\pm\}0\.41ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)23\.923\.9minUpd\.%1\.15%1\.15\\%[SA:IMDB](https://huggingface.co/datasets/stanfordnlp/imdb)Acc±\{\\pm\}Std82\.97±0\.3482\.97\{\\pm\}0\.34ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)163\.6163\.6minUpd\.%0\.78%0\.78\\%[QA:MMLU](https://huggingface.co/datasets/cais/mmlu)Acc±\{\\pm\}Std72\.83±1\.9772\.83\{\\pm\}1\.97ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)131\.7131\.7minUpd\.%0\.68%0\.68\\%Forgetting✓[IMDB](https://huggingface.co/datasets/stanfordnlp/imdb),✓[HotpotQA](https://huggingface.co/datasets/hotpotqa/hotpot_qa),✓[XSum](https://huggingface.co/datasets/EdinburghNLP/xsum),✓[Dolly\-15k](https://huggingface.co/datasets/databricks/databricks-dolly-15k),✓[CNN/DM](https://huggingface.co/datasets/abisee/cnn_dailymail),✓[Alpaca](https://huggingface.co/datasets/tatsu-lab/alpaca),✓[HumanEval](https://huggingface.co/datasets/openai/openai_humaneval),✓[MBPP](https://huggingface.co/datasets/Muennighoff/mbpp),✗[MMLU](https://huggingface.co/datasets/cais/mmlu),✗[SST\-2](https://huggingface.co/datasets/stanfordnlp/sst2),✗[SQuAD](https://huggingface.co/datasets/rajpurkar/squad),Stoppingη\\etaSA:SST\-2η\\eta=0\.15, SA:IMDBη\\eta=0\.2, QA:MMLUη\\eta=0\.25TDA\-High3\-MistralModelMistral\-7B\-v0\.3Prior task[QA:GSM8K](https://huggingface.co/datasets/openai/gsm8k)Freezing strategyCosine\-WassersteinH0H\_\{0\}High\-33Frozen IDsVV:\{V0,V10,V12\}\\\{V\_\{0\},V\_\{10\},V\_\{12\}\\\}OO:\{O0,O1,O2\}\\\{O\_\{0\},O\_\{1\},O\_\{2\}\\\}Transferred to[SA:SST\-2](https://huggingface.co/datasets/stanfordnlp/sst2)Acc±\{\\pm\}Std96\.21±0\.7096\.21\{\\pm\}0\.70ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)24\.024\.0minUpd\.%2\.44%2\.44\\%[SA:IMDB](https://huggingface.co/datasets/stanfordnlp/imdb)Acc±\{\\pm\}Std79\.76±0\.3079\.76\{\\pm\}0\.30ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)137\.7137\.7minUpd\.%2\.92%2\.92\\%[QA:MMLU](https://huggingface.co/datasets/cais/mmlu)Acc±\{\\pm\}Std60\.68±3\.0660\.68\{\\pm\}3\.06ETA \([H200](https://www.nvidia.com/en-us/data-center/h200/)\)110\.3110\.3minUpd\.%2\.98%2\.98\\%Forgetting✓[MMLU](https://huggingface.co/datasets/cais/mmlu),✓[SST\-2](https://huggingface.co/datasets/stanfordnlp/sst2),✓[IMDB](https://huggingface.co/datasets/stanfordnlp/imdb),✓[HotpotQA](https://huggingface.co/datasets/hotpotqa/hotpot_qa),✓[XSum](https://huggingface.co/datasets/EdinburghNLP/xsum),✓[CNN/DM](https://huggingface.co/datasets/abisee/cnn_dailymail),✓[HumanEval](https://huggingface.co/datasets/openai/openai_humaneval),✓[MBPP](https://huggingface.co/datasets/Muennighoff/mbpp)✗[SQuAD](https://huggingface.co/datasets/rajpurkar/squad),✗[Dolly\-15k](https://huggingface.co/datasets/databricks/databricks-dolly-15k),✗[Alpaca](https://huggingface.co/datasets/tatsu-lab/alpaca),Stoppingη\\etaSA:SST\-2η\\eta=0\.2, SA:IMDBη\\eta=0\.15, QA:MMLUη\\eta=0\.2

Figure 8:Reusable Freezing Profiles\.We define a reusable freezing profile learned from a prior\-task fine\-tuning run\. Each model card summarizes the model, prior task, candidate matrix set, freezing strategy, selected frozen IDs, and outcome on target datasets\. We report three profiles used in our experiments: TDA\-High3\-LLaMA, TDA\-High3\-Qwen, and TDA\-High3\-Mistral\.
## Appendix CExperimental Setup and Hyperparameters

Tables[3](https://arxiv.org/html/2607.16637#A3.T3)and[4](https://arxiv.org/html/2607.16637#A3.T4)summarize the training and evaluation setup used in our experiments\. Table[3](https://arxiv.org/html/2607.16637#A3.T3)reports task\-level training settings, model\-specific learning rates, and LoRA configurations, while Table[4](https://arxiv.org/html/2607.16637#A3.T4)reports the decoding and checkpoint\-scoring settings\.

Table 3:Training hyperparameters used in our fine\-tuning experiments\. Freeze denotes the selective tuning setting used by the Eltwise\- and TDA\-based methods\.SettingQA:GSM8KSA:SST\-2SA:IMDBQA:MMLUTask\-level training settingsEpochs66666666Train subset7,4737\{,\}47320,00020\{,\}00025,00025\{,\}00020,00020\{,\}000Batch size1616646432323232Gradient accumulation11111111Max sequence length5125121024102410241024512512Precisionbf16bf16bf16bf16SchedulercosinecosinecosinecosineOptimizerAdamWAdamWAdamWAdamWWarmup Ratio0\.030\.030\.030\.03Max grad norm1\.01\.01\.01\.01\.01\.01\.01\.0Gradient checkpointingenabledenabledenabledenabledSeed42/123/45642/123/45642/123/45642/123/45642/123/45642/123/45642/123/45642/123/456Model\-specific learning ratesQwen3\-8B\-Base full FT LR2×10−52\{\\times\}10^\{\-5\}1×10−51\{\\times\}10^\{\-5\}1×10−51\{\\times\}10^\{\-5\}1×10−51\{\\times\}10^\{\-5\}LLaMA\-3\.1\-8B full FT LR2×10−52\{\\times\}10^\{\-5\}2×10−52\{\\times\}10^\{\-5\}2×10−52\{\\times\}10^\{\-5\}2×10−52\{\\times\}10^\{\-5\}Mistral\-7B\-v0\.3 full FT LR5×10−65\{\\times\}10^\{\-6\}5×10−65\{\\times\}10^\{\-6\}5×10−65\{\\times\}10^\{\-6\}5×10−65\{\\times\}10^\{\-6\}LoRA LR2×10−42\{\\times\}10^\{\-4\}2×10−42\{\\times\}10^\{\-4\}2×10−42\{\\times\}10^\{\-4\}2×10−42\{\\times\}10^\{\-4\}Spectrumsame as full FT LRsame as full FT LRsame as full FT LRsame as full FT LRDropBPsame as LoRA LRsame as LoRA FT LRsame as LoRA FT LRsame as LoRA FT LRFreeze LRsame as full FT LRsame as full FT LRsame as full FT LRsame as full FT LRLoRA configurationRank / alpha / dropout16/32/0\.0516/32/0\.0516/32/0\.0516/32/0\.0516/32/0\.0516/32/0\.0516/32/0\.0516/32/0\.05Target modulesq,k,v,oq,k,v,oq,k,v,oq,k,v,oq,k,v,oq,k,v,oq,k,v,oq,k,v,o

Table 4:Evaluation settings used forSA:SST\-2,SA:IMDB,QA:MMLUcheckpoint scoring in the transferred\-knowledge experiments\.SettingSA:SST\-2SA:IMDBQA:MMLUSplitvalidationtestvalidationDecodinggreedygreedygreedySamplingdisableddisableddisabledMax new tokens444422Prompt max length25625651251210241024MetricAccuracyAccuracyAccuracySubset reportingthree fixed foldsthree fixed foldsthree fixed folds

## Appendix DMatrix Types, Weight Changes, and Freezing Profiles

To quantify projection\-level changes, we compare each epoch\-6 fine\-tuned matrix with its pretrained counterpart\. For each projection matrixW∈\{K,Q,V,O\}W\\in\\\{K,Q,V,O\\\}, we compute mean element\-wise relative drift:

d​\(W\)=1n​∑i=1n\|W6,i−W0,i\|\|W0,i\|,d\(W\)=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\frac\{\|W\_\{6,i\}\-W\_\{0,i\}\|\}\{\|W\_\{0,i\}\|\},whereW0W\_\{0\}andW6W\_\{6\}denote the pretrained and epoch\-6 weights,nnis the number of matrix entries\.

We justify our matrix selection design by ranking all candidate matrices by their average distance to the pretrained model\. We compute the average rank of each matrix family \(KK,QQ,VV, andOO\) and count how manyKKorQQmatrices appear among the top distance matrices\. As Table[5](https://arxiv.org/html/2607.16637#A4.T5)and Table[6](https://arxiv.org/html/2607.16637#A4.T6)show, empirically,KKandQQoccupy low positions in the ranking and are largely absent from the top\-ranked set, which supports restricting selective tuning toVVandOO\.

Table 5:Normalized average projection ranks \(rank divided by total count\)\. Lower values indicate larger drift\.LLaMA\-3\.1\-8B/Mistral\-7B\-v0\.3are normalized by128128,Qwen3\-8B\-Baseby144144\.DatasetStatisticLLaMA\-3\.1\-8BMistral\-7B\-v0\.3Qwen3\-8B\-BaseQA:GSM8KAvg K rank0\.850\.710\.29Avg Q rank0\.580\.480\.33Avg V rank0\.260\.410\.75Avg O rank0\.320\.410\.65SA:IMDBAvg K rank0\.790\.610\.41Avg Q rank0\.510\.530\.44Avg V rank0\.350\.440\.62Avg O rank0\.360\.430\.55QA:MMLUAvg K rank0\.720\.660\.43Avg Q rank0\.540\.520\.42Avg V rank0\.370\.430\.64Avg O rank0\.390\.410\.53SA:SST2Avg K rank0\.760\.610\.39Avg Q rank0\.510\.510\.44Avg V rank0\.360\.440\.60Avg O rank0\.390\.450\.59Table 6:Top\-NNprojection counts under mean element\-wise relative drift\. Each cell reportsK​Q​\(K\+Q\)\|V​O​\(V\+O\)K\\ Q\\ \(K\{\+\}Q\)\\ \|\\ V\\ O\\ \(V\{\+\}O\)among the top\-NNmost changed projections\.LLaMA\-3\.1\-8B/Mistral\-7B\-v0\.3runs contain 128 projection matrices from 32 layers, whileQwen3\-8B\-Baseruns contain 144 projection matrices from 36 layers\.DatasetTop\-NNLLaMA\-3\.1\-8BMistral\-7B\-v0\.3Qwen3\-8B\-BaseQA:GSM8KTop\-101 1 \(2\)\|\|6 2 \(8\)1 4 \(5\)\|\|3 2 \(5\)8 2 \(10\)\|\|0 0 \(0\)Top\-201 1 \(2\)\|\|15 3 \(18\)3 7 \(10\)\|\|7 3 \(10\)14 6 \(20\)\|\|0 0 \(0\)Top\-301 1 \(2\)\|\|21 7 \(28\)3 10 \(13\)\|\|12 5 \(17\)16 14 \(30\)\|\|0 0 \(0\)SA:IMDBTop\-101 1 \(2\)\|\|6 2 \(8\)0 0 \(0\)\|\|7 3 \(10\)6 4 \(10\)\|\|0 0 \(0\)Top\-201 1 \(2\)\|\|12 6 \(18\)1 0 \(1\)\|\|13 6 \(19\)9 11 \(20\)\|\|0 0 \(0\)Top\-301 2 \(3\)\|\|14 13 \(27\)1 1 \(2\)\|\|14 14 \(28\)15 13 \(28\)\|\|0 2 \(2\)QA:MMLUTop\-101 1 \(2\)\|\|6 2 \(8\)1 0 \(1\)\|\|6 3 \(9\)6 4 \(10\)\|\|0 0 \(0\)Top\-201 1 \(2\)\|\|13 5 \(18\)1 0 \(1\)\|\|12 7 \(19\)10 7 \(17\)\|\|1 2 \(3\)Top\-301 1 \(2\)\|\|15 13 \(28\)1 0 \(1\)\|\|14 15 \(29\)10 12 \(22\)\|\|4 4 \(8\)SA:SST2Top\-101 1 \(2\)\|\|6 2 \(8\)0 0 \(0\)\|\|7 3 \(10\)8 2 \(10\)\|\|0 0 \(0\)Top\-201 1 \(2\)\|\|11 7 \(18\)0 0 \(0\)\|\|12 8 \(20\)13 7 \(20\)\|\|0 0 \(0\)Top\-301 2 \(3\)\|\|14 13 \(27\)0 1 \(1\)\|\|15 14 \(29\)17 11 \(28\)\|\|1 1 \(2\)
## Appendix EModel Weight\-Change Statistics

We report full\-model weight\-change statistics by comparing each fine\-tuned checkpoint against its pretrained initialization\. Changed Params\. quantifies how widely the checkpoint differs from the pretrained model by counting parameters whose absolute update exceeds a fixed threshold:

\|wfinal−wbase\|\>10−6\.\|w\_\{\\mathrm\{final\}\}\-w\_\{\\mathrm\{base\}\}\|\>10^\{\-6\}\.Mean Rel\.Δ\\Deltameasures the average relative magnitude of the parameter update:

Mean​Rel\.Δ=mean​\(\|wfinal−wbase\|\|wbase\|\)\.\\mathrm\{Mean\\ Rel\.\}\\ \\Delta=\\mathrm\{mean\}\\left\(\\frac\{\|w\_\{\\mathrm\{final\}\}\-w\_\{\\mathrm\{base\}\}\|\}\{\|w\_\{\\mathrm\{base\}\}\|\}\\right\)\.Together, these statistics capture both the spread of parameter changes across the model and the average size of those changes relative to the pretrained weights\.

