Diminishing Returns of Data in Sparse Autoencoders: Functional Behavior does not Guarantee Feature Reproducibility

# Diminishing Returns of Data in Sparse Autoencoders: Functional Behavior does not Guarantee Feature Reproducibility
Source: [https://arxiv.org/html/2610.20680](https://arxiv.org/html/2610.20680)
Madelyn MathaiR\. Venkatesh BabuAffiliation:Indian Institute of ScienceChirag AgarwalAffiliation:University of Virginia

###### Abstract

Sparse autoencoders \(SAEs\) are typically trained on hundreds of millions to billions of activation tokens, yet little work has tested whether this scale is necessary\. We study how much activation data SAEs need by training four SAE architectures on three language models with budgets ranging from 5M tokens to a 500M\-token reference\. We measure two notions of convergence: functional behavior on standard evaluations and one\-to\-one reproducibility of learned features\. Functional data requirements are strongly metric\-dependent: reconstruction often approaches the full\-data reference at substantially smaller budgets, whereas feature reproducibility continues to improve with data\. For Gemma\-2\-2B BatchTopK, the fraction of strictly matched features rises from 2% at 5M tokens to 29% at 50M\. Data selection also matters: random sampling better preserves the original activation distribution and generally favors reconstruction, whereas diversity\-based sampling frequently improves AutoInterp\. A theoretical analysis separates statistical consistency of subsampled training from the stronger identifiability and optimization conditions required for one\-to\-one feature recovery\. Our results show that SAE training can often be made far more data\-efficient when budgets are matched to the goal, and feature\-level claims are backed by explicit reproducibility checks, paving the way toward cheaper and more reliable SAE research\. The code can be found on[here](https://github.com/AikyamLab/data-sae)\.

## 1Introduction

Sparse autoencoders \(SAEs\) have become a widely used tool for mechanistic interpretability, aiming to decompose dense model activations into sparse, interpretable features\[[1](https://arxiv.org/html/2610.20680#bib.bib3),[2](https://arxiv.org/html/2610.20680#bib.bib1),[3](https://arxiv.org/html/2610.20680#bib.bib7)\]\. A central motivation is*polysemanticity*: individual neurons can respond to multiple unrelated concepts, making neuron\-level interpretation difficult\[[4](https://arxiv.org/html/2610.20680#bib.bib8),[5](https://arxiv.org/html/2610.20680#bib.bib9)\]\. SAEs address this by learning an encoder that maps activations to sparse latent features and a decoder that reconstructs original activations\[[6](https://arxiv.org/html/2610.20680#bib.bib10)\]\. Recent work has scaled SAEs to frontier models such as GPT\-4\[[7](https://arxiv.org/html/2610.20680#bib.bib5)\]and Claude 3 Sonnet\[[8](https://arxiv.org/html/2610.20680#bib.bib4)\], motivating training on increasinglylarge collections of activations\.

This raises a basic question:*how much activation data does an SAE actually need?*We study thisunexploredquestion in an online training setting by varying activation exposure and measuring two notions of convergence:functional behaviorasks how reconstruction and downstream SAE metrics change with activation budget, whilefeature convergenceasks when the learned features themselves become reproducible under one\-to\-one matching\. To clarify the distinction, we develop a theoretical framework of when additional data will lead to feature recovery and show that reducing the amount of activation data alone is not sufficient to guarantee dictionary convergence\. This distinction matters because a central motivation for SAEs is not merely good reconstruction but reliable feature discovery\. Prior work has questioned whether independently trained SAEs recover the same latent structure\[[9](https://arxiv.org/html/2610.20680#bib.bib22),[10](https://arxiv.org/html/2610.20680#bib.bib24),[11](https://arxiv.org/html/2610.20680#bib.bib23)\], with[Paulo and Belrose \[9\]](https://arxiv.org/html/2610.20680#bib.bib22)using Hungarian matching\[[12](https://arxiv.org/html/2610.20680#bib.bib25)\]and encoder\-decoder agreement to quantify feature correspondence across dictionaries\.

Our work suggests that large activation budgets are not universally required for high\-quality SAE behavior\. Our main contributions are: 1\) We derive a theoretical characterization of feature recovery under data reduction, and separate statistical consistency from the additional requirements of feature identifiability and successful optimization; 2\) across three language models, four SAE architectures, multiple configurations, and activation budgets we present a systematic study of online SAE data requirements; 3\) we demonstrate a distinction between functional behavior and feature convergence, where functional SAE metrics exhibit distinct data\-scaling behavior, while one\-to\-one feature reproducibility continues to improve substantially with additional activation exposure; and 4\) we show that data selection changes SAE quality in metric\-dependent ways: random sampling better preserves reconstruction and intervention\-level metrics, while diversity\-based sampling shifts training toward high\-norm activations and frequently improves Autointerp\[[13](https://arxiv.org/html/2610.20680#bib.bib17)\]\.

To the best of our knowledge, our findings provide thefirst systematic evidencethat SAE training can besubstantially moredata\-efficient than current large\-scale practice\. For reconstruction and several functional evaluations, small activation budgets \(5M\-token\) recover much of the behavior of a 500M\-token SAE\. In contrast, reproducible feature recovery becomes increasingly reliable with larger activation budgets, highlighting SAE’s data requirements depend on the evaluation objective\.