Table 7:Model weight\-change statistics relative to the pretrained checkpoint\. TDA\-High3 uses86\.25%86\.25\\%less parameter updates than LoRA across the 12 LLM\-dataset settings\.MethodQwen3\-8B\-BaseLLaMA\-3\.1\-8BMistral\-7B\-v0\.3TotalParams\.ChangedParams\.Upd\.\(%\)Mean Rel\.Δ\\Delta\(%\)TotalParams\.ChangedParams\.Upd\.\(%\)Mean Rel\.Δ\\Delta\(%\)TotalParams\.ChangedParams\.Upd\.\(%\)Mean Rel\.Δ\\Delta\(%\)QA:GSM8KFull8\.19B1\.198B14\.6312\.578\.03B2\.211B27\.5426\.127\.25B2\.280B31\.4625\.57LoRA8\.19B1\.508B18\.413\.898\.03B1\.341B16\.709\.137\.25B1\.340B18\.4936\.57TDA\-High38\.19B0\.100B1\.221\.658\.03B0\.212B2\.643\.177\.25B0\.204B2\.823\.02TDA\-High68\.19B0\.091B1\.121\.558\.03B0\.187B2\.332\.897\.25B0\.183B2\.522\.79TDA\-High98\.19B0\.082B1\.001\.458\.03B0\.165B2\.052\.607\.25B0\.162B2\.232\.55Eltwise\-High38\.19B0\.099B1\.211\.628\.03B0\.220B2\.743\.177\.25B0\.213B2\.943\.11Eltwise\-High68\.19B0\.091B1\.111\.528\.03B0\.196B2\.442\.887\.25B0\.194B2\.672\.85Eltwise\-High98\.19B0\.082B1\.011\.418\.03B0\.175B2\.182\.637\.25B0\.167B2\.302\.53SA:SST\-2Full8\.19B1\.066B13\.018\.788\.03B1\.757B21\.8811\.077\.25B1\.812B25\.0012\.56LoRA8\.19B0\.903B11\.021\.018\.03B1\.335B16\.622\.037\.25B1\.334B18\.4115\.87TDA\-High38\.19B0\.094B1\.150\.938\.03B0\.162B2\.021\.277\.25B0\.177B2\.441\.44SA:IMDBFull8\.19B0\.683B8\.346\.698\.03B2\.004B24\.9516\.877\.25B2\.078B28\.6716\.88LoRA8\.19B1\.505B18\.383\.118\.03B1\.338B16\.675\.047\.25B1\.340B18\.4939\.57TDA\-High38\.19B0\.064B0\.780\.788\.03B0\.189B2\.361\.887\.25B0\.211B2\.921\.96QA:MMLUFull8\.19B0\.770B9\.397\.168\.03B2\.513B31\.2921\.727\.25B2\.564B35\.3821\.22LoRA8\.19B1\.508B18\.414\.028\.03B1\.339B16\.675\.337\.25B1\.340B18\.4934\.39TDA\-High38\.19B0\.056B0\.680\.868\.03B0\.200B2\.492\.067\.25B0\.216B2\.982\.13

## Appendix FMagnitude Changes in Models

Entrywise weight change\.We first report entrywise weight movement as a magnitude\-based baseline\. For each fine\-tuned matrix, we compare the epoch\-6 weights against the pretrained weights and summarize the raw mean absolute change for theVVandOOprojections\. The entrywise plots in Figure[9](https://arxiv.org/html/2607.16637#A6.F9)provide a direct view of parameter movement across datasets and model families\. This magnitude\-based view is later compared with the topology\-based distance used for matrix ranking and freezing\.

VVProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x9.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x10.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x11.png)OOProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x12.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x13.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x14.png)LLaMA\-3\.1\-8BQwen3\-8B\-BaseMistral\-7B\-v0\.3Figure 9:Epoch\-6 Eltwise distances for theVVandOOprojection matrices underH0H\_\{0\}\.
## Appendix GWasserstein Distances in Models

Wasserstein distance\.We also report the Wasserstein\-based structural distance used by our topological analysis\. As defined later in Definition[4\.1](https://arxiv.org/html/2607.16637#S4.Thmtheorem1), each projection matrix is treated as a row cloud and compared to its pretrained version through the Wasserstein distance between persistence diagrams\. The Wasserstein plots in Figure[10](https://arxiv.org/html/2607.16637#A7.F10)show this epoch\-6 structural change for theVVandOOprojections under the same model and dataset ordering as the entrywise plots\.

VVProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x15.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x16.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x17.png)OOProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x18.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x19.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x20.png)LLaMA\-3\.1\-8BQwen3\-8B\-BaseMistral\-7B\-v0\.3Figure 10:Epoch\-6 Wasserstein distances for theVVandOOprojection matrices underH0H\_\{0\}\.
## Appendix HFreezing Pseudocode

Algorithms[1](https://arxiv.org/html/2607.16637#alg1)–[3](https://arxiv.org/html/2607.16637#alg3)summarize the fullTopoTunerselection pipeline\. Algorithm[1](https://arxiv.org/html/2607.16637#alg1)computes projection\-specific layer orderings using Wasserstein topological drift, Algorithm[2](https://arxiv.org/html/2607.16637#alg2)converts these orderings into frozenK/Q/V/OK/Q/V/Olayer sets under the Low or High regime, and Algorithm[3](https://arxiv.org/html/2607.16637#alg3)applies the resulting sets by freezing all parameters and re\-enabling gradients only for the selected unfrozenK/Q/V/OK/Q/V/Oprojections\.

Algorithm 1Computing TDA\-Based Wasserstein Layer Orderings1:Pretrained model

fθ\(0\)f\_\{\\theta^\{\(0\)\}\}; fully fine\-tuned model

fθ\(T\)f\_\{\\theta^\{\(T\)\}\}; projection set

𝒫=\{K,Q,V,O\}\\mathcal\{P\}=\\\{K,Q,V,O\\\}; Wasserstein order

pp
2:Ordered layer lists

πK,πQ,πV,πO\\pi\_\{K\},\\pi\_\{Q\},\\pi\_\{V\},\\pi\_\{O\}
3:for

P∈𝒫P\\in\\mathcal\{P\}do

4:foreach transformer layer

ℓ\\elldo

5:Extract pretrained matrix

Wℓ,P\(0\)W\_\{\\ell,P\}^\{\(0\)\}and fine\-tuned matrix

Wℓ,P\(T\)W\_\{\\ell,P\}^\{\(T\)\}
6:

Xℓ,P\(0\)←rows​\(Wℓ,P\(0\)\)X\_\{\\ell,P\}^\{\(0\)\}\\leftarrow\\mathrm\{rows\}\(W\_\{\\ell,P\}^\{\(0\)\}\),

Xℓ,P\(T\)←rows​\(Wℓ,P\(T\)\)X\_\{\\ell,P\}^\{\(T\)\}\\leftarrow\\mathrm\{rows\}\(W\_\{\\ell,P\}^\{\(T\)\}\)
7:

Dℓ,P\(0\)←PD​\(Xℓ,P\(0\)\)D\_\{\\ell,P\}^\{\(0\)\}\\leftarrow\\mathrm\{PD\}\(X\_\{\\ell,P\}^\{\(0\)\}\),

Dℓ,P\(T\)←PD​\(Xℓ,P\(T\)\)D\_\{\\ell,P\}^\{\(T\)\}\\leftarrow\\mathrm\{PD\}\(X\_\{\\ell,P\}^\{\(T\)\}\)
8:

sℓ,P←Wp​\(Dℓ,P\(0\),Dℓ,P\(T\)\)s\_\{\\ell,P\}\\leftarrow W\_\{p\}\\left\(D\_\{\\ell,P\}^\{\(0\)\},D\_\{\\ell,P\}^\{\(T\)\}\\right\)⊳\\trianglerightWasserstein distance between persistence diagrams

9:endfor

10:

πP←argsortℓ​\(sℓ,P\)\\pi\_\{P\}\\leftarrow\\mathrm\{argsort\}\_\{\\ell\}\(s\_\{\\ell,P\}\)⊳\\trianglerightlow to high Wasserstein score

11:endfor

12:return

πK,πQ,πV,πO\\pi\_\{K\},\\pi\_\{Q\},\\pi\_\{V\},\\pi\_\{O\}

Algorithm 2Constructing Frozen Layer Sets from Wasserstein Layer Orderings1:Ordered layer lists

πK,πQ,πV,πO\\pi\_\{K\},\\pi\_\{Q\},\\pi\_\{V\},\\pi\_\{O\}sorted from low to high Wasserstein score; number of layers

LL; freezing budget

BB; regime

r∈\{Low,High\}r\\in\\\{\\textsc\{Low\},\\textsc\{High\}\\\}
2:Frozen layer sets

ℱK,ℱQ,ℱV,ℱO\\mathcal\{F\}\_\{K\},\\mathcal\{F\}\_\{Q\},\\mathcal\{F\}\_\{V\},\\mathcal\{F\}\_\{O\}
3:for

P∈\{K,Q,V,O\}P\\in\\\{K,Q,V,O\\\}do

4:if

r=Lowr=\\textsc\{Low\}then

5:

ℱP←\{πP​\[0\],…,πP​\[B−1\]\}\\mathcal\{F\}\_\{P\}\\leftarrow\\\{\\pi\_\{P\}\[0\],\\ldots,\\pi\_\{P\}\[B\-1\]\\\}
6:elseif

r=Highr=\\textsc\{High\}then

7:

ℱP←\{πP​\[L−B\],…,πP​\[L−1\]\}\\mathcal\{F\}\_\{P\}\\leftarrow\\\{\\pi\_\{P\}\[L\-B\],\\ldots,\\pi\_\{P\}\[L\-1\]\\\}
8:endif

9:endfor

10:return

ℱK,ℱQ,ℱV,ℱO\\mathcal\{F\}\_\{K\},\\mathcal\{F\}\_\{Q\},\\mathcal\{F\}\_\{V\},\\mathcal\{F\}\_\{O\}

Algorithm 3SelectiveK/Q/V/OK/Q/V/OTuning from Frozen Layer Sets1:Model

fθf\_\{\\theta\}; frozen layer sets

ℱK,ℱQ,ℱV,ℱO\\mathcal\{F\}\_\{K\},\\mathcal\{F\}\_\{Q\},\\mathcal\{F\}\_\{V\},\\mathcal\{F\}\_\{O\}
2:Model with selected

K/Q/V/OK/Q/V/Oparameters trainable

3:for allparameters

p∈θp\\in\\thetado

4:

p\.requires​\_​grad←Falsep\.\\mathrm\{requires\\\_grad\}\\leftarrow\\mathrm\{False\}
5:endfor

6:for allnamed parameters

\(n,p\)\(n,p\)in

fθf\_\{\\theta\}do

7:Parse transformer layer index

ℓ\\ellfrom parameter name

nn
8:if

nncontainsk\_projand

ℓ∉ℱK\\ell\\notin\\mathcal\{F\}\_\{K\}then

9:

p\.requires​\_​grad←Truep\.\\mathrm\{requires\\\_grad\}\\leftarrow\\mathrm\{True\}
10:elseif

nncontainsq\_projand

ℓ∉ℱQ\\ell\\notin\\mathcal\{F\}\_\{Q\}then

11:

p\.requires​\_​grad←Truep\.\\mathrm\{requires\\\_grad\}\\leftarrow\\mathrm\{True\}
12:elseif

nncontainsv\_projand

ℓ∉ℱV\\ell\\notin\\mathcal\{F\}\_\{V\}then

13:

p\.requires​\_​grad←Truep\.\\mathrm\{requires\\\_grad\}\\leftarrow\\mathrm\{True\}
14:elseif

nncontainso\_projand

ℓ∉ℱO\\ell\\notin\\mathcal\{F\}\_\{O\}then

15:

p\.requires​\_​grad←Truep\.\\mathrm\{requires\\\_grad\}\\leftarrow\\mathrm\{True\}
16:endif

17:endfor

18:Keep all non\-

K/Q/V/OK/Q/V/Oparameters frozen

19:return

fθf\_\{\\theta\}

## Appendix ITopological Drift Across Models and Datasets

We use the topological distance to initialization and topological drift defined in Definition[4\.1](https://arxiv.org/html/2607.16637#S4.Thmtheorem1)and Definition[4\.2](https://arxiv.org/html/2607.16637#S4.Thmtheorem2)to visualize how attention projection matrices change during fine\-tuning\. Figures[11](https://arxiv.org/html/2607.16637#A9.F11)\-[14](https://arxiv.org/html/2607.16637#A9.F14)show this quantity for theVVandOOprojections across datasets and models\. The drift\-bar plots summarize epoch\-to\-epoch changes, while the line plots show the Wasserstein distance from initialization across layers for each epoch\.

VVProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x21.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x22.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x23.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x24.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x25.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x26.png)OOProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x27.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x28.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x29.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x30.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x31.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x32.png)LLaMA\-3\.1\-8BQwen3\-8B\-BaseMistral\-7B\-v0\.3Figure 11:Topological drift of theVVandOOprojection onQA:GSM8KunderH0H\_\{0\}\. The top row shows epoch\-to\-epoch drift bars, and the bottom row shows Wasserstein distance curves across layers\.VVProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x33.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x34.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x35.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x36.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x37.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x38.png)OOProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x39.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x40.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x41.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x42.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x43.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x44.png)LLaMA\-3\.1\-8BQwen3\-8B\-BaseMistral\-7B\-v0\.3Figure 12:Topological drift of theVVandOOprojection onQA:MMLUunderH0H\_\{0\}\. The top row shows epoch\-to\-epoch drift bars, and the bottom row shows Wasserstein distance curves across layers\.VVProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x45.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x46.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x47.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x48.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x49.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x50.png)OOProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x51.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x52.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x53.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x54.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x55.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x56.png)LLaMA\-3\.1\-8BQwen3\-8B\-BaseMistral\-7B\-v0\.3Figure 13:Topological drift of theVVprojection onSA:IMDBunderH0H\_\{0\}\. The top row shows epoch\-to\-epoch drift bars, and the bottom row shows Wasserstein distance curves across layers\.VVProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x57.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x58.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x59.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x60.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x61.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x62.png)OOProjection![Refer to caption](https://arxiv.org/html/2607.16637v1/x63.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x64.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x65.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x66.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x67.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x68.png)LLaMA\-3\.1\-8BQwen3\-8B\-BaseMistral\-7B\-v0\.3Figure 14:Topological drift of theVVandOOprojection onSA:SST\-2underH0H\_\{0\}\. The top row shows epoch\-to\-epoch drift bars, and the bottom row shows Wasserstein distance curves across layers\.
## Appendix JFine\-tuning with Perfect Knowledge

Table[8](https://arxiv.org/html/2607.16637#A10.T8)reports the perfect\-knowledgeQA:GSM8Kresults forQwen3\-8B\-Base,LLaMA\-3\.1\-8B, andMistral\-7B\-v0\.3\. Across all three models, selective freezing recovers much of the full\-fine\-tuning performance while training only about6%6\\%to8%8\\%of the weights\. For the best selective configurations inQwen3\-8B\-Base, TDA\-High33reaches87\.62±0\.1887\.62\\pm 0\.18with8\.45%8\.45\\%trainable parameters, close to LoRA at87\.89±0\.9087\.89\\pm 0\.90and above full fine\-tuning at86\.27±0\.7586\.27\\pm 0\.75\. ForLLaMA\-3\.1\-8B, TDA\-High66reaches61\.17±0\.7561\.17\\pm 0\.75with6\.79%6\.79\\%trainable parameters, slightly above full fine\-tuning at60\.70±0\.8760\.70\\pm 0\.87and above LoRA at56\.90±1\.9256\.90\\pm 1\.92\. ForMistral\-7B\-v0\.3, TDA\-High33reaches52\.87±0\.4652\.87\\pm 0\.46with8\.39%8\.39\\%trainable parameters, compared with full fine\-tuning at59\.59±0\.8559\.59\\pm 0\.85and LoRA at49\.28±1\.9449\.28\\pm 1\.94\.

Table 8:Perfect Knowledge\.Fine\-tuning models onQA:GSM8Kwith selectedVVandOOfreezing\. Wasserstein distances useH0H\_\{0\}\. Projection IDs and results of otherkkvalues are given in Table[9](https://arxiv.org/html/2607.16637#A10.T9)\.Qwen3\-8B\-BaseLlama\-3\.1\-8BMistral\-7B\-v0\.3MethodTrain\.%Upd\. %Acc\.±\\pmStd \(↑\\uparrow\)Time \(min\)Train\.%Upd\.Acc\.±\\pmStd \(↑\\uparrow\)Time \(min\)Train\.%Upd\.Acc\.±\\pmStd \(↑\\uparrow\)Time \(min\)Full Fine\-tuning10014\.6386\.27±0\.7586\.27\\pm 0\.7547\.310027\.5460\.70±0\.8760\.70\\pm 0\.8738\.710031\.4659\.59±0\.8559\.59\\pm 0\.8543\.3PEFTLoRA0\.1918\.4187\.89±0\.90\\mathbf\{87\.89\\pm 0\.90\}35\.40\.1716\.7056\.90±1\.9256\.90\\pm 1\.9228\.60\.1918\.4949\.28±1\.9449\.28\\pm 1\.9433\.5TDA\-based tuning \(ours\)High38\.451\.2287\.62±0\.18¯\\underline\{87\.62\\pm 0\.18\}34\.57\.572\.6459\.33±0\.46¯\\underline\{59\.33\\pm 0\.46\}26\.28\.392\.8252\.87±0\.46\\mathbf\{52\.87\\pm 0\.46\}30\.5High67\.681\.1286\.91±0\.4686\.91\\pm 0\.4634\.46\.792\.3361\.17±0\.75\\mathbf\{61\.17\\pm 0\.75\}25\.76\.82\.5252\.64±0\.99¯\\underline\{52\.64\\pm 0\.99\}30\.5High96\.911\.0086\.91±0\.1686\.91\\pm 0\.1634\.46\.012\.0558\.97±0\.7658\.97\\pm 0\.7625\.55\.212\.2351\.40±0\.9251\.40\\pm 0\.9230\.3

These results show two patterns\. First, selective freezing yields large reductions in updated parameters with competitive performance across all three models\. Second, TDA\-based freezing gives the best selective result for forQwen3\-8B\-BaseandMistral\-7B\-v0\.3, and tied inLLaMA\-3\.1\-8B\. For more details, see Tables[9](https://arxiv.org/html/2607.16637#A10.T9),[10](https://arxiv.org/html/2607.16637#A10.T10), and[11](https://arxiv.org/html/2607.16637#A10.T11)\.

Table 9:LLaMA\-3\.1\-8BonQA:GSM8Kwith separateVVandOOfreezing\. Among selective freezing settings, the best configuration is tied at TDA\-High66\(%61\.17\) and Eltwise\-Low99\(%61\.17\)\.MethodFrozen LayersTrainable %Avg\. Accuracy±\\pmStd \(↑\\uparrow\)Time \(min\)BaselinesFull–10010060\.70±0\.8760\.70\\pm 0\.8738\.7LoRA–0\.170\.1756\.90±1\.9256\.90\\pm 1\.9228\.6Magnitude\-based Tuning \(ours\)EltwiseLow3:V:\[9,10,21\]∣O:\[29,31,23\]V\{:\}\[9,10,21\]\\mid O\{:\}\[29,31,23\]7\.577\.5759\.80±2\.0059\.80\\pm 2\.0026\.4EltwiseLow6:V:\[9,10,21,29,8,18\]∣O:\[29,31,23,27,7,4\]V\{:\}\[9,10,21,29,8,18\]\\mid O\{:\}\[29,31,23,27,7,4\]6\.796\.7959\.33±1\.9359\.33\\pm 1\.9326\.3EltwiseLow9:V:\[9,10,21,29,8,18,7,2,5\]∣O:\[29,31,23,27,7,4,3,28,12\]V\{:\}\[9,10,21,29,8,18,7,2,5\]\\mid O\{:\}\[29,31,23,27,7,4,3,28,12\]6\.016\.0161\.17±1\.42\\mathbf\{61\.17\\pm 1\.42\}26\.2EltwiseLow12:V:\[9,10,21,29,8,18,7,2,5,30,17,23\]∣O:\[29,31,23,27,7,4,3,28,12,9,6,14\]V\{:\}\[9,10,21,29,8,18,7,2,5,30,17,23\]\\mid O\{:\}\[29,31,23,27,7,4,3,28,12,9,6,14\]5\.225\.2260\.57±1\.1660\.57\\pm 1\.1626\.3EltwiseLow15:V:\[9,10,21,29,8,18,7,2,5,30,17,23,19,16,4\]∣O:\[29,31,23,27,7,4,3,28,12,9,6,14,8,1,19\]V\{:\}\[9,10,21,29,8,18,7,2,5,30,17,23,19,16,4\]\\mid O\{:\}\[29,31,23,27,7,4,3,28,12,9,6,14,8,1,19\]4\.444\.4460\.50±1\.0460\.50\\pm 1\.0426\.2EltwiseHigh3:V:\[27,13,20\]∣O:\[26,18,25\]V\{:\}\[27,13,20\]\\mid O\{:\}\[26,18,25\]7\.577\.5759\.73±2\.0659\.73\\pm 2\.0626\.2EltwiseHigh6:V:\[25,6,28,27,13,20\]∣O:\[20,16,0,26,18,25\]V\{:\}\[25,6,28,27,13,20\]\\mid O\{:\}\[20,16,0,26,18,25\]6\.796\.7959\.60±0\.3559\.60\\pm 0\.3526\.3EltwiseHigh9:V:\[24,15,14,25,6,28,27,13,20\]∣O:\[30,13,10,20,16,0,26,18,25\]V\{:\}\[24,15,14,25,6,28,27,13,20\]\\mid O\{:\}\[30,13,10,20,16,0,26,18,25\]6\.016\.0159\.47±0\.9759\.47\\pm 0\.9726\.2EltwiseHigh12:V:\[12,0,1,24,15,14,25,6,28,27,13,20\]∣O:\[15,17,21,30,13,10,20,16,0,26,18,25\]V\{:\}\[12,0,1,24,15,14,25,6,28,27,13,20\]\\mid O\{:\}\[15,17,21,30,13,10,20,16,0,26,18,25\]5\.225\.2258\.27±0\.3158\.27\\pm 0\.3126\.2EltwiseHigh15:V:\[26,11,31,12,0,1,24,15,14,25,6,28,27,13,20\]∣O:\[24,22,5,15,17,21,30,13,10,20,16,0,26,18,25\]V\{:\}\[26,11,31,12,0,1,24,15,14,25,6,28,27,13,20\]\\mid O\{:\}\[24,22,5,15,17,21,30,13,10,20,16,0,26,18,25\]4\.444\.4458\.30±1\.6058\.30\\pm 1\.6026\.1TDA\-based Tuning \(ours\)H​0H0Low3:V:\[29,26,30\]∣O:\[31,30,26\]V\{:\}\[29,26,30\]\\mid O\{:\}\[31,30,26\]7\.577\.5758\.17±0\.4258\.17\\pm 0\.4225\.4H​0H0Low6:V:\[26,27,28,29,30,31\]∣O:\[24,25,26,29,30,31\]V\{:\}\[26,27,28,29,30,31\]\\mid O\{:\}\[24,25,26,29,30,31\]6\.796\.7957\.43±0\.2557\.43\\pm 0\.2525\.4H​0H0Low9:V:\[22,23,25,26,27,28,29,30,31\]∣O:\[21,23,24,25,26,27,29,30,31\]V\{:\}\[22,23,25,26,27,28,29,30,31\]\\mid O\{:\}\[21,23,24,25,26,27,29,30,31\]6\.016\.0158\.60±0\.5658\.60\\pm 0\.5625\.6H​0H0Low12:V:\[15,21,22,23,24,25,26,27,28,29,30,31\]∣O:\[20,21,22,23,24,25,26,27,28,29,30,31\]V\{:\}\[15,21,22,23,24,25,26,27,28,29,30,31\]\\mid O\{:\}\[20,21,22,23,24,25,26,27,28,29,30,31\]5\.225\.2258\.43±1\.4058\.43\\pm 1\.4025\.8H​0H0Low15:V:\[29,26,30,28,31,27,25,22,23,15,24,21,17,3,1\]∣O:\[31,30,26,29,25,24,27,21,23,28,22,20,16,4,19\]V\{:\}\[29,26,30,28,31,27,25,22,23,15,24,21,17,3,1\]\\mid O\{:\}\[31,30,26,29,25,24,27,21,23,28,22,20,16,4,19\]4\.444\.4458\.43±0\.5858\.43\\pm 0\.5825\.1H​0H0High3:V:\[8,2,0\]∣O:\[11,10,13\]V\{:\}\[8,2,0\]\\mid O\{:\}\[11,10,13\]7\.577\.5759\.33±0\.4659\.33\\pm 0\.4626\.2H​0H0High6:V:\[0,2,7,8,10,11\]∣O:\[0,5,9,10,11,13\]V\{:\}\[0,2,7,8,10,11\]\\mid O\{:\}\[0,5,9,10,11,13\]6\.796\.7961\.17±0\.75\\mathbf\{61\.17\\pm 0\.75\}25\.7H​0H0High9:V:\[0,2,5,6,7,8,10,11,16\]∣O:\[0,2,5,8,9,10,11,12,13\]V\{:\}\[0,2,5,6,7,8,10,11,16\]\\mid O\{:\}\[0,2,5,8,9,10,11,12,13\]6\.016\.0158\.97±0\.7658\.97\\pm 0\.7625\.5H​0H0High12:V:\[0,2,5,6,7,8,10,11,12,13,16,18\]∣O:\[0,1,2,5,6,7,8,9,10,11,12,13\]V\{:\}\[0,2,5,6,7,8,10,11,12,13,16,18\]\\mid O\{:\}\[0,1,2,5,6,7,8,9,10,11,12,13\]5\.225\.2258\.37±0\.6058\.37\\pm 0\.6025\.9H​0H0High15:V:\[0,2,4,5,6,7,8,9,10,11,12,13,14,16,18\]∣O:\[0,1,2,3,5,6,7,8,9,10,11,12,13,15,18\]V\{:\}\[0,2,4,5,6,7,8,9,10,11,12,13,14,16,18\]\\mid O\{:\}\[0,1,2,3,5,6,7,8,9,10,11,12,13,15,18\]4\.444\.4457\.80±1\.1857\.80\\pm 1\.1826\.0