## 2Related Work

Sparse autoencoders and evaluation\.Since the standard SAE\[[1](https://arxiv.org/html/2610.20680#bib.bib3),[2](https://arxiv.org/html/2610.20680#bib.bib1)\], several architectural variants have been proposed, including JumpReLU\[[14](https://arxiv.org/html/2610.20680#bib.bib11)\], TopK and BatchTopK\[[7](https://arxiv.org/html/2610.20680#bib.bib5),[15](https://arxiv.org/html/2610.20680#bib.bib12)\], and Matryoshka SAEs\[[16](https://arxiv.org/html/2610.20680#bib.bib13)\]\. Evaluation has likewise moved beyond reconstruction loss to feature absorption\[[17](https://arxiv.org/html/2610.20680#bib.bib15)\], sparse probing\[[18](https://arxiv.org/html/2610.20680#bib.bib18),[7](https://arxiv.org/html/2610.20680#bib.bib5)\], unlearning\[[19](https://arxiv.org/html/2610.20680#bib.bib21)\], and automated interpretability\[[20](https://arxiv.org/html/2610.20680#bib.bib16),[13](https://arxiv.org/html/2610.20680#bib.bib17)\], which[Karvonen et al\. \[21\]](https://arxiv.org/html/2610.20680#bib.bib14)compiled into SAEBench, now the standard benchmark for SAE evaluation\.

Feature stability\.A separate line of work asks whether the features SAEs recover are stable\.[Paulo and Belrose \[9\]](https://arxiv.org/html/2610.20680#bib.bib22)find that SAEs trained on the same data with different seeds learn substantially different features, casting doubt on whether any single SAE identifies a canonical set\. Yet[Lan et al\. \[11\]](https://arxiv.org/html/2610.20680#bib.bib23)show that the spaces these features span remain similar, even across language models\.[Gerasimov et al\. \[10\]](https://arxiv.org/html/2610.20680#bib.bib24)reconcile these findings: Seed\-unstable features carry little reconstruction or predictive signal and cluster in reproducible low\-rank subspaces, suggesting basis ambiguity rather than noise\. We complement this work by asking how feature reproducibility depends on the amount of activation data\.

Data selection\.Outside interpretability, data selection is a well\-studied route to efficiency: coreset methods seek a small subset that preserves the training signal of the full dataset\[[22](https://arxiv.org/html/2610.20680#bib.bib26)\], and[Coleman et al\. \[23\]](https://arxiv.org/html/2610.20680#bib.bib27)show that cheap proxy models can select such subsets an order of magnitude faster without hurting final accuracy\. A common diversity\-based strategy is farthest point sampling, which greedily adds the point farthest from the current selection\[[24](https://arxiv.org/html/2610.20680#bib.bib29),[25](https://arxiv.org/html/2610.20680#bib.bib30)\]and has been applied to core\-set selection for deep networks\[[26](https://arxiv.org/html/2610.20680#bib.bib31)\]\. Closest to our setting,[Yang et al\. \[27\]](https://arxiv.org/html/2610.20680#bib.bib28)use SAE features to prune instruction\-tuning data, but they select the language model’s training data rather than the SAE’s\. We instead ask whether the choice of activations matters beyond their quantity\.

Research gap:Standard practice trains SAEs on hundreds of millions of tokens\[[21](https://arxiv.org/html/2610.20680#bib.bib14),[28](https://arxiv.org/html/2610.20680#bib.bib6)\]\. To our knowledge, no prior work systematically tests whether this scale is necessary, or whether a selected subset of activations can match SAEs trained on far larger activation budgets on established metrics\.

## 3When Should Feature Matching Agree?

Letθ^1,N\\widehat\{\\theta\}\_\{1,N\}denote an SAE trained on the full activation sample andθ^q,N\\widehat\{\\theta\}\_\{q,N\}denote an SAE trained on a subsample of value\-independent random activations with fractionq∈\(0,1\)q\\in\(0,1\)\. Both runs use the same SAE training objective\. LetR⁡\(θ\)=𝔼X∼PX​\[ℓθ​\(X\)\]R\(\\theta\)=\\mathbb\{E\}\_\{X\\sim P\_\{X\}\}\[\\ell\_\{\\theta\}\(X\)\]denote the corresponding population objective, whereℓθ​\(X\)\\ell\_\{\\theta\}\(X\)represents the training loss the SAE computes on a single activation X, using parametersθ\\theta, andR^1,N​\(θ\)=1N​∑t=1Nℓθ​\(Xt\)\\widehat\{R\}\_\{1,N\}\(\\theta\)=\\frac\{1\}\{N\}\\sum\_\{t=1\}^\{N\}\\ell\_\{\\theta\}\(X\_\{t\}\)denote full sample empirical risk\. We define the indices retained by value\-independent random sampling usingSq,N⊆\{1,…,N\}S\_\{q,N\}\\subseteq\\\{1,\\ldots,N\\\}and the subsample empirical risk usingR^q,N​\(θ\)=1\|Sq,N\|​∑t∈Sq,Nℓθ​\(Xt\)\.\\widehat\{R\}\_\{q,N\}\(\\theta\)=\\frac\{1\}\{\|S\_\{q,N\}\|\}\\sum\_\{t\\in S\_\{q,N\}\}\\ell\_\{\\theta\}\(X\_\{t\}\)\.Finally,E⁡\(θ\)=\[e1​\(θ\),…,ep​\(θ\)\]E\(\\theta\)=\[e\_\{1\}\(\\theta\),\\ldots,e\_\{p\}\(\\theta\)\]andD⁡\(θ\)=\[d1​\(θ\),…,dp​\(θ\)\]D\(\\theta\)=\[d\_\{1\}\(\\theta\),\\ldots,d\_\{p\}\(\\theta\)\]denote the encoder\- and decoder\-side feature system, respectively\. We assume that features are normalized before cosine\-based comparison\. Let𝒮p\\mathcal\{S\}\_\{p\}denote the set ofp×pp\\times ppermutation matrices, so that forP∈𝒮pP\\in\\mathcal\{S\}\_\{p\},E¯​\(θ′\)​P\\bar\{E\}\(\\theta^\{\\prime\}\)PisE¯​\(θ′\)\\bar\{E\}\(\\theta^\{\\prime\}\)with its columns reordered\. For two SAE parameter settingsθ\\thetaandθ′\\theta^\{\\prime\}, define the encoder and decoder one\-to\-one distances as

dE​\(θ,θ′\)\\displaystyle d\_\{E\}\(\\theta,\\theta^\{\\prime\}\)=minP∈𝒮p⁡‖E¯​\(θ\)−E¯​\(θ′\)​P‖F,\\displaystyle=\\min\_\{P\\in\\mathcal\{S\}\_\{p\}\}\\left\\\|\\bar\{E\}\(\\theta\)\-\\bar\{E\}\(\\theta^\{\\prime\}\)P\\right\\\|\_\{F\},\(1\)dD​\(θ,θ′\)\\displaystyle d\_\{D\}\(\\theta,\\theta^\{\\prime\}\)=minP∈𝒮p⁡‖D¯​\(θ\)−D¯​\(θ′\)​P‖F,\\displaystyle=\\min\_\{P\\in\\mathcal\{S\}\_\{p\}\}\\left\\\|\\bar\{D\}\(\\theta\)\-\\bar\{D\}\(\\theta^\{\\prime\}\)P\\right\\\|\_\{F\},\(2\)where∥⋅∥F\\\|\\cdot\\\|\_\{F\}is the Frobenius norm, which is exactly zero when the two systems have the same feature directions\.

Assumptions\.We require four conditions:\(i\) Common population objective:full\- and reduced\-data runs use the same architecture, preprocessing, sparsity mechanism, loss weights, and population weighting;\(ii\) Statistical consistency:both empirical objectives converge uniformly to the same population risk;\(iii\) Optimization consistency:training approaches the empirical optimum as the number of observed activations increases; and\(iv\) Feature identifiability:all population minimizers recover the same encoder and decoder feature systems up to a common latent permutation, with positive separation from alternative feature systems\.

###### Theorem 1\(Asymptotic encoder and decoder feature alignment\)\.

Under the assumptions above, for every fixedq\>0q\>0,

dE​\(θ^q,N,θ^1,N\)\\displaystyle d\_\{E\}\(\\widehat\{\\theta\}\_\{q,N\},\\widehat\{\\theta\}\_\{1,N\}\)→𝑝0,\\displaystyle\\xrightarrow\{p\}0,\(3\)dD​\(θ^q,N,θ^1,N\)\\displaystyle d\_\{D\}\(\\widehat\{\\theta\}\_\{q,N\},\\widehat\{\\theta\}\_\{1,N\}\)→𝑝0\.\\displaystyle\\xrightarrow\{p\}0\.\(4\)Equivalently,

minPE∈𝒮p⁡‖E¯q,N−E¯1,N​PE‖F\\displaystyle\\min\_\{P\_\{E\}\\in\\mathcal\{S\}\_\{p\}\}\\left\\\|\\bar\{E\}\_\{q,N\}\-\\bar\{E\}\_\{1,N\}P\_\{E\}\\right\\\|\_\{F\}→𝑝0,\\displaystyle\\xrightarrow\{p\}0,\(5\)minPD∈𝒮p⁡‖D¯q,N−D¯1,N​PD‖F\\displaystyle\\min\_\{P\_\{D\}\\in\\mathcal\{S\}\_\{p\}\}\\left\\\|\\bar\{D\}\_\{q,N\}\-\\bar\{D\}\_\{1,N\}P\_\{D\}\\right\\\|\_\{F\}→𝑝0\.\\displaystyle\\xrightarrow\{p\}0\.\(6\)

Proof sketch\.Under assumptions \(i\)–\(ii\), the full\- and reduced\-data empirical risks converge uniformly toRR\. By \(iii\), both trained SAEs approach the minimizers ofRR, and by \(iv\) those minimizers share a single feature system up to permutation\. The result follows by the triangle inequality\.

###### Corollary 1\(Agreement of encoder and decoder matchings\)\.

LetPE\(N\)P\_\{E\}^\{\(N\)\}andPD\(N\)P\_\{D\}^\{\(N\)\}denote the permutations that maximize total cosine similarity between the two SAEs’ encoder and decoder directions, respectively:

PE\(N\)\\displaystyle P\_\{E\}^\{\(N\)\}=arg⁡max⁡∑j=1pP∈𝒮p⁡cos⁡\(eq,N,j,e1,N,P⁡\(j\)\),\\displaystyle=\\arg\\max\_\{P\\in\\mathcal\{S\}\_\{p\}\}\\ \\sum\_\{j=1\}^\{p\}\\cos\\\!\\big\(e\_\{q,N,j\},\\,e\_\{1,N,P\(j\)\}\\big\),\(7\)PD\(N\)\\displaystyle P\_\{D\}^\{\(N\)\}=arg⁡max⁡∑j=1pP∈𝒮p⁡cos⁡\(dq,N,j,d1,N,P⁡\(j\)\)\.\\displaystyle=\\arg\\max\_\{P\\in\\mathcal\{S\}\_\{p\}\}\\ \\sum\_\{j=1\}^\{p\}\\cos\\\!\\big\(d\_\{q,N,j\},\\,d\_\{1,N,P\(j\)\}\\big\)\.\(8\)Suppose that at the population optimum the encoder and decoder directions of each feature share the same index, and that the optimal one\-to\-one matching is unique for both encoder and decoder cosine sums, exceeding every other permutation by a fixed marginδ\>0\\delta\>0\. Then, under the assumptions of Theorem[1](https://arxiv.org/html/2610.20680#Thmtheorem1),

Pr⁡\(PE\(N\)=PD\(N\)\)⟶1as​N→∞\.\\Pr\\\!\\big\(P\_\{E\}^\{\(N\)\}=P\_\{D\}^\{\(N\)\}\\big\)\\longrightarrow 1\\qquad\\text\{as \}N\\to\\infty\.