Table 10:Qwen3\-8B\-BaseonQA:GSM8Kwith separateVVandOOfreezing\. Among selective freezing settings, the best configuration is TDA\-High33\(%87\.62\), followed by Eltwise\-High99\(%87\.43\) and Eltwise\-Low1212\(%87\.40\)\.MethodFrozen V/O LayersTrainable %Avg\. Accuracy±\\pmStd \(↑\\uparrow\)Time \(min\)BaselinesFull–10010086\.27±0\.7586\.27\\pm 0\.7547\.3LoRA–0\.190\.1987\.89±0\.9087\.89\\pm 0\.9035\.4Magnitude\-based Tuning \(ours\)EltwiseLow3:V:\[11,24,8\]∣O:\[32,34,24\]V\{:\}\[11,24,8\]\\mid O\{:\}\[32,34,24\]8\.458\.4587\.30±0\.3687\.30\\pm 0\.3634\.6EltwiseLow6:V:\[11,24,8,12,18,33\]∣O:\[32,34,24,10,27,33\]V\{:\}\[11,24,8,12,18,33\]\\mid O\{:\}\[32,34,24,10,27,33\]7\.687\.6887\.23±0\.1287\.23\\pm 0\.1234\.5EltwiseLow9:V:\[11,24,8,12,18,33,5,34,15\]∣O:\[32,34,24,10,27,33,19,6,4\]V\{:\}\[11,24,8,12,18,33,5,34,15\]\\mid O\{:\}\[32,34,24,10,27,33,19,6,4\]6\.916\.9187\.23±0\.2187\.23\\pm 0\.2134\.4EltwiseLow12:V:\[11,24,8,12,18,33,5,34,15,29,9,35\]∣O:\[32,34,24,10,27,33,19,6,4,13,11,20\]V\{:\}\[11,24,8,12,18,33,5,34,15,29,9,35\]\\mid O\{:\}\[32,34,24,10,27,33,19,6,4,13,11,20\]6\.146\.1487\.40±0\.44\\mathbf\{87\.40\\pm 0\.44\}34\.2EltwiseLow15:V:\[11,24,8,12,18,33,5,34,15,29,9,35,0,2,4\]∣O:\[32,34,24,10,27,33,19,6,4,13,11,20,28,29,9\]V\{:\}\[11,24,8,12,18,33,5,34,15,29,9,35,0,2,4\]\\mid O\{:\}\[32,34,24,10,27,33,19,6,4,13,11,20,28,29,9\]5\.385\.3886\.80±0\.1786\.80\\pm 0\.1734\.2EltwiseHigh3:V:\[32,22,26\]∣O:\[35,2,30\]V\{:\}\[32,22,26\]\\mid O\{:\}\[35,2,30\]8\.458\.4587\.33±0\.1587\.33\\pm 0\.1534\.6EltwiseHigh6:V:\[28,3,13,32,22,26\]∣O:\[21,5,15,35,2,30\]V\{:\}\[28,3,13,32,22,26\]\\mid O\{:\}\[21,5,15,35,2,30\]7\.687\.6887\.17±0\.1587\.17\\pm 0\.1534\.5EltwiseHigh9:V:\[1,19,31,28,3,13,32,22,26\]∣O:\[14,8,26,21,5,15,35,2,30\]V\{:\}\[1,19,31,28,3,13,32,22,26\]\\mid O\{:\}\[14,8,26,21,5,15,35,2,30\]6\.916\.9187\.43±0\.57\\mathbf\{87\.43\\pm 0\.57\}34\.4EltwiseHigh12:V:\[17,27,10,1,19,31,28,3,13,32,22,26\]∣O:\[17,22,16,14,8,26,21,5,15,35,2,30\]V\{:\}\[17,27,10,1,19,31,28,3,13,32,22,26\]\\mid O\{:\}\[17,22,16,14,8,26,21,5,15,35,2,30\]6\.146\.1486\.83±0\.2986\.83\\pm 0\.2934\.2EltwiseHigh15:V:\[25,20,23,17,27,10,1,19,31,28,3,13,32,22,26\]∣O:\[1,3,25,17,22,16,14,8,26,21,5,15,35,2,30\]V\{:\}\[25,20,23,17,27,10,1,19,31,28,3,13,32,22,26\]\\mid O\{:\}\[1,3,25,17,22,16,14,8,26,21,5,15,35,2,30\]5\.385\.3886\.37±0\.6586\.37\\pm 0\.6534\.1TDA\-based Tuning \(ours\)H​0H0Low3:V:\[25,10,4\]∣O:\[6,34,29\]V\{:\}\[25,10,4\]\\mid O\{:\}\[6,34,29\]8\.458\.4586\.40±0\.3886\.40\\pm 0\.3834\.7H​0H0Low6:V:\[25,10,4,6,12,17\]∣O:\[6,34,29,25,28,4\]V\{:\}\[25,10,4,6,12,17\]\\mid O\{:\}\[6,34,29,25,28,4\]7\.687\.6886\.91±0\.2786\.91\\pm 0\.2734\.6H​0H0Low9:V:\[25,10,4,6,12,17,8,15,7\]∣O:\[6,34,29,25,28,4,30,26,31\]V\{:\}\[25,10,4,6,12,17,8,15,7\]\\mid O\{:\}\[6,34,29,25,28,4,30,26,31\]6\.916\.9186\.86±0\.4286\.86\\pm 0\.4234\.4H​0H0Low12:V:\[25,10,4,6,12,17,8,15,7,19,26,34\]∣O:\[6,34,29,25,28,4,30,26,31,32,9,35\]V\{:\}\[25,10,4,6,12,17,8,15,7,19,26,34\]\\mid O\{:\}\[6,34,29,25,28,4,30,26,31,32,9,35\]6\.146\.1486\.30±0\.3586\.30\\pm 0\.3534\.0H​0H0Low15:V:\[25,10,4,6,12,17,8,15,7,19,26,34,24,28,30\]∣O:\[6,34,29,25,28,4,30,26,31,32,9,35,27,24,5\]V\{:\}\[25,10,4,6,12,17,8,15,7,19,26,34,24,28,30\]\\mid O\{:\}\[6,34,29,25,28,4,30,26,31,32,9,35,27,24,5\]5\.385\.3886\.33±0\.1986\.33\\pm 0\.1934\.1H​0H0High3:V:\[1,5,0\]∣O:\[18,17,15\]V\{:\}\[1,5,0\]\\mid O\{:\}\[18,17,15\]8\.458\.4587\.62±0\.18\\mathbf\{87\.62\\pm 0\.18\}34\.5H​0H0High6:V:\[31,29,1,5,0,13\]∣O:\[14,19,17,18,15,16\]V\{:\}\[31,29,1,5,0,13\]\\mid O\{:\}\[14,19,17,18,15,16\]7\.687\.6886\.91±0\.4686\.91\\pm 0\.4634\.4H​0H0High9:V:\[32,3,21,13,31,29,1,5,0\]∣O:\[20,13,22,16,14,19,17,18,15\]V\{:\}\[32,3,21,13,31,29,1,5,0\]\\mid O\{:\}\[20,13,22,16,14,19,17,18,15\]6\.916\.9186\.91±0\.1686\.91\\pm 0\.1634\.4H​0H0High12:V:\[16,33,35,32,3,21,13,31,29,1,5,0\]∣O:\[2,11,0,20,13,22,16,14,19,17,18,15\]V\{:\}\[16,33,35,32,3,21,13,31,29,1,5,0\]\\mid O\{:\}\[2,11,0,20,13,22,16,14,19,17,18,15\]6\.146\.1487\.10±0\.1287\.10\\pm 0\.1234\.0H​0H0High15:V:\[14,20,11,16,33,35,32,3,21,13,31,29,1,5,0\]∣O:\[1,12,23,2,11,0,20,13,22,16,14,19,17,18,15\]V\{:\}\[14,20,11,16,33,35,32,3,21,13,31,29,1,5,0\]\\mid O\{:\}\[1,12,23,2,11,0,20,13,22,16,14,19,17,18,15\]5\.385\.3886\.88±0\.3186\.88\\pm 0\.3134\.1

Table 11:Mistral\-7B\-v0\.3onQA:GSM8Kwith separateVVandOOfreezing\. Among selective freezing settings, the best configuration is TDA\-Low6 \(54\.31%\), followed by Eltwise\-High6 \(53\.96%\)\.MethodFrozen LayersTrainable %Avg\. Accuracy±\\pmStd \(↑\\uparrow\)Time \(min\)BaselinesFull–10010059\.59±0\.8559\.59\\pm 0\.8543\.3LoRA–0\.190\.1949\.28±1\.9449\.28\\pm 1\.9433\.5Magnitude\-based Tuning \(ours\)EltwiseLow3:V:\[6,10,26\]∣O:\[14,18,19\]V\{:\}\[6,10,26\]\\mid O\{:\}\[14,18,19\]8\.398\.3953\.58±0\.5553\.58\\pm 0\.5530\.5EltwiseLow6:V:\[1,6,10,14,19,26\]∣O:\[11,14,18,19,27,29\]V\{:\}\[1,6,10,14,19,26\]\\mid O\{:\}\[11,14,18,19,27,29\]6\.86\.853\.78±1\.0953\.78\\pm 1\.0930\.5EltwiseLow9:V:\[1,3,6,7,10,11,14,19,26\]∣O:\[1,11,13,14,18,19,21,27,29\]V\{:\}\[1,3,6,7,10,11,14,19,26\]\\mid O\{:\}\[1,11,13,14,18,19,21,27,29\]5\.215\.2153\.52±0\.7753\.52\\pm 0\.7730\.4EltwiseLow12:V:\[1,3,6,7,10,11,13,14,17,19,21,26\]∣O:\[1,2,8,11,13,14,18,19,21,27,29,30\]V\{:\}\[1,3,6,7,10,11,13,14,17,19,21,26\]\\mid O\{:\}\[1,2,8,11,13,14,18,19,21,27,29,30\]3\.623\.6252\.29±0\.6152\.29\\pm 0\.6130\.3EltwiseLow15:V:\[1,2,3,6,7,10,11,13,14,17,19,21,23,26,31\]∣O:\[1,2,5,8,11,13,14,15,18,19,21,25,27,29,30\]V\{:\}\[1,2,3,6,7,10,11,13,14,17,19,21,23,26,31\]\\mid O\{:\}\[1,2,5,8,11,13,14,15,18,19,21,25,27,29,30\]2\.032\.0351\.91±0\.4251\.91\\pm 0\.4230\.1EltwiseHigh3:V:\[8,9,30\]∣O:\[4,6,12\]V\{:\}\[8,9,30\]\\mid O\{:\}\[4,6,12\]8\.398\.3953\.90±2\.63\\mathbf\{53\.90\\pm 2\.63\}31\.3EltwiseHigh6:V:\[4,8,9,16,29,30\]∣O:\[4,6,7,12,28,31\]V\{:\}\[4,8,9,16,29,30\]\\mid O\{:\}\[4,6,7,12,28,31\]6\.86\.853\.96±1\.02\\mathbf\{53\.96\\pm 1\.02\}30\.4EltwiseHigh9:V:\[0,4,8,9,16,18,25,29,30\]∣O:\[0,3,4,6,7,12,22,28,31\]V\{:\}\[0,4,8,9,16,18,25,29,30\]\\mid O\{:\}\[0,3,4,6,7,12,22,28,31\]5\.215\.2152\.29±1\.5452\.29\\pm 1\.5430\.3EltwiseHigh12:V:\[0,4,8,9,12,16,18,20,24,25,29,30\]∣O:\[0,3,4,6,7,10,12,16,22,23,28,31\]V\{:\}\[0,4,8,9,12,16,18,20,24,25,29,30\]\\mid O\{:\}\[0,3,4,6,7,10,12,16,22,23,28,31\]3\.623\.6253\.73±1\.0553\.73\\pm 1\.0530\.3EltwiseHigh15:V:\[0,4,5,8,9,12,16,18,20,22,24,25,28,29,30\]∣O:\[0,3,4,6,7,9,10,12,16,17,20,22,23,28,31\]V\{:\}\[0,4,5,8,9,12,16,18,20,22,24,25,28,29,30\]\\mid O\{:\}\[0,3,4,6,7,9,10,12,16,17,20,22,23,28,31\]2\.032\.0353\.60±0\.2353\.60\\pm 0\.2330\.1TDA\-based Tuning \(ours\)H​0H0Low3:V:\[24,25,29\]∣O:\[25,28,31\]V\{:\}\[24,25,29\]\\mid O\{:\}\[25,28,31\]8\.398\.3953\.93±1\.06\\mathbf\{53\.93\\pm 1\.06\}30\.4H​0H0Low6:V:\[22,24,25,26,29,30\]∣O:\[22,25,28,29,30,31\]V\{:\}\[22,24,25,26,29,30\]\\mid O\{:\}\[22,25,28,29,30,31\]6\.86\.854\.31±1\.64\\mathbf\{54\.31\\pm 1\.64\}30\.5H​0H0Low9:V:\[22,23,24,25,26,27,28,29,30\]∣O:\[21,22,24,25,26,28,29,30,31\]V\{:\}\[22,23,24,25,26,27,28,29,30\]\\mid O\{:\}\[21,22,24,25,26,28,29,30,31\]5\.215\.2153\.88±0\.2353\.88\\pm 0\.2330\.4H​0H0Low12:V:\[16,20,22,23,24,25,26,27,28,29,30,31\]∣O:\[17,21,22,23,24,25,26,27,28,29,30,31\]V\{:\}\[16,20,22,23,24,25,26,27,28,29,30,31\]\\mid O\{:\}\[17,21,22,23,24,25,26,27,28,29,30,31\]3\.623\.6252\.79±1\.0152\.79\\pm 1\.0130\.1H​0H0Low15:V:\[16,18,19,20,21,22,23,24,25,26,27,28,29,30,31\]∣O:\[3,6,17,20,21,22,23,24,25,26,27,28,29,30,31\]V\{:\}\[16,18,19,20,21,22,23,24,25,26,27,28,29,30,31\]\\mid O\{:\}\[3,6,17,20,21,22,23,24,25,26,27,28,29,30,31\]2\.032\.0352\.82±0\.5652\.82\\pm 0\.5630\.0H​0H0High3:V:\[0,10,12\]∣O:\[0,1,2\]V\{:\}\[0,10,12\]\\mid O\{:\}\[0,1,2\]8\.398\.3952\.87±0\.4652\.87\\pm 0\.4630\.5H​0H0High6:V:\[0,8,9,10,11,12\]∣O:\[0,1,2,9,11,16\]V\{:\}\[0,8,9,10,11,12\]\\mid O\{:\}\[0,1,2,9,11,16\]6\.86\.852\.64±0\.9952\.64\\pm 0\.9930\.5H​0H0High9:V:\[0,2,6,8,9,10,11,12,13\]∣O:\[0,1,2,9,10,11,13,15,16\]V\{:\}\[0,2,6,8,9,10,11,12,13\]\\mid O\{:\}\[0,1,2,9,10,11,13,15,16\]5\.215\.2151\.40±0\.9251\.40\\pm 0\.9230\.3H​0H0High12:V:\[0,2,5,6,7,8,9,10,11,12,13,14\]∣O:\[0,1,2,5,7,9,10,11,12,13,15,16\]V\{:\}\[0,2,5,6,7,8,9,10,11,12,13,14\]\\mid O\{:\}\[0,1,2,5,7,9,10,11,12,13,15,16\]3\.623\.6250\.64±0\.9550\.64\\pm 0\.9530\.2H​0H0High15:V:\[0,1,2,4,5,6,7,8,9,10,11,12,13,14,15\]∣O:\[0,1,2,4,5,7,8,9,10,11,12,13,15,16,19\]V\{:\}\[0,1,2,4,5,6,7,8,9,10,11,12,13,14,15\]\\mid O\{:\}\[0,1,2,4,5,7,8,9,10,11,12,13,15,16,19\]2\.032\.0348\.60±0\.6048\.60\\pm 0\.6030\.0

## Appendix KTransferred\-Knowledge Accuracy Curves

Figure[15](https://arxiv.org/html/2607.16637#A11.F15)shows the epoch\-wise accuracy trajectories forQA:MMLU,SA:SST\-2, andSA:IMDBin the transferred\-knowledge setting\. The curves compare full fine\-tuning, LoRA, andTopoTunerafter reusing the freezing profile learned fromQA:GSM8K\. These trajectories provide a per\-epoch view of model adaptation across the transferred evaluation tasks\.