###### Corollary 2\(Assignment agreement is weaker than strong feature recovery\)\.

Encoder–decoder assignment agreement does not imply high encoder and decoder cosine similarity at a finite data budget\. In particular,

PE\(N\)​\(j\)=PD\(N\)​\(j\)=kP\_\{E\}^\{\(N\)\}\(j\)=P\_\{D\}^\{\(N\)\}\(j\)=konly states that featurekkis independently selected as the optimal one\-to\-one counterpart under both encoder and decoder matching\. It does not implycos⁡\(eq,N,j,e1,N,k\)\>τ\\cos\\left\(e\_\{q,N,j\},e\_\{1,N,k\}\\right\)\>\\tauorcos⁡\(dq,N,j,d1,N,k\)\>τ\\cos\\left\(d\_\{q,N,j\},d\_\{1,N,k\}\\right\)\>\\taufor a fixed thresholdτ\\tau\. Thus, substantial agreement between encoder\- and decoder\-based Hungarian assignments can coexist with a much smaller fraction of strongly aligned features\.

Interpretation\.Theorem[1](https://arxiv.org/html/2610.20680#Thmtheorem1)characterizes conditions under which reducing activation data preserves the asymptotic feature system\. It also clarifies that, at finite budgets, assignment agreement is weaker than strong directional recovery: encoder and decoder matchings may select the same reference counterpart even when their cosine similarities remain moderate\. This distinction motivates our empirical comparison of counterpart agreement and strict shared\-feature recovery in Sec\.[6](https://arxiv.org/html/2610.20680#S6)\.

## 4Experimental Setup

We study SAE data requirements along two axes:*activation quantity*and*activation selection*\. Unless stated otherwise, we follow the corresponding SAEBench\[[21](https://arxiv.org/html/2610.20680#bib.bib14)\]training and evaluation configurations so that small\-data runs are directly comparable to the large\-data reference\.

Models and SAE Configurations\.We evaluate three large language models \(LLMs\) \- Gemma\-2\-2B\[[29](https://arxiv.org/html/2610.20680#bib.bib33)\]\(layer 12\), Pythia\-160M\[[30](https://arxiv.org/html/2610.20680#bib.bib32)\]\(layer 6\), and Qwen3\-8B\[[31](https://arxiv.org/html/2610.20680#bib.bib34)\]\(layer 17\), and four SAE architectures: BatchTopK\[[15](https://arxiv.org/html/2610.20680#bib.bib12)\], Matryoshka BatchTopK\[[16](https://arxiv.org/html/2610.20680#bib.bib13)\], JumpReLU\[[14](https://arxiv.org/html/2610.20680#bib.bib11)\], and ReLU\[[1](https://arxiv.org/html/2610.20680#bib.bib3),[2](https://arxiv.org/html/2610.20680#bib.bib1)\], across multiple configurations\. Within each comparison, we hold the LLM, activation site, dictionary width, sparsity setting, and optimization hyperparameters fixed, varying only the activation budget or selection strategy\.

Online SAE Training\.SAEs are trained on FineWeb\[[32](https://arxiv.org/html/2610.20680#bib.bib35)\]in an*online*activation\-generation setting\[[33](https://arxiv.org/html/2610.20680#bib.bib2)\]\. Text is streamed through a frozen LLM, target\-layer activations are buffered, and SAE mini\-batches are drawn from the selected activations before the buffer is refreshed with new text\. Selected activations are used once, avoiding a fixed precomputed dataset\. Across budgets, we fix the500500M training configuration and optimization steps\[[33](https://arxiv.org/html/2610.20680#bib.bib2)\], varying only the retained activation fraction to isolate data scale from training exposure\.

Training\-Data Budgets\.The reference configuration is designed for500500M activation tokens using\[[33](https://arxiv.org/html/2610.20680#bib.bib2)\]framework\. For reduced\-data experiments, we retain a fractionq∈\{0\.01,0\.05,0\.10,1\.0\}q\\in\\\{0\.01,0\.05,0\.10,1\.0\\\}of the activations generated at each buffer refresh\. Thus, each fresh activation pool contributes1%1\\%,5%5\\%,10%10\\%, or100%100\\%of its activations to SAE training\. The selected activations are consumed once before a new activation pool is generated from the online data stream\. Therefore,qqcontrols the amount of fresh activation data retained from each online buffer, rather than the size of a fixed replay dataset\.

Data sampling\.We reduce the activation set using two sampling strategies: random sampling and farthest\-point sampling \(FPS\), which explicitly promotes geometric diversity\.

Evaluation\.We evaluate all SAEs using SAEBench\[[21](https://arxiv.org/html/2610.20680#bib.bib14)\]default configurations and metrics, including reconstruction, sparse probing\[[18](https://arxiv.org/html/2610.20680#bib.bib18),[7](https://arxiv.org/html/2610.20680#bib.bib5)\], absorption\[[17](https://arxiv.org/html/2610.20680#bib.bib15)\], SCR\[[34](https://arxiv.org/html/2610.20680#bib.bib19)\], TPP\[[21](https://arxiv.org/html/2610.20680#bib.bib14)\], RAVEL\[[35](https://arxiv.org/html/2610.20680#bib.bib20)\], automated interpretability\[[20](https://arxiv.org/html/2610.20680#bib.bib16),[13](https://arxiv.org/html/2610.20680#bib.bib17)\], and unlearning\[[19](https://arxiv.org/html/2610.20680#bib.bib21)\]\. We measure feature reproducibility against the500500M reference using one\-to\-one Hungarian matching of encoder and decoder features\[[9](https://arxiv.org/html/2610.20680#bib.bib22),[12](https://arxiv.org/html/2610.20680#bib.bib25)\]\. Matching similarity and encoder\-decoder assignment agreement distinguish*feature convergence*from*functional convergence*