SA: SST\-2![Refer to caption](https://arxiv.org/html/2607.16637v1/x69.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x70.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x71.png)SA: IMDB![Refer to caption](https://arxiv.org/html/2607.16637v1/x72.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x73.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x74.png)QA: MMLU![Refer to caption](https://arxiv.org/html/2607.16637v1/x75.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x76.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x77.png)LLaMA\-3\.1\-8BQwen3\-8B\-BaseMistral\-7B\-v0\.3Figure 15:Accuracies on each epoch for all the models across datasets
## Appendix LSpectrum and DropBP

#### Spectrum\.

Spectrum\[[26](https://arxiv.org/html/2607.16637#bib.bib18)\]is a parameter\-selection baseline that uses signal\-to\-noise ratio \(SNR\) statistics to decide which parameters should remain trainable\. In our implementation, we first tested several top\-N%N\\%SNR configurations, includingN∈\{50,10,3\}N\\in\\\{50,10,3\\\}, to understand how the selected trainable budget changes with the masking ratio\. We report theN=5N=5setting because its updatable parameter fraction is closest toTopoTuner, making the comparison more controlled in terms of effective parameter updates\.

#### DropBP\.

DropBP\[[35](https://arxiv.org/html/2607.16637#bib.bib17)\]is a compute\- and memory\-oriented training baseline that reduces backward computation during fine\-tuning\. Since DropBP can be combined with either full fine\-tuning or LoRA, we considered both DropBP\+Full and DropBP\+LoRA as possible baselines\. We use DropBP\+LoRA in the main comparison because its updatable parameter fraction is closer toTopoTuner, making the LoRA\-based DropBP setting the fairer comparison\.

#### Experimental protocol\.

For both baselines, we keep the experimental setup aligned with the rest of our transfer experiments\. Spectrum is run with selective full\-parameter fine\-tuning over the SNR\-selected unfrozen weights, while DropBP is run with LoRA adapters on the attention projections\. We use the same model\-dataset grid, training data regime, and evaluation pipeline as our mainSA:SST\-2,SA:IMDB, andQA:MMLUexperiments\. The results are provided in Table[2](https://arxiv.org/html/2607.16637#S6.T2)\. Experiment setups and hyperparameter details are provided in the Appendix[C](https://arxiv.org/html/2607.16637#A3)\.

## Appendix MFine\-tuning with Dynamic Stopping

![Refer to caption](https://arxiv.org/html/2607.16637v1/x78.png)![Refer to caption](https://arxiv.org/html/2607.16637v1/x79.png)SA:IMDBQA:MMLUFigure 16:Topology\-based stopping epochs under full fine\-tuning\. The plots show selected stopping epochs acrossη\\etathresholds forSA:IMDBandQA:MMLUacross the three model families\.
## Appendix NCross\-task Representation Geometry

For each model, projectionp∈\{K,Q,V,O\}p\\in\\\{K,Q,V,O\\\}, layerℓ\\ell, and dataset pair\(Da,Db\)\(D\_\{a\},D\_\{b\}\), we compute the Wasserstein distance between their final\-epoch persistence diagrams:

dp,ℓ​\(Da,Db\)=W​\(P​DDa,p,ℓ\(6\),P​DDb,p,ℓ\(6\)\)\.d\_\{p,\\ell\}\(D\_\{a\},D\_\{b\}\)=W\\\!\\left\(PD^\{\(6\)\}\_\{D\_\{a\},p,\\ell\},PD^\{\(6\)\}\_\{D\_\{b\},p,\\ell\}\\right\)\.We then normalize this distance within each layer and projection:

sp,ℓ​\(Da,Db\)=1−dp,ℓ​\(Da,Db\)maxDi,Dj⁡dp,ℓ​\(Di,Dj\)\.s\_\{p,\\ell\}\(D\_\{a\},D\_\{b\}\)=1\-\\frac\{d\_\{p,\\ell\}\(D\_\{a\},D\_\{b\}\)\}\{\\max\_\{D\_\{i\},D\_\{j\}\}d\_\{p,\\ell\}\(D\_\{i\},D\_\{j\}\)\}\.Thus,11indicates identical diagrams and0indicates the most different dataset pair for that layer and projection\. We averagesp,ℓs\_\{p,\\ell\}over layers to obtain the reported cross\-dataset similarity\.

Tables[12](https://arxiv.org/html/2607.16637#A14.T12)–[14](https://arxiv.org/html/2607.16637#A14.T14)report Wasserstein similarities between the task\-induced representations\. Across models,SA:IMDB,SA:SST\-2, andQA:MMLUshow higher mutual similarity than their similarities withQA:GSM8K\. This separation is most visible forQwen3\-8B\-Base, whereSA:IMDB–QA:MMLUreaches0\.50/0\.44/0\.45/0\.380\.50/0\.44/0\.45/0\.38acrossK/Q/V/OK/Q/V/O, whileSA:SST\-2–QA:GSM8Kis near zero\.LLaMA\-3\.1\-8BandMistral\-7B\-v0\.3show the same pattern: NLU\-style tasks are more aligned with one another, whereasQA:GSM8Kremains the least aligned task in most projection families\. These results suggest that mathematical reasoning induces a different topological adaptation pattern than the sentiment and general QA\-style tasks\.

Table 12:Cross\-dataset topological similarity for full\-finetunedLLaMA\-3\.1\-8B\. Each entry reports normalized Wasserstein similarity between final\-epoch persistence diagrams, averaged over 32 layers for theK/Q/V/OK/Q/V/Oprojections\.SA:IMDB\(K/Q/V/O\)\(K/Q/V/O\)SA:SST\-2\(K/Q/V/O\)\(K/Q/V/O\)QA:MMLU\(K/Q/V/O\)\(K/Q/V/O\)QA:GSM8K\(K/Q/V/O\)\(K/Q/V/O\)SA:IMDB1/1/1/11/1/1/1SA:SST\-20\.36/0\.29/0\.33/0\.280\.36/0\.29/0\.33/0\.281/1/1/11/1/1/1QA:MMLU0\.14/0\.15/0\.13/0\.120\.14/0\.15/0\.13/0\.120\.23/0\.19/0\.18/0\.160\.23/0\.19/0\.18/0\.161/1/1/11/1/1/1QA:GSM8K0\.10/0\.11/0\.09/0\.080\.10/0\.11/0\.09/0\.080\.17/0\.10/0\.12/0\.110\.17/0\.10/0\.12/0\.110\.01/0\.07/0\.02/0\.020\.01/0\.07/0\.02/0\.021/1/1/11/1/1/1

Table 13:Cross\-dataset topological similarity for full\-finetunedQwen3\-8B\-Base\. Each entry reports normalized Wasserstein similarity between final\-epoch persistence diagrams, averaged over 36 layers for theK/Q/V/OK/Q/V/Oprojections\.SA:IMDB\(K/Q/V/O\)\(K/Q/V/O\)SA:SST\-2\(K/Q/V/O\)\(K/Q/V/O\)QA:MMLU\(K/Q/V/O\)\(K/Q/V/O\)QA:GSM8K\(K/Q/V/O\)\(K/Q/V/O\)SA:IMDB1/1/1/11/1/1/1SA:SST\-20\.41/0\.35/0\.38/0\.330\.41/0\.35/0\.38/0\.331/1/1/11/1/1/1QA:MMLU0\.50/0\.44/0\.45/0\.380\.50/0\.44/0\.45/0\.380\.36/0\.31/0\.33/0\.280\.36/0\.31/0\.33/0\.281/1/1/11/1/1/1QA:GSM8K0\.06/0\.05/0\.05/0\.050\.06/0\.05/0\.05/0\.050\.01/0\.00/0\.00/0\.010\.01/0\.00/0\.00/0\.010\.06/0\.05/0\.05/0\.050\.06/0\.05/0\.05/0\.051/1/1/11/1/1/1

Table 14:Cross\-dataset topological similarity for full\-finetunedMistral\-7B\-v0\.3\. Each entry reports normalized Wasserstein similarity between final\-epoch persistence diagrams, averaged over 32 layers for theK/Q/V/OK/Q/V/Oprojections\.SA:IMDB\(K/Q/V/O\)\(K/Q/V/O\)SA:SST\-2\(K/Q/V/O\)\(K/Q/V/O\)QA:MMLU\(K/Q/V/O\)\(K/Q/V/O\)QA:GSM8K\(K/Q/V/O\)\(K/Q/V/O\)SA:IMDB1/1/1/11/1/1/1SA:SST\-20\.29/0\.27/0\.32/0\.270\.29/0\.27/0\.32/0\.271/1/1/11/1/1/1QA:MMLU0\.12/0\.12/0\.13/0\.120\.12/0\.12/0\.13/0\.120\.17/0\.14/0\.18/0\.150\.17/0\.14/0\.18/0\.151/1/1/11/1/1/1QA:GSM8K0\.07/0\.04/0\.08/0\.050\.07/0\.04/0\.08/0\.050\.11/0\.05/0\.12/0\.100\.11/0\.05/0\.12/0\.100\.02/0\.02/0\.01/0\.030\.02/0\.02/0\.01/0\.031/1/1/11/1/1/1

## Appendix OCatastrophic Forgetting Evaluation

The retention results show a second advantage of selective tuning\. Tables[15](https://arxiv.org/html/2607.16637#A15.T15)[17](https://arxiv.org/html/2607.16637#A15.T17)evaluate the models on 11 different datasets after prior fine\-tuning onQA:GSM8K\. The clearest retention gains appear forLLaMA\-3\.1\-8BandQwen3\-8B\-Base\. For both models,TopoTunerimproves over LoRA onQA:MMLU,SA:SST\-2,SA:IMDB,IR:SQuAD, andIR:HotpotQA, showing stronger preservation on question\-answering, sentiment\-analysis, and information\-retrieval evaluations\.

Qwen3\-8B\-Baseshows a more mixed pattern\. LoRA remains stronger on several generation\-style metrics, especially summarization, instruction following, and code generation\. However,TopoTunerstays very close to LoRA on most metrics and still improves over LoRA onSA:IMDBandIR:HotpotQA\. Overall, these results suggest that selective fine\-tuning reduces forgetting most clearly on question\-answering, sentiment\-analysis, and information\-retrieval tasks, while its effect on generation\-style evaluations depend on architecture\.

Table 15:Catastrophic forgetting evaluation onQA:GSM8Kfine\-tunedLLaMA\-3\.1\-8B\. Metrics are avg\.±\\pmstd over three fixed, non\-overlapping evaluation subsets\. Base and full finetuned models are shown as reference rows\.Boldindicates the better result between LoRA andTopoTunerfor each metric\.Question AnsweringSentiment AnalysisInformation RetrievalQA:MMLUSA:SST\-2SA:IMDBIR:SQuAD v1\.1IR:HotpotQAModelAcc\.Acc\.Acc\.EMF1EMF1Base model54\.5±1\.2554\.5\\pm 1\.2594\.0±0\.7294\.0\\pm 0\.7289\.8±0\.3889\.8\\pm 0\.3876\.0±0\.3276\.0\\pm 0\.3286\.9±0\.2086\.9\\pm 0\.2023\.1±1\.2523\.1\\pm 1\.2531\.9±0\.6431\.9\\pm 0\.64Full finetuned52\.9±0\.4452\.9\\pm 0\.4494\.3±0\.8094\.3\\pm 0\.8089\.0±0\.4789\.0\\pm 0\.4774\.4±0\.3974\.4\\pm 0\.3985\.0±0\.1585\.0\\pm 0\.1521\.3±1\.4021\.3\\pm 1\.4029\.4±0\.7629\.4\\pm 0\.76LoRA51\.8±1\.0151\.8\\pm 1\.0192\.7±0\.5292\.7\\pm 0\.5289\.3±0\.4889\.3\\pm 0\.4872\.3±0\.4972\.3\\pm 0\.4983\.6±0\.6683\.6\\pm 0\.6619\.2±1\.0319\.2\\pm 1\.0327\.2±0\.9727\.2\\pm 0\.97TopoTuner53\.6±0\.51\\mathbf\{53\.6\\pm 0\.51\}94\.4±0\.19\\mathbf\{94\.4\\pm 0\.19\}90\.3±0\.36\\mathbf\{90\.3\\pm 0\.36\}76\.9±0\.24\\mathbf\{76\.9\\pm 0\.24\}86\.0±0\.48\\mathbf\{86\.0\\pm 0\.48\}21\.8±1\.12\\mathbf\{21\.8\\pm 1\.12\}30\.2±0\.71\\mathbf\{30\.2\\pm 0\.71\}

SummarizationCode GenerationSU:XSumSU:CNN/DMCG:HumanEvalCG:MBPPModelR1R2RLR1R2RLpass@1pass@1Base model25\.2±0\.0525\.2\\pm 0\.056\.7±0\.086\.7\\pm 0\.0818\.8±0\.1018\.8\\pm 0\.1031\.7±0\.2931\.7\\pm 0\.2913\.6±0\.1813\.6\\pm 0\.1820\.6±0\.2320\.6\\pm 0\.230\.0±0\.000\.0\\pm 0\.002\.7±1\.352\.7\\pm 1\.35Full finetuned14\.8±0\.1614\.8\\pm 0\.164\.1±0\.114\.1\\pm 0\.1111\.7±0\.1711\.7\\pm 0\.1729\.5±0\.3429\.5\\pm 0\.3412\.4±0\.2412\.4\\pm 0\.2418\.9±0\.2818\.9\\pm 0\.281\.8±1\.851\.8\\pm 1\.853\.5±3\.073\.5\\pm 3\.07LoRA27\.2±0\.21\\mathbf\{27\.2\\pm 0\.21\}7\.8±0\.22\\mathbf\{7\.8\\pm 0\.22\}21\.6±0\.35\\mathbf\{21\.6\\pm 0\.35\}29\.8±0\.27\\mathbf\{29\.8\\pm 0\.27\}12\.6±0\.22\\mathbf\{12\.6\\pm 0\.22\}19\.2±0\.19\\mathbf\{19\.2\\pm 0\.19\}1\.2±1\.051\.2\\pm 1\.054\.7±3\.48\\mathbf\{4\.7\\pm 3\.48\}TopoTuner27\.2±0\.17\\mathbf\{27\.2\\pm 0\.17\}7\.7±0\.087\.7\\pm 0\.0821\.0±0\.2121\.0\\pm 0\.2127\.4±0\.4527\.4\\pm 0\.4511\.8±0\.2511\.8\\pm 0\.2517\.7±0\.3417\.7\\pm 0\.341\.8±3\.21\\mathbf\{1\.8\\pm 3\.21\}3\.1±1\.333\.1\\pm 1\.33

Instruction FollowingIF:Dolly\-15kIF:AlpacaModelR1R2RLMETEORR1R2RLMETEORBase model38\.0±0\.3238\.0\\pm 0\.3219\.3±0\.2519\.3\\pm 0\.2529\.9±0\.2529\.9\\pm 0\.2527\.227\.232\.6±0\.2732\.6\\pm 0\.2715\.0±0\.2515\.0\\pm 0\.2526\.1±0\.1826\.1\\pm 0\.1825\.725\.7Full finetuned38\.5±0\.1738\.5\\pm 0\.1719\.4±0\.1419\.4\\pm 0\.1430\.5±0\.0930\.5\\pm 0\.0927\.127\.139\.4±0\.2139\.4\\pm 0\.2119\.0±0\.2819\.0\\pm 0\.2832\.3±0\.1732\.3\\pm 0\.1727\.227\.2LoRA38\.0±0\.31\\mathbf\{38\.0\\pm 0\.31\}19\.5±0\.14\\mathbf\{19\.5\\pm 0\.14\}30\.5±0\.20\\mathbf\{30\.5\\pm 0\.20\}27\.027\.037\.0±0\.15\\mathbf\{37\.0\\pm 0\.15\}18\.2±0\.16\\mathbf\{18\.2\\pm 0\.16\}30\.7±0\.13\\mathbf\{30\.7\\pm 0\.13\}26\.0\\mathbf\{26\.0\}TopoTuner37\.2±0\.0937\.2\\pm 0\.0918\.9±0\.2618\.9\\pm 0\.2629\.4±0\.1829\.4\\pm 0\.1827\.5\\mathbf\{27\.5\}31\.9±0\.2731\.9\\pm 0\.2714\.3±0\.2714\.3\\pm 0\.2725\.7±0\.2825\.7\\pm 0\.2825\.525\.5

Table 16:Catastrophic forgetting evaluation onQA:GSM8Kfine\-tunedQwen3\-8B\-Base\. Metrics are avg\.±\\pmstd over three fixed, non\-overlapping evaluation subsets\. Base and full finetuned models are shown as reference rows\.Boldindicates the better result between LoRA andTopoTunerfor each metric\.Question AnsweringSentiment AnalysisInformation RetrievalQA:MMLUSA:SST\-2SA:IMDBIR:SQuAD v1\.1IR:HotpotQAModelAcc\.Acc\.Acc\.EMF1EMF1Base model64\.2±0\.7864\.2\\pm 0\.7894\.0±0\.7994\.0\\pm 0\.7990\.0±0\.2690\.0\\pm 0\.2667\.2±0\.5767\.2\\pm 0\.5782\.9±0\.3882\.9\\pm 0\.3821\.7±0\.9421\.7\\pm 0\.9430\.4±0\.6030\.4\\pm 0\.60Full finetuned64\.4±0\.6664\.4\\pm 0\.6693\.9±0\.9993\.9\\pm 0\.9989\.6±0\.2589\.6\\pm 0\.2575\.2±0\.1975\.2\\pm 0\.1987\.3±0\.4487\.3\\pm 0\.4421\.9±0\.4021\.9\\pm 0\.4030\.4±0\.0530\.4\\pm 0\.05LoRA64\.4±0\.58\\mathbf\{64\.4\\pm 0\.58\}94\.4±0\.79\\mathbf\{94\.4\\pm 0\.79\}89\.9±0\.3589\.9\\pm 0\.3577\.8±0\.19\\mathbf\{77\.8\\pm 0\.19\}88\.7±0\.20\\mathbf\{88\.7\\pm 0\.20\}22\.0±0\.3922\.0\\pm 0\.3930\.6±0\.2430\.6\\pm 0\.24TopoTuner64\.3±0\.6364\.3\\pm 0\.6393\.8±0\.9193\.8\\pm 0\.9190\.3±0\.22\\mathbf\{90\.3\\pm 0\.22\}74\.4±0\.2574\.4\\pm 0\.2586\.9±0\.3686\.9\\pm 0\.3622\.2±0\.64\\mathbf\{22\.2\\pm 0\.64\}30\.7±0\.16\\mathbf\{30\.7\\pm 0\.16\}

SummarizationCode GenerationSU:XSumSU:CNN/DMCG:HumanEvalCG:MBPPModelR1R2RLR1R2RLpass@1pass@1Base model25\.9±0\.0925\.9\\pm 0\.096\.4±0\.046\.4\\pm 0\.0418\.2±0\.0318\.2\\pm 0\.0338\.5±0\.1338\.5\\pm 0\.1316\.2±0\.0816\.2\\pm 0\.0824\.8±0\.0924\.8\\pm 0\.096\.1±1\.136\.1\\pm 1\.130\.4±0\.670\.4\\pm 0\.67Full finetuned25\.8±0\.0825\.8\\pm 0\.086\.4±0\.046\.4\\pm 0\.0418\.1±0\.0718\.1\\pm 0\.0738\.6±0\.1338\.6\\pm 0\.1316\.2±0\.1016\.2\\pm 0\.1024\.8±0\.0824\.8\\pm 0\.084\.9±2\.184\.9\\pm 2\.180\.8±0\.680\.8\\pm 0\.68LoRA26\.3±0\.08\\mathbf\{26\.3\\pm 0\.08\}6\.6±0\.07\\mathbf\{6\.6\\pm 0\.07\}18\.5±0\.04\\mathbf\{18\.5\\pm 0\.04\}38\.6±0\.06\\mathbf\{38\.6\\pm 0\.06\}16\.3±0\.09\\mathbf\{16\.3\\pm 0\.09\}24\.9±0\.08\\mathbf\{24\.9\\pm 0\.08\}8\.6±2\.22\\mathbf\{8\.6\\pm 2\.22\}1\.2±1\.18\\mathbf\{1\.2\\pm 1\.18\}TopoTuner26\.0±0\.0426\.0\\pm 0\.046\.5±0\.026\.5\\pm 0\.0218\.3±0\.0518\.3\\pm 0\.0538\.6±0\.16\\mathbf\{38\.6\\pm 0\.16\}16\.3±0\.12\\mathbf\{16\.3\\pm 0\.12\}24\.9±0\.09\\mathbf\{24\.9\\pm 0\.09\}7\.3±3\.747\.3\\pm 3\.740\.8±0\.680\.8\\pm 0\.68

Instruction FollowingIF:Dolly\-15kIF:AlpacaModelR1R2RLMETEORR1R2RLMETEORBase model36\.8±0\.2136\.8\\pm 0\.2117\.6±0\.1617\.6\\pm 0\.1627\.5±0\.1627\.5\\pm 0\.1627\.027\.037\.6±0\.2237\.6\\pm 0\.2217\.9±0\.2017\.9\\pm 0\.2028\.8±0\.1328\.8\\pm 0\.1329\.229\.2Full finetuned37\.2±0\.2937\.2\\pm 0\.2918\.0±0\.1718\.0\\pm 0\.1727\.9±0\.2027\.9\\pm 0\.2027\.527\.537\.3±0\.1837\.3\\pm 0\.1817\.5±0\.1517\.5\\pm 0\.1528\.5±0\.0628\.5\\pm 0\.0628\.928\.9LoRA38\.0±0\.33\\mathbf\{38\.0\\pm 0\.33\}18\.7±0\.14\\mathbf\{18\.7\\pm 0\.14\}28\.7±0\.17\\mathbf\{28\.7\\pm 0\.17\}28\.0\\mathbf\{28\.0\}38\.3±0\.22\\mathbf\{38\.3\\pm 0\.22\}18\.3±0\.26\\mathbf\{18\.3\\pm 0\.26\}29\.4±0\.17\\mathbf\{29\.4\\pm 0\.17\}29\.5\\mathbf\{29\.5\}TopoTuner37\.3±0\.2437\.3\\pm 0\.2418\.1±0\.1618\.1\\pm 0\.1628\.0±0\.1428\.0\\pm 0\.1427\.327\.337\.8±0\.2637\.8\\pm 0\.2618\.0±0\.3118\.0\\pm 0\.3129\.0±0\.2129\.0\\pm 0\.2129\.329\.3

Table 17:Catastrophic forgetting evaluation onQA:GSM8Kfine\-tunedMistral\-7B\-v0\.3\. Metrics are avg\.±\\pmstd over three fixed, non\-overlapping evaluation subsets\. Base and full finetuned models are shown as reference rows\.Boldindicates the better result between LoRA andTopoTunerfor each metric\.Question AnsweringSentiment AnalysisInformation RetrievalQA:MMLUSA:SST\-2SA:IMDBIR:SQuAD v1\.1IR:HotpotQAModelAcc\.Acc\.Acc\.EMF1EMF1Base model54\.5±0\.4154\.5\\pm 0\.4189\.0±0\.6189\.0\\pm 0\.6161\.4±0\.4861\.4\\pm 0\.4871\.3±1\.0271\.3\\pm 1\.0283\.9±0\.7283\.9\\pm 0\.7223\.6±1\.0823\.6\\pm 1\.0831\.8±0\.8031\.8\\pm 0\.80Full finetuned52\.9±0\.3352\.9\\pm 0\.3389\.0±1\.8489\.0\\pm 1\.8460\.2±0\.5060\.2\\pm 0\.5073\.2±0\.6473\.2\\pm 0\.6484\.4±0\.5484\.4\\pm 0\.5421\.6±1\.2821\.6\\pm 1\.2829\.0±0\.8329\.0\\pm 0\.83LoRA50\.2±0\.7950\.2\\pm 0\.7970\.6±2\.8170\.6\\pm 2\.8159\.2±0\.5659\.2\\pm 0\.5669\.8±0\.9869\.8\\pm 0\.9881\.6±0\.8081\.6\\pm 0\.8018\.7±0\.8818\.7\\pm 0\.8826\.5±0\.6626\.5\\pm 0\.66TopoTuner53\.2±0\.33\\mathbf\{53\.2\\pm 0\.33\}89\.9±0\.86\\mathbf\{89\.9\\pm 0\.86\}60\.6±0\.56\\mathbf\{60\.6\\pm 0\.56\}70\.2±0\.74\\mathbf\{70\.2\\pm 0\.74\}83\.0±0\.63\\mathbf\{83\.0\\pm 0\.63\}22\.0±1\.34\\mathbf\{22\.0\\pm 1\.34\}29\.9±0\.86\\mathbf\{29\.9\\pm 0\.86\}

SummarizationCode GenerationSU:XSumSU:CNN/DMCG:HumanEvalCG:MBPPModelR1R2RLR1R2RLpass@1pass@1Base model22\.2±0\.1022\.2\\pm 0\.106\.2±0\.126\.2\\pm 0\.1216\.4±0\.1716\.4\\pm 0\.1739\.5±0\.1639\.5\\pm 0\.1617\.4±0\.1417\.4\\pm 0\.1425\.0±0\.1425\.0\\pm 0\.141\.8±1\.821\.8\\pm 1\.823\.5±1\.153\.5\\pm 1\.15Full finetuned22\.2±0\.1622\.2\\pm 0\.167\.2±0\.117\.2\\pm 0\.1116\.9±0\.2116\.9\\pm 0\.2139\.6±0\.1839\.6\\pm 0\.1817\.5±0\.2017\.5\\pm 0\.2025\.0±0\.1425\.0\\pm 0\.141\.8±1\.851\.8\\pm 1\.851\.6±1\.771\.6\\pm 1\.77LoRA29\.2±0\.23\\mathbf\{29\.2\\pm 0\.23\}8\.9±0\.08\\mathbf\{8\.9\\pm 0\.08\}22\.4±0\.24\\mathbf\{22\.4\\pm 0\.24\}36\.5±0\.1836\.5\\pm 0\.1815\.5±0\.2215\.5\\pm 0\.2223\.6±0\.1423\.6\\pm 0\.140\.0±0\.000\.0\\pm 0\.000\.4±0\.670\.4\\pm 0\.67TopoTuner27\.7±0\.1427\.7\\pm 0\.148\.5±0\.158\.5\\pm 0\.1520\.6±0\.2620\.6\\pm 0\.2639\.8±0\.21\\mathbf\{39\.8\\pm 0\.21\}17\.6±0\.22\\mathbf\{17\.6\\pm 0\.22\}25\.1±0\.16\\mathbf\{25\.1\\pm 0\.16\}2\.4±1\.04\\mathbf\{2\.4\\pm 1\.04\}1\.9±1\.34\\mathbf\{1\.9\\pm 1\.34\}

Instruction FollowingIF:Dolly\-15kIF:AlpacaModelR1R2RLMETEORR1R2RLMETEORBase model28\.7±0\.2128\.7\\pm 0\.2112\.8±0\.0512\.8\\pm 0\.0521\.7±0\.1321\.7\\pm 0\.1324\.424\.426\.8±0\.2326\.8\\pm 0\.2311\.3±0\.1811\.3\\pm 0\.1820\.0±0\.1520\.0\\pm 0\.1524\.224\.2Full finetuned35\.5±0\.1335\.5\\pm 0\.1317\.3±0\.0217\.3\\pm 0\.0227\.9±0\.0727\.9\\pm 0\.0726\.726\.732\.8±0\.0832\.8\\pm 0\.0814\.8±0\.0414\.8\\pm 0\.0425\.4±0\.0225\.4\\pm 0\.0226\.526\.5LoRA34\.7±0\.29\\mathbf\{34\.7\\pm 0\.29\}17\.1±0\.15\\mathbf\{17\.1\\pm 0\.15\}28\.2±0\.12\\mathbf\{28\.2\\pm 0\.12\}24\.724\.735\.0±0\.14\\mathbf\{35\.0\\pm 0\.14\}16\.3±0\.19\\mathbf\{16\.3\\pm 0\.19\}28\.3±0\.10\\mathbf\{28\.3\\pm 0\.10\}25\.4\\mathbf\{25\.4\}TopoTuner33\.4±0\.3533\.4\\pm 0\.3516\.2±0\.0916\.2\\pm 0\.0926\.4±0\.1726\.4\\pm 0\.1726\.4\\mathbf\{26\.4\}29\.1±0\.2529\.1\\pm 0\.2512\.8±0\.1512\.8\\pm 0\.1522\.3±0\.1822\.3\\pm 0\.1825\.225\.2

Taken together, the experiments support four claims\. First, fine\-tuning changes are small in magnitude but structured in location, which makes naive layer\-order freezing ineffective\. Second, Wasserstein ranking and norm ranking each recover useful selective\-tuning configurations, but neither signal alone induces a universally dominant ordering\. Third, topology reveals cross\-task geometry that norm does not express, withQA:GSM8Kseparated cleanly fromSA:IMDB,SA:SST\-2, andQA:MMLUin Wasserstein space\. Fourth, selectiveV/OV/Otuning can preserve substantial target\-task performance while reducing trainable parameters and retaining more out\-of\-domain capability than full fine\-tuning\. This combination of efficiency, retention, and task\-geometry analysis is the central empirical message of the paper\.