## 5Functional Data Requirements Are Metric Dependent

Figure 1:Functional data scaling for a BatchTopK SAE with1616k latents\.SAEBench metrics are reported for SAEs trained with \{5M, 25M, 50M, 500M\} activation tokens\. While smaller activation budgets approaches the 500M reference for reconstruction, probing, AutoInterp, and intervention\-metrics \(e, h\) exhibit heterogeneous, non\-monotonic dependence on activation exposure\.Figure 2:Functional data scaling for a standard ReLU SAE with6565k latents\.SAEBench metrics are reported for SAEs trained with \{5M, 25M, 50M, 500M\} activation tokens\. Reconstruction remains close to the500500M reference at smaller activation budgets, whereas feature\-level metrics show metric\-dependent and frequently non\-monotonic responses to increasing activation exposure\.Here, we show how much activation data is required to reproduce the functional behavior of a large\-data SAE\. We compare SAEs trained on randomly sampled activation budgets of55M,2525M, and5050M tokens against corresponding500500M\-token references\. We focus on two representative configurations: a BatchTopK SAE with1616k latents and a standard ReLU SAE with6565k latents trained on layer 12 residual\-stream activations of Gemma\-2\-2B\. Figs\.[1](https://arxiv.org/html/2610.20680#S5.F1)\-[2](https://arxiv.org/html/2610.20680#S5.F2)report the corresponding SAEBench metrics across activation budgets\. Results for additional SAE architectures, dictionary widths, and sparsities are provided in Appendix[A](https://arxiv.org/html/2610.20680#A1)\.

a\. Reconstruction\.The loss measures how well the SAE reconstruction preserves the model’s next\-token prediction behavior relative to the original model and a zero\-ablation baseline\.

Observation:The information required for reconstruction is learned early\.In Figs\.[1](https://arxiv.org/html/2610.20680#S5.F1)\-[2](https://arxiv.org/html/2610.20680#S5.F2), we find that all reduced\-data SAEs remain close to the500500M reference\. While the55M setting shows only a modest gap,2525M and5050M SAEs are generally very close to the reference, highlighting that substantially less than500500M activation tokens are sufficient to retain the LLM’s behavior captured by SAE reconstruction\.

b\.kk\-Sparse Probing \(SP\)\.It asks whether a pre\-specified concept can be recovered from a small number of SAE latents\. Following SAEBench, we reportk=1k=1, so the metric specifically probes whether concept information is concentrated in an individual latent\.

Observation:Concept localization is largely preserved at smaller data budgets\.Across sparsity settings and dictionary widths, reduced\-data SAEs remain close to the500500M reference\. Sparse\-probing accuracy is broadly consistent across55M\-500500M budgets, with smaller budgets sometimes outperforming the500500M reference, showing that concept localization is well preserved with substantially less data\.

c\. Feature Absorption \(FA\)\.It measures a complementary failure mode: whether a concept that should be represented by its main feature is instead partially captured by other latents\. We report1−1\-mean absorption; higher is better\.

Observation:More data does not systematically reduce concept fragmentation\.FA is more sensitive to data budget than SP, yet reduced\-data SAEs frequently outperform the500500M reference\. Together, SP and FA show that strong concept localization and feature alignment can be achieved with substantially smaller data budgets\.

d\. Automated Interpretability\.AutoInterp asks an LLM to infer a semantic description of a latent from its activating examples and evaluates whether that description predicts activation on held\-out sequences, probing the semantic description of individual features rather than reconstruction quality\.

Observation:Feature description is not monotonically determined by activation quantity\.AutoInterp varies moderately with activation budget, but its ordering depends on SAE configuration, including sparsity and dictionary width\. Reduced\-data SAEs outperform the500500M reference in many regimes and underperform it in others\. In several settings, the trend is even reversed, with smaller activation budgets yielding higher AutoInterp scores\. Thus, increasing activation exposuredoes notconsistently improve the semantic description of individual features\.

e\. Spurious Correlation Removal \(SCR\)\.It evaluates whether ablating a small set of concept\-related latents removes a spurious correlation while preserving the intended classification signal\.

Observation:SCR shows moderate but non\-monotonic sensitivity to data\.SCR varies more with activation budget than reconstruction, but increasing data does not consistently improve intervention specificity, and the performance of reduced\-data SAEs depends on the width and the architecture\.

f\. Targeted Probe Perturbation \(TPP\)\.It asks whether ablating latents associated with one class primarily affects that class while leaving others unchanged\.

Observation:TPP is remarkably insensitive to activation budgetAcross both dictionary sizes, TPP remains close to the500500M reference despite the large difference in activation exposure\. The ordering of data budgets is non\-monotonic, but the absolute deviations are small, suggesting that this measure of class\-specific feature separation requires relatively little additional data\.

g\. RAVEL\.It evaluates whether an intervention successfully changes a targeted attribute while preserving unrelated attributes\.

Observation:RAVEL is non\-monotonic across activation budgets\.It varies with activation budget and architecture, indicating that effective intervention can be achieved without requiring consistently larger activation exposure\.

h\. Unlearning\.It suppresses latents associated with a target domain and measures whether target knowledge can be removed while unrelated performance is preserved\.

Observation:Unlearning is the most configuration\-sensitive evaluationIt varies substantially with activation budget, architecture, and sparsity, highlighting the importance of these factors for unlearning performance\. Reduced\-data SAEs span a wide performance range relative to the500500M reference, highlighting that effective unlearning can emerge even with substantially smaller activation budgets\. To strengthen all the observations above, we additonally add a paired statistical test in Table[26](https://arxiv.org/html/2610.20680#A2.T26)\.

In total, these results raise a natural next question: if all other hyperparameters are held constant, are the observed volatile differences primarily a consequence of using less data, or do they also depend on*which*activations are randomly selected? To separate data quantity from data selection, we next compare random sampling with diversity\-based activation selection at matched budgets\.

Does Data Selection Matter?Here, we ask whether SAE data requirements are determined only by*how much*data is used, or also by*which*activations are selected\. To separate these effects, we compare uniform random sampling \(RS\) with Farthest Point Sampling \(FPS\)\[[24](https://arxiv.org/html/2610.20680#bib.bib29),[25](https://arxiv.org/html/2610.20680#bib.bib30)\]at matched activation budgets\. While RS retains activations independently of their values and therefore preserves the frequency structure of the activation stream in expectation, FPS selects activations to increase geometric coverage, preferentially retaining points that are far from those already selected\. We follow the evaluation configuration of Figs\.[1](https://arxiv.org/html/2610.20680#S5.F1)\-[2](https://arxiv.org/html/2610.20680#S5.F2)and compare both selection strategies\. Results for additional SAE architectures are reported in Appendix[A](https://arxiv.org/html/2610.20680#A1)\.