## Appendix POverfitting Analysis

We analyze overfitting behavior across full fine\-tuning, LoRA, TDA\-High3, TDA\-Low3, Eltwise\-High3, and Eltwise\-Low3 forLLaMA\-3\.1\-8BTable[18](https://arxiv.org/html/2607.16637#A16.T18)reports the corresponding generalization gaps\. ForLLaMA\-3\.1\-8B, full fine\-tuning obtains the highest final test accuracy, but it also shows the largest train\-test gap\. In contrast, TDA\-High3 provides the best selective test accuracy while yielding the smallest train\-test gap\. Overall, these results support theTopoTunercan act as a structural regularizer, preserving most of the useful adaptation while avoiding the stronger overfitting behavior observed in full fine\-tuning\.

Table 18:Overfitting analysis onLLaMA\-3\.1\-8Bacross fine\-tuning methods\. Train\-validation and train\-test gaps are computed from epoch\-6 accuracy values\.Boldindicates the best result among LoRA, Eltwise, and TDA variants\. Smaller gaps indicate better generalization stability\.MethodTrainValTestTrain\-Val GapTrain\-Test GapLLaMA\-3\.1\-8BFull FT71\.6171\.6158\.7458\.7459\.5959\.5912\.8712\.8712\.0212\.02LoRA62\.5062\.5058\.51\\mathbf\{58\.51\}57\.3257\.323\.99\\mathbf\{3\.99\}5\.185\.18Eltwise\-High361\.6261\.6256\.0856\.0858\.07\\mathbf\{58\.07\}5\.545\.543\.553\.55Eltwise\-Low362\.0362\.0355\.6255\.6256\.7956\.796\.416\.415\.245\.24TDA\-High361\.1561\.1556\.5356\.5358\.07\\mathbf\{58\.07\}4\.614\.613\.08\\mathbf\{3\.08\}TDA\-Low362\.53\\mathbf\{62\.53\}57\.1457\.1457\.3257\.325\.395\.395\.215\.21
## Appendix QProofs

Stability motivation\.The usefulness ofΔi\\Delta\_\{i\}rests on the standard stability principle of persistent homology: small perturbations in the underlying metric induce small perturbations in the resulting diagrams\. In our setting, this means that if fine\-tuning changes pairwise distances between rows only slightly, then the corresponding topological drift must also remain small\. The next proposition makes this statement explicit\.

###### Proposition Q\.1\(Stability of topological sensitivity\)\.

LetX=\{x1,…,xd\}X=\\\{x\_\{1\},\\dots,x\_\{d\}\\\}andX~=\{x~1,…,x~d\}\\tilde\{X\}=\\\{\\tilde\{x\}\_\{1\},\\dots,\\tilde\{x\}\_\{d\}\\\}be the row clouds ofWiW\_\{i\}andW~i\\tilde\{W\}\_\{i\}, indexed consistently\. Letddandd~\\tilde\{d\}be the metrics used to construct the Vietoris–Rips filtrations ofXXandX~\\tilde\{X\}\. If‖d−d~‖∞:=maxa,b⁡\|d​\(a,b\)−d~​\(a,b\)\|≤η\\\|d\-\\tilde\{d\}\\\|\_\{\\infty\}:=\\max\_\{a,b\}\|d\(a,b\)\-\\tilde\{d\}\(a,b\)\|\\leq\\eta, then for everyk≥0k\\geq 0we havedB​\(P​Dk​\(X\),P​Dk​\(X~\)\)≤ηd\_\{B\}\(PD\_\{k\}\(X\),PD\_\{k\}\(\\tilde\{X\}\)\)\\leq\\eta, and thereforeWp​\(P​Dk​\(X\),P​Dk​\(X~\)\)≤Nk1/p​ηW\_\{p\}\(PD\_\{k\}\(X\),PD\_\{k\}\(\\tilde\{X\}\)\)\\leq N\_\{k\}^\{1/p\}\\eta, wheredBd\_\{B\}is the bottleneck distance andNkN\_\{k\}is the maximum number of off\-diagonal points in the two diagrams\.

###### Proof\.

For any simplexS⊆\{1,…,d\}S\\subseteq\\\{1,\\dots,d\\\}, letϕ​\(S\)=maxa,b∈S⁡d​\(a,b\)\\phi\(S\)=\\max\_\{a,b\\in S\}d\(a,b\)andϕ~​\(S\)=maxa,b∈S⁡d~​\(a,b\)\\tilde\{\\phi\}\(S\)=\\max\_\{a,b\\in S\}\\tilde\{d\}\(a,b\)\. The assumption‖d−d~‖∞≤η\\\|d\-\\tilde\{d\}\\\|\_\{\\infty\}\\leq\\etaimplies\|ϕ​\(S\)−ϕ~​\(S\)\|≤η\|\\phi\(S\)\-\\tilde\{\\phi\}\(S\)\|\\leq\\etafor every simplexSS\. HenceVRε​\(X\)⊆VRε\+η​\(X~\)\\mathrm\{VR\}\_\{\\varepsilon\}\(X\)\\subseteq\\mathrm\{VR\}\_\{\\varepsilon\+\\eta\}\(\\tilde\{X\}\)andVRε​\(X~\)⊆VRε\+η​\(X\)\\mathrm\{VR\}\_\{\\varepsilon\}\(\\tilde\{X\}\)\\subseteq\\mathrm\{VR\}\_\{\\varepsilon\+\\eta\}\(X\)for allε≥0\\varepsilon\\geq 0, so the two filtrations areη\\eta\-interleaved\. By the stability theorem for persistence diagrams under interleavings\[[4](https://arxiv.org/html/2607.16637#bib.bib19)\], it follows thatdB​\(P​Dk​\(X\),P​Dk​\(X~\)\)≤ηd\_\{B\}\(PD\_\{k\}\(X\),PD\_\{k\}\(\\tilde\{X\}\)\)\\leq\\eta\. Finally, a bottleneck matching with costdBd\_\{B\}moves at mostNkN\_\{k\}off\-diagonal points by at mostdBd\_\{B\}in∥⋅∥∞\\\|\\cdot\\\|\_\{\\infty\}, so the totalpp\-cost is at mostNk​dBpN\_\{k\}d\_\{B\}^\{p\}\. Taking theppth root yieldsWp​\(P​Dk​\(X\),P​Dk​\(X~\)\)≤Nk1/p​dB≤Nk1/p​ηW\_\{p\}\(PD\_\{k\}\(X\),PD\_\{k\}\(\\tilde\{X\}\)\)\\leq N\_\{k\}^\{1/p\}d\_\{B\}\\leq N\_\{k\}^\{1/p\}\\eta\. ∎

We use this stability result in the next section to justify when a prior\-task topological profile remains stable under transfer to a new task\.

We consider both highest\-drift and lowest\-drift selections\. For a budgetbb, we defineℱpriorTDA​\-​H​\(b\)\\mathcal\{F\}\_\{\\mathrm\{prior\}\}^\{\\mathrm\{TDA\\text\{\-\}H\}\}\(b\)as thebbmatrices with the largest values ofSiS\_\{i\}, andℱpriorTDA​\-​L​\(b\)\\mathcal\{F\}\_\{\\mathrm\{prior\}\}^\{\\mathrm\{TDA\\text\{\-\}L\}\}\(b\)as thebbmatrices with the smallest values\. In the magnitude\-based baseline, we analogously rank matrices usingMiM\_\{i\}\. These selected matrices are frozen from the start of fine\-tuning on the target task𝒯∗\\mathcal\{T\}^\{\\ast\}\. This setting tests whether structural drift observed in one prior dataset can serve as a reusable fine\-tuning profile for selective adaptation to other datasets, and whether topology\-based rankings provide a different freezing signal from simple relative weight changes\.

###### Theorem Q\.2\(Stability of reusable freezing profiles\)\.

Fix a pretrained transformerfθ\(0\)f\_\{\\theta^\{\(0\)\}\}and a monitored matrix setℐ\\mathcal\{I\}\. Let𝒯\\mathcal\{T\}be a prior task and let𝒯∗\\mathcal\{T\}^\{\\ast\}be a target task\. For each task, letθ𝒯\(t\)=\{Wi,𝒯\(t\)\}i=1m\\theta\_\{\\mathcal\{T\}\}^\{\(t\)\}=\\\{W\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\\\}\_\{i=1\}^\{m\}andθ𝒯∗\(t\)=\{Wi,𝒯∗\(t\)\}i=1m\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}=\\\{W\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\\\}\_\{i=1\}^\{m\}denote the fine\-tuned parameters afterttoptimization steps, both initialized atθ\(0\)\\theta^\{\(0\)\}\. LetL𝒯​\(θ\)L\_\{\\mathcal\{T\}\}\(\\theta\)andL𝒯∗​\(θ\)L\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\)be the corresponding population fine\-tuning objectives, and writeG𝒯​\(θ\)=∇θL𝒯​\(θ\)G\_\{\\mathcal\{T\}\}\(\\theta\)=\\nabla\_\{\\theta\}L\_\{\\mathcal\{T\}\}\(\\theta\)andG𝒯∗​\(θ\)=∇θL𝒯∗​\(θ\)G\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\)=\\nabla\_\{\\theta\}L\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\)\. Assume both trajectories remain in a ballB​\(θ\(0\),R\)B\(\\theta^\{\(0\)\},R\), both gradient fields areLL\-Lipschitz on this ball, andsupθ∈B​\(θ\(0\),R\)‖G𝒯​\(θ\)−G𝒯∗​\(θ\)‖2≤ϵ\\sup\_\{\\theta\\in B\(\\theta^\{\(0\)\},R\)\}\\\|G\_\{\\mathcal\{T\}\}\(\\theta\)\-G\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\)\\\|\_\{2\}\\leq\\epsilon\. If both tasks are fine\-tuned with the same gradient descent update and step sizeα\\alpha, then for everyt≤tmaxt\\leq t\_\{\\max\}we have‖θ𝒯\(t\)−θ𝒯∗\(t\)‖2≤Rt\\\|\\theta\_\{\\mathcal\{T\}\}^\{\(t\)\}\-\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\\\|\_\{2\}\\leq R\_\{t\}, whereRt:=α​ϵ​∑r=0t−1\(1\+α​L\)rR\_\{t\}:=\\alpha\\epsilon\\sum\_\{r=0\}^\{t\-1\}\(1\+\\alpha L\)^\{r\}\.

Assume that the row\-cloud metricδi\\delta\_\{i\}is fixed across tasks and stable under row perturbations\. For eachi∈ℐi\\in\\mathcal\{I\}and fixed homological dimensionkk, defineDi,𝒯\(t\):=𝒲​a​s​sp​\(PDk​\(Xi\(0\)\),PDk​\(Xi,𝒯\(t\)\)\)D\_\{i,\\mathcal\{T\}\}^\{\(t\)\}:=\\mathcal\{W\}\{ass\}\_\{p\}\(\\mathrm\{PD\}\_\{k\}\(X\_\{i\}^\{\(0\)\}\),\\mathrm\{PD\}\_\{k\}\(X\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\)\)andDi,𝒯∗\(t\):=𝒲​a​s​sp​\(PDk​\(Xi\(0\)\),PDk​\(Xi,𝒯∗\(t\)\)\)D\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}:=\\mathcal\{W\}\{ass\}\_\{p\}\(\\mathrm\{PD\}\_\{k\}\(X\_\{i\}^\{\(0\)\}\),\\mathrm\{PD\}\_\{k\}\(X\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\)\)\. Then\|Di,𝒯\(t\)−Di,𝒯∗\(t\)\|≤2​Ni,k1/p​Rt\|D\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\-D\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\|\\leq 2N\_\{i,k\}^\{1/p\}R\_\{t\}, whereNi,kN\_\{i,k\}is the maximum number of off\-diagonal points in the two persistence diagrams for matrixiiin dimensionkk\.

For Scenario A, takek=0k=0and define the prior and target ranking scores bySiprior:=Di,𝒯\(tmax\)S\_\{i\}^\{\\mathrm\{prior\}\}:=D\_\{i,\\mathcal\{T\}\}^\{\(t\_\{\\max\}\)\}andSi∗:=Di,𝒯∗\(tmax\)S\_\{i\}^\{\\ast\}:=D\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\_\{\\max\}\)\}\. LetS\(1\)prior≥⋯≥S\(\|ℐ\|\)priorS\_\{\(1\)\}^\{\\mathrm\{prior\}\}\\geq\\cdots\\geq S\_\{\(\|\\mathcal\{I\}\|\)\}^\{\\mathrm\{prior\}\}be the sorted prior scores, and letgH​\(b\):=S\(b\)prior−S\(b\+1\)priorg\_\{H\}\(b\):=S\_\{\(b\)\}^\{\\mathrm\{prior\}\}\-S\_\{\(b\+1\)\}^\{\\mathrm\{prior\}\}be the high\-drift margin for freezing budgetbb\. Letδmax:=2​maxi∈ℐ⁡Ni,01/p​Rtmax\\delta\_\{\\max\}:=2\\max\_\{i\\in\\mathcal\{I\}\}N\_\{i,0\}^\{1/p\}R\_\{t\_\{\\max\}\}\. IfgH​\(b\)\>2​δmaxg\_\{H\}\(b\)\>2\\delta\_\{\\max\}, thenFpriorTDA−H​\(b\)=F∗TDA−H​\(b\)F\_\{\\mathrm\{prior\}\}^\{\\mathrm\{TDA\-H\}\}\(b\)=F\_\{\\ast\}^\{\\mathrm\{TDA\-H\}\}\(b\)\. The same statement holds forFpriorTDA−L​\(b\)F\_\{\\mathrm\{prior\}\}^\{\\mathrm\{TDA\-L\}\}\(b\)after sorting the scores in increasing order and using the corresponding low\-drift margin\.