Figure 3:Random sampling \(RS\) vs\. FPS for a BatchTopK SAE with1616k latents\.At each activation budget, paired bars compare FPS and RS, while the green dashed line denotes the corresponding500500M reference\. Selection strategy induces metric\-dependent tradeoffs: RS better preserves reconstruction and several intervention metrics, whereas FPS often improves AutoInterp\.Fig\.[3](https://arxiv.org/html/2610.20680#S5.F3)shows that the choice of activation subset matters even at a fixed budget: RS and FPS do not produce uniformly better or worse SAEs, but instead favor different properties of the learned representation\.

i\) Random sampling preserves reconstruction\.Across dictionary sizes and budgets, loss recovered under RS stays closer to the500500M reference than under FPS, most visibly at small budgets\. Geometric coverage alone thus preserves the model’s predictive behavior less faithfully than representative sampling\.

ii\) Concept\- and intervention\-level metrics favor RS\.SCR, TPP, and RAVEL also favor RS across budgets, in some cases exceeding the500500M reference, while SP and FA show no consistent winner\.

iii\) FPS yields higher AutoInterp\.FPS outperforms RS on AutoInterp at every budget in both configurations, and in 79% of matched comparisons across all architectures, models, and budgets\. FPS shifts training toward geometrically distinct, high\-norm activations and is associated with higher AutoInterp but lower reconstruction scores\. FPS also outperforms RS on unlearning at some budgets \(5M and 50M in Fig\.[3](https://arxiv.org/html/2610.20680#S5.F3)\), though less consistently than for AutoInterp\.

Data Quality Is Objective Dependent\.The matched\-budget comparison shows that reducing SAE training data is not characterized by a single notion of data quality\. RS better preserves several properties of the full activation stream, most clearly reconstruction, while FPS can improve metrics like AutoInterp and, in some settings of unlearning\. Thus,which activations are selected matters in addition to how many are selected\.A subset that is representative of the original activation distribution and a subset that maximizes geometric diversity optimize*different*notions of SAE quality\.

Why does FPS produce this tradeoff?To characterize what each selector retains, we compared the L2\-norm distribution of residual\-stream activations selected for SAE training by RS vs\. FPS, with all selectors applied to identical activation pools drawn from the 500M\-token stream\. RS reproduces the stream’s distribution within sampling error at every budget, so its effect is purely a reduction in data quantity\. FPS instead concentrates training on a higher\-norm region of activation space, shifting the distribution further right as the subset shrinks from 10% to 1% of 500M reference \(see Table[1](https://arxiv.org/html/2610.20680#S5.T1)\)\.

Table 1:FPS reshapes the training\-data distribution; random selection does not\.\*Random selection is distribution\-neutral at every budget \(KL≤0\.0003\\leq 0\.0003bits\); FPS\-induced divergence grows with subset selectivity\. Results for Qwen3\-8B and Pythia\-160M are in Table[24](https://arxiv.org/html/2610.20680#A2.T24)in the appendix\**Δ\\Deltamean norm*is the relative shift in mean activationL2L\_\{2\}norm;*KL*is the base\-2 divergence between binned norm distributions; and*Mean\-dir\. cosine*measures mean\-direction alignment\. All selectors use identical activation pools\. Values are mean±\\pmSD across pool refreshes \(50 for norm statistics; 15 for direction statistics\); values shown as±0\.000\\pm 0\.000have SD<10−4<10^\{\-4\}\.

## 6Functional Similarity Does Not Imply Feature Reproducibility

So far, increasing activation budget does not uniformly improve SAE function: reconstruction often approaches the500500M reference early, while other metrics remain heterogeneous and non\-monotonic\. We now ask a stricter question:*does functional similarity imply recovery of the same features?*

Following prior work on SAE feature reproducibility\[[9](https://arxiv.org/html/2610.20680#bib.bib22)\], we compare reduced\-data SAEs against the corresponding500500M\-token reference using one\-to\-one Hungarian matching\. Encoder and decoder features are matched independently by cosine similarity\. A latent is*shared*at the thresholdτ\\tauif both matchings select the same reference latent and both cosine similarities exceedτ\\tau\. Unless otherwise stated, we useτ=0\.7\\tau=0\.7as a strict one\-to\-one feature\-recovery criterion\.

Counterpart Agreement Emerges Before Strong Feature Recovery\.Even at small activation budgets, reduced\-data SAEs are not learning completely unrelated dictionaries\. For Gemma\-2\-2B BatchTopK at55M tokens, the mean matched decoder cosine similarity is0\.4540\.454, and independently computed encoder and decoder matchings identify the same reference counterpart for66\.6%66\.6\\%of latents\. However, only2\.20%2\.20\\%of latents satisfy the stricter shared\-feature criterion atτ=0\.7\\tau=0\.7\.

This gap distinguishes two different notions of correspondence\.*Counterpart agreement*asks whether the encoder and decoder identify the same reference feature, whereas*strong feature recovery*additionally requires the corresponding feature directions themselves to be highly aligned\. Reduced\-data SAEs can therefore recover recognizable counterparts well before those counterparts become strongly reproducible\.

Feature Reproducibility Improves Systematically with Data\.Table[2](https://arxiv.org/html/2610.20680#S6.T2)shows that feature recovery strengthens consistently with activation budget\. For Gemma\-2\-2B BatchTopK, encoder\-decoder agreement rises from66\.61%66\.61\\%at55M to83\.47%83\.47\\%at5050M, while strict shared\-feature recovery atτ=0\.7\\tau=0\.7increases from2\.20%2\.20\\%to28\.86%28\.86\\%\. The same trend holds for ReLU SAEs, showing that counterpart agreement emerges earlier than strong one\-to\-one feature recovery\. Additional results are in Table[25](https://arxiv.org/html/2610.20680#A2.T25)in the appendix\.

Table 2:Counterpart agreement and strict feature correspondence\.*Enc\.\-dec\. agree*is the fraction of features for which independently computed encoder and decoder Hungarian matchings select the same500500M reference latent\.*Shared*additionally requires both encoder and decoder cosine similarities to exceedτ=0\.7\\tau=0\.7\. We find a similar observation for both Qwen3\-8b and Pythia\-160m, shown in Table[25](https://arxiv.org/html/2610.20680#A2.T25)in the appendix\.Qualitative Examples Illustrate the Same Distinction\.Table[3](https://arxiv.org/html/2610.20680#S6.T3)presents representative Gemma\-2\-2B ReLU examples spanning low, medium, and high mean encoder\-decoder alignment\. For each 5M\-500M SAE pair, Hungarian matching aligns features using encoder\-decoder cosine similarity, while Gemma\-based AutoInterp provides balanced\-accuracy scores\. We retain pairs for which the encoder\- and decoder\-based matches agree, and both AutoInterp scores are positive and select one representative pair from each of the alignment bins for qualitative analysis\. At low and medium alignment, matched features can have substantially different semantic descriptions despite both matching criteria identifying them as corresponding\. In contrast, the highly aligned example captures essentially the same semantic pattern across the two SAEs:Comparative adjectives and adverbs, of\- ten followed by than\. This qualitative behavior is consistent with the quantitative results above\. Counterpart identity alone is a weak notion of feature recovery; strongly aligned matched pairs provide a much stricter indication that the same feature has been recovered across data budgets\.

Table 3:Representative matched Gemma\-2\-2B ReLU features between the55M SAE and the500500M reference across increasing mean encoder\-decoder alignment\. The matched latents are \(2782↔27822782\\leftrightarrow 2782\), \(411↔2629411\\leftrightarrow 2629\),\(1507↔15071507\\leftrightarrow 1507\)\.Functional Similarity Does Not Imply Feature Reproducibility\.By2525M–5050M tokens, reconstruction and several functional SAE metrics are already close to their500500M reference values in many regimes, yet feature recovery remains incomplete\. Table[2](https://arxiv.org/html/2610.20680#S6.T2)shows that increasing activation exposure consistently strengthens one\-to\-one feature correspondence across models and SAE architectures, even when functional metrics do not improve monotonically with data\. Thus, functional similarity can emerge at smaller budgets, while feature reproducibility continues to improve with additional activation exposure, making them distinct notions of SAE convergence\.

## 7Conclusion

In this work, we ask a simple but practically important question:*How much activation data does an SAE actually need?*We study this across four SAE architectures and three language models, analyzing data requirements along three axes: theory, functional performance, and feature reproducibility\. First, we develop a theoretical framework characterizing when reduced\-data and full\-data SAEs should recover the same feature system, showing that convergence to the same population objective is not sufficient for one\-to\-one feature recovery; identifiability and optimization conditions are also required\. Second, training SAEs on1%1\\%,5%5\\%,10%10\\%, and100%100\\%of the available online activation data, we find that many SAEBench metrics approach the large\-data reference well before the full budget\. Third, feature\-level analysis reveals a different regime: encoder\- and decoder\-based Hungarian matchings agree for a substantial fraction of features even at small budgets, but stricter one\-to\-one similarity criteria show continued improvement with more data\. Counterpart agreement thus emerges earlier than strong feature reproducibility\.

Together, these results suggest that SAE data requirements are objective\-dependent\. Smaller activation budgets can often preserve reconstruction and aggregate functional behavior, while reproducible recovery of individual features may require substantially more data\. We therefore recommend matching SAE training scale to the intended use: avoid defaulting to the largest available activation corpus for functional evaluation, but explicitly assess feature reproducibility when individual latents are used as scientific objects\.

Limitations\.Our study relies heavily on the training and evaluation implementations provided by SAEBench\. All functional metrics are computed using the corresponding SAEBench evaluation scripts, and our SAE training pipeline follows the same underlying framework\. Consequently, our conclusions inherit any assumptions, implementation choices, and metric\-specific limitations of these tools\. Additionally, the current SAEBench evaluation framework supports RAVEL and unlearning evaluations only for Gemma models\. Consequently, we are currently unable to evaluate the Qwen and Pythia models using these metrics\.

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## Appendix AExperimental details and additional Functional Data\-Scaling Results

We evaluate four SAE architectures: standard ReLU, BatchTopK, JumpReLU, and Matryoshka BatchTopK\. Across all configurations, SAE batch size20482048, dictionary widths16,38416\{,\}384and65,53665\{,\}536,10001000warmup steps, and50005000sparsity\-warmup steps\. Decay begins after80%80\\%of training, TopK\-stylekk\-annealing ends after1%1\\%of the total training steps, BOS tokens are removed, and activations exceeding10×10\\timesthe reference activation norm are filtered\. The default full\-data training budget is500500M activation tokens\. For activation extraction, Pythia\-160M uses LM batch size3232, context length10241024, andfloat32; Gemma\-2\-2B uses batch size44, context length10241024, andbfloat16; and Qwen3\-8B uses batch size1616, context length20482048, andbfloat16\. For Gemma\-2\-2B and Pythia\-160M, we evaluate both 16K and 65K dictionary widths\. For Qwen3\-8B, we use only the 65K width across all SAE architectures\.

The trainer index used in our evaluation tables identifies the architecture\-specific sparsity operating point\. We use theL1L\_\{1\}sparsity penalty over\{0\.012,0\.015,0\.020,0\.030,0\.040,0\.060\}\.\\\{0\.012,\\,0\.015,\\,0\.020,\\,0\.030,\\,0\.040,\\,0\.060\\\}\.At dictionary width1616K, trainers00\-55correspond to these values in order; at width6464K, trainers66\-1111follow the same ordering\.