###### Proof\.

Letet:=‖θ𝒯\(t\)−θ𝒯∗\(t\)‖2e\_\{t\}:=\\\|\\theta\_\{\\mathcal\{T\}\}^\{\(t\)\}\-\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\\\|\_\{2\}\. Since both runs start from the same pretrained model,e0=0e\_\{0\}=0\. The two update equations areθ​𝒯\(t\+1\)=θ𝒯\(t\)−α​G𝒯​\(θ𝒯\(t\)\)\\theta\{\\mathcal\{T\}\}^\{\(t\+1\)\}=\\theta\_\{\\mathcal\{T\}\}^\{\(t\)\}\-\\alpha G\_\{\\mathcal\{T\}\}\(\\theta\_\{\\mathcal\{T\}\}^\{\(t\)\}\)andθ𝒯∗\(t\+1\)=θ𝒯∗\(t\)−α​G𝒯∗​\(θ𝒯∗\(t\)\)\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\+1\)\}=\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\-\\alpha G\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\)\. Thereforeet\+1≤‖θ𝒯\(t\)−θ𝒯∗\(t\)−α​\(G𝒯​\(θ𝒯\(t\)\)−G𝒯​\(θ𝒯∗\(t\)\)\)‖2\+α​‖G​𝒯​\(θ𝒯∗\(t\)\)−G𝒯∗​\(θ𝒯∗\(t\)\)‖2e\_\{t\+1\}\\leq\\\|\\theta\_\{\\mathcal\{T\}\}^\{\(t\)\}\-\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\-\\alpha\(G\_\{\\mathcal\{T\}\}\(\\theta\_\{\\mathcal\{T\}\}^\{\(t\)\}\)\-G\_\{\\mathcal\{T\}\}\(\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\)\)\\\|\_\{2\}\+\\alpha\\\|G\{\\mathcal\{T\}\}\(\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\)\-G\_\{\\mathcal\{T\}^\{\\ast\}\}\(\\theta\_\{\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\)\\\|\_\{2\}\. By Lipschitzness and task\-gradient closeness,e​t\+1≤\(1\+α​L\)​et\+α​ϵe\{t\+1\}\\leq\(1\+\\alpha L\)e\_\{t\}\+\\alpha\\epsilon\. Iterating this recurrence giveset≤α​ϵ​∑r=0t−1\(1\+α​L\)r=Rte\_\{t\}\\leq\\alpha\\epsilon\\sum\_\{r=0\}^\{t\-1\}\(1\+\\alpha L\)^\{r\}=R\_\{t\}\.

For a monitored matrixWiW\_\{i\}, the block difference is bounded by the full parameter difference, so\|Wi,𝒯\(t\)−Wi,𝒯∗\(t\)\|​F≤Rt\|W\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\-W\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\|F\\leq R\_\{t\}\. Hence every row satisfies‖xa,𝒯\(i,t\)−xa,𝒯∗\(i,t\)‖2≤Rt\\\|x\_\{a,\\mathcal\{T\}\}^\{\(i,t\)\}\-x\_\{a,\\mathcal\{T\}^\{\\ast\}\}^\{\(i,t\)\}\\\|\_\{2\}\\leq R\_\{t\}\. Since the chosen row\-cloud metric is fixed across tasks and stable under row perturbations, we have‖δi,𝒯\(t\)−δi,𝒯∗\(t\)‖∞≤2​Rt\\\|\\delta\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\-\\delta\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\\\|\_\{\\infty\}\\leq 2R\_\{t\}\.

Applying Proposition[4\.3](https://arxiv.org/html/2607.16637#S4.Thmtheorem3)withη=2​Rt\\eta=2R\_\{t\}gives𝒲​a​s​sp​\(PDk​\(Xi,𝒯\(t\)\),PDk​\(Xi,𝒯∗\(t\)\)\)≤2​Ni,k1/p​Rt\\mathcal\{W\}\{ass\}\_\{p\}\(\\mathrm\{PD\}\_\{k\}\(X\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\),\\mathrm\{PD\}\_\{k\}\(X\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\)\)\\leq 2N\_\{i,k\}^\{1/p\}R\_\{t\}\. Since both distancesDi,𝒯\(t\)D\_\{i,\\mathcal\{T\}\}^\{\(t\)\}andDi,𝒯∗\(t\)D\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}are measured from the same pretrained diagramPDk​\(Xi\(0\)\)\\mathrm\{PD\}\_\{k\}\(X\_\{i\}^\{\(0\)\}\), the triangle inequality for𝒲​a​s​sp\\mathcal\{W\}\{ass\}\_\{p\}gives\|Di,𝒯\(t\)−Di,𝒯∗\(t\)\|≤𝒲​a​s​sp​\(PDk​\(Xi,𝒯\(t\)\),PDk​\(Xi,𝒯∗\(t\)\)\)≤2​Ni,k1/p​Rt\|D\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\-D\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\|\\leq\\mathcal\{W\}\{ass\}\_\{p\}\(\\mathrm\{PD\}\_\{k\}\(X\_\{i,\\mathcal\{T\}\}^\{\(t\)\}\),\\mathrm\{PD\}\_\{k\}\(X\_\{i,\\mathcal\{T\}^\{\\ast\}\}^\{\(t\)\}\)\)\\leq 2N\_\{i,k\}^\{1/p\}R\_\{t\}\.

Setδmax:=2​maxi∈ℐ⁡Ni,01/p​Rtmax\\delta\_\{\\max\}:=2\\max\_\{i\\in\\mathcal\{I\}\}N\_\{i,0\}^\{1/p\}R\_\{t\_\{\\max\}\}\. Then every Scenario A ranking score changes by at mostδmax\\delta\_\{\\max\}between the prior task and the target task\. If the high\-drift gapgH​\(b\)g\_\{H\}\(b\)is larger than2​δmax2\\delta\_\{\\max\}, no matrix outside the prior top\-bbset can pass a matrix inside that set after moving all scores by at mostδmax\\delta\_\{\\max\}\. HenceFpriorTDA−H​\(b\)=F∗TDA−H​\(b\)F\_\{\\mathrm\{prior\}\}^\{\\mathrm\{TDA\-H\}\}\(b\)=F\_\{\\ast\}^\{\\mathrm\{TDA\-H\}\}\(b\)\. The low\-drift claim is identical after sorting in increasing order\. ∎

Theorem[5\.1](https://arxiv.org/html/2607.16637#S5.Thmtheorem1)gives a sufficient condition for reusable freezing profiles while avoiding the claim that all datasets produce identical fine\-tuning dynamics\. It states that, for a fixed pretrained modelfθ\(0\)f\_\{\\theta^\{\(0\)\}\}, the profile learned on a prior task remains valid on a target task when the two tasks induce close gradient fields in the local fine\-tuning region and the prior ranking has a nonzero margin\. The fixed model matters because the local Jacobians, attention geometry, and row\-cloud geometry are inherited fromθ\(0\)\\theta^\{\(0\)\}\. The dataset changes the coefficients of the update, but under the gradient\-field condition it cannot move the monitored matrices far enough to change their topological ranking\. Proposition[4\.3](https://arxiv.org/html/2607.16637#S4.Thmtheorem3)then converts this parameter\-level closeness into persistence\-diagram closeness, and the margin condition converts persistence\-diagram closeness into equality of the selected freezing profile\.

Connection to existing evidence\.\.This theorem is consistent with prior evidence that fine\-tuning remains constrained near a pretrained initialization\. Intrinsic\-dimension results show that language\-model adaptation can often be described by a low\-dimensional update subspace\[[1](https://arxiv.org/html/2607.16637#bib.bib26)\]\. LoRA shows that strong adaptation can be achieved through low\-rank updates to pretrained weight matrices\[[13](https://arxiv.org/html/2607.16637#bib.bib32)\], while BitFit shows that changing only bias parameters can be competitive in several language\-understanding settings\[[41](https://arxiv.org/html/2607.16637#bib.bib23)\]\. These results support the premise that downstream adaptation does not use the full parameter space uniformly\. Task arithmetic and model soups give a more direct precedent for reusing fine\-tuning changes from a shared initialization: task vectors behave as reusable weight\-space objects, and independently fine\-tuned models from the same initialization can often be averaged successfully\[[16](https://arxiv.org/html/2607.16637#bib.bib25),[36](https://arxiv.org/html/2607.16637#bib.bib24)\]\. The topological part of our argument relies on the standard stability theorem for persistence diagrams\[[4](https://arxiv.org/html/2607.16637#bib.bib19)\], and previous neural\-network studies have already used persistent homology to compare trained models and layer representations\[[24](https://arxiv.org/html/2607.16637#bib.bib40),[25](https://arxiv.org/html/2607.16637#bib.bib41)\]\. In this setting,TopoTuneradds a matrix\-level conclusion: if two tasks remain in the same local adaptation regime offθ\(0\)f\_\{\\theta^\{\(0\)\}\}, then their Wasserstein drift scores remain close, and the same high\-drift or low\-drift freezing profile can be reused\.

## NeurIPS Paper Checklist

1. 1\.Claims
2. Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
3. Answer:\[Yes\]
4. Justification: The abstract and introduction state the method, scope, baselines, and empirical outcomes, and these align with the experimental and theoretical results in Sections[4](https://arxiv.org/html/2607.16637#S4)\-[6](https://arxiv.org/html/2607.16637#S6)\.
5. 2\.Limitations
6. Question: Does the paper discuss the limitations of the work performed by the authors?
7. Answer:\[Yes\]
8. Justification: We provide the limitations in Section[6\.3](https://arxiv.org/html/2607.16637#S6.SS3)
9. 3\.Theory assumptions and proofs
10. Question: For each theoretical result, does the paper provide the full set of assumptions and a complete \(and correct\) proof?
11. Answer:\[Yes\]
12. Justification: Theoretical results include clearly stated assumptions and complete proofs in Sections[4](https://arxiv.org/html/2607.16637#S4)\-[5](https://arxiv.org/html/2607.16637#S5), with propositions and theorems properly defined and referenced\.
13. 4\.Experimental result reproducibility
14. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper \(regardless of whether the code and data are provided or not\)?
15. Answer:\[Yes\]
16. Justification: The paper provides datasets, models, training procedure, evaluation metrics, and checkpointing details in Section[6](https://arxiv.org/html/2607.16637#S6)and Appendix[J](https://arxiv.org/html/2607.16637#A10)sufficient to reproduce the main results\.
17. 5\.Open access to data and code
18. Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
19. Answer:\[Yes\]
20. Justification: The implementation is provided through an anonymized repository and experiments use public datasets and open\-source pretrained models\.
21. 6\.Experimental setting/details
22. Question: Does the paper specify all the training and test details \(e\.g\., data splits, hyperparameters, how they were chosen, type of optimizer\) necessary to understand the results?
23. Answer:\[Yes\]
24. Justification: The paper specifies training details, dataset splits, metrics, hyperparameters, and evaluation protocol in Section[6](https://arxiv.org/html/2607.16637#S6)and Appendix[C](https://arxiv.org/html/2607.16637#A3)\.
25. 7\.Experiment statistical significance
26. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
27. Answer:\[Yes\]
28. Justification: Tables[2](https://arxiv.org/html/2607.16637#S6.T2)and[9](https://arxiv.org/html/2607.16637#A10.T9)report accuracy with standard deviations computed from three fixed, non\-overlapping subsets, indicating variability of results\.
29. 8\.Experiments compute resources
30. Question: For each experiment, does the paper provide sufficient information on the computer resources \(type of compute workers, memory, time of execution\) needed to reproduce the experiments?
31. Answer:\[Yes\]
32. Justification: The paper reports GPU type, wall\-clock times, memory, storage, or total compute required across experiments\.
33. 9\.Code of ethics
35. Answer:\[Yes\]
36. Justification: The work uses public datasets and pretrained models, preserves anonymity, and does not involve human subjects or sensitive data\.
37. 10\.Broader impacts
38. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
39. Answer:\[Yes\]
40. Justification: The paper briefly mentions broader impacts and safety and alignment in Appendix[A](https://arxiv.org/html/2607.16637#A1)but does not provide a structured discussion of positive and negative societal impacts\.
41. 11\.Safeguards
42. Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse \(e\.g\., pre\-trained language models, image generators, or scraped datasets\)?
43. Answer:\[N/A\]\.
44. Justification: The work does not release high\-risk models or datasets that require special safeguards\.
45. 12\.Licenses for existing assets
46. Question: Are the creators or original owners of assets \(e\.g\., code, data, models\), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
47. Answer:\[Yes\]
48. Justification: We use publicly available datasets, pretrained models, and libraries \(including GSM8K, MMLU, IMDB, SST\-2, HotpotQA, SQuAD v1\.1, XSum, CNN/DailyMail, Dolly\-15k, Alpaca, HumanEval, MBPP; LLaMA\-3\.1\-8B, Mistral\-7B\-v0\.3, Qwen3\-8B\-Base; PyTorch, Ripser, Gudhi\), and we follow their respective licenses and terms of use as specified in their official releases\.
49. 13\.New assets
50. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
51. Answer:\[Yes\]
52. Justification: The method and code are documented and released via an anonymized repository with sufficient detail for use\.
53. 14\.Crowdsourcing and research with human subjects
54. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
55. Answer:\[N/A\]\.
56. Justification: The paper does not involve human subjects or crowdsourcing\.
57. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
58. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
59. Answer:\[N/A\]\.
60. Justification: The work does not involve human subjects requiring IRB approval\.
61. 16\.Declaration of LLM usage
62. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
63. Answer:\[No\]
64. Justification: The paper does not use LLMs as an important component in its design\.