For BatchTopK, we usek∈\{80,160\}k\\in\\\{80,160\\\}\. Trainers00and11correspond tok=80k=80andk=160k=160at width1616K, while trainers22and33correspond to the same two operating points at width6464K\. The trainer usesauxk\_alpha=1/32\\texttt\{auxk\\\_alpha\}=1/32,threshold\_beta=0\.999\\texttt\{threshold\\\_beta\}=0\.999, and begins threshold tracking after10001000steps\.

For JumpReLU, we use the target activation countL0target∈\{80,160\}L\_\{0\}^\{\\mathrm\{target\}\}\\in\\\{80,160\\\}\. Trainers00and11correspond to targetL0=80L\_\{0\}=80and160160at width1616K, while trainers22and33correspond to the same targets at width6464K\. The sparsity penalty is fixed to1\.01\.0and the JumpReLU bandwidth to0\.0010\.001\.

For Matryoshka BatchTopK, we usek∈\{80,160\}k\\in\\\{80,160\\\}\. Trainers00and11correspond tok=80k=80and160160at width1616K, and trainers22and33correspond to the same values at width6464K\. The nested groups use fractions\{132,116,18,14,1732\},\\left\\\{\\frac\{1\}\{32\},\\,\\frac\{1\}\{16\},\\,\\frac\{1\}\{8\},\\,\\frac\{1\}\{4\},\\,\\frac\{17\}\{32\}\\right\\\},withauxk\_alpha=1/32\\texttt\{auxk\\\_alpha\}=1/32,threshold\_beta=0\.999\\texttt\{threshold\\\_beta\}=0\.999, and threshold tracking beginning after10001000steps\.

Table 4:SAE evaluation across activation budgets and selection strategies for Gemma\-2\-2B, ReLU \(16k\)\. Random and FPS are compared at 5M,25M, and 50M activation budgets against the corresponding 500M baseline\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowUnlearning↑\\uparrowSparsity\(L0\)↓\\downarrow05MRandom0\.9930\.9590\.7030\.2530\.0290\.7560\.6570\.0061400\.2FPS0\.9850\.9740\.7030\.1940\.0210\.7500\.6390\.0091298\.825MRandom0\.9950\.9740\.7040\.2610\.0310\.7870\.6530\.0021353\.2FPS0\.9920\.9830\.7180\.2230\.0220\.8020\.6340\.0041265\.150MRandom0\.9950\.9790\.7150\.2640\.0300\.7970\.6560\.0091346\.5FPS0\.9930\.9550\.7120\.2300\.0240\.7720\.6250\.0041204\.0500MBaseline0\.9950\.9470\.7120\.2680\.0300\.8050\.6410\.0091289\.515MRandom0\.9900\.9500\.6960\.2530\.0250\.7910\.6320\.006997\.1FPS0\.9790\.9710\.7050\.1910\.0160\.7500\.6220\.002907\.325MRandom0\.9920\.9680\.7130\.2750\.0280\.7750\.6220\.024962\.3FPS0\.9880\.9550\.7200\.2160\.0240\.7910\.6130\.034882\.450MRandom0\.9900\.9460\.7310\.2620\.0290\.8050\.6280\.011960\.8FPS0\.9900\.9210\.7360\.2370\.0250\.8050\.6140\.004840\.4500MBaseline0\.9930\.9420\.7340\.2610\.0300\.8080\.6370\.008914\.025MRandom0\.9820\.9630\.7350\.2530\.0240\.8070\.5700\.015600\.4FPS0\.9700\.9540\.7360\.1970\.0120\.7690\.5650\.006534\.225MRandom0\.9880\.9040\.7350\.2630\.0240\.7930\.5960\.002590\.1FPS0\.9800\.8700\.7560\.2120\.0210\.8000\.5610\.006526\.850MRandom0\.9880\.9250\.7450\.2720\.0250\.7990\.5970\.011590\.0FPS0\.9840\.8420\.7470\.2390\.0200\.7960\.5870\.009501\.0500MBaseline0\.9900\.8900\.7490\.2770\.0270\.8050\.5940\.015563\.835MRandom0\.9750\.8760\.7700\.2480\.0180\.7930\.5430\.004269\.5FPS0\.9520\.8840\.7740\.1740\.0100\.7460\.5140\.024230\.725MRandom0\.9820\.8400\.7780\.2590\.0200\.7820\.5480\.006275\.9FPS0\.9690\.7490\.7860\.1880\.0140\.7880\.5490\.009242\.650MRandom0\.9790\.7990\.7660\.2830\.0230\.8030\.5220\.017279\.0FPS0\.9720\.7620\.7940\.2060\.0160\.7910\.5570\.008234\.9500MBaseline0\.9840\.7910\.7850\.2830\.0210\.7960\.5540\.023271\.445MRandom0\.9690\.8270\.8000\.2210\.0150\.7630\.4890\.015157\.7FPS0\.9340\.8150\.8070\.1450\.0080\.7680\.4750\.073133\.425MRandom0\.9750\.7690\.8010\.2380\.0150\.7630\.5010\.032163\.3FPS0\.9570\.7320\.8120\.1760\.0090\.7690\.4880\.023143\.850MRandom0\.9740\.7730\.8070\.2370\.0160\.7720\.5030\.023165\.9FPS0\.9620\.7190\.8110\.1940\.0120\.7850\.4760\.049141\.1500MBaseline0\.9770\.7460\.8020\.2730\.0170\.8000\.5210\.045162\.655MRandom0\.9560\.8350\.8290\.1970\.0090\.7610\.3740\.02683\.4FPS0\.9060\.8440\.8160\.1110\.0070\.7590\.4190\.04171\.125MRandom0\.9590\.7710\.8290\.1900\.0110\.7620\.4340\.03884\.0FPS0\.9390\.7360\.8260\.1410\.0060\.7440\.4570\.14375\.050MRandom0\.9640\.7630\.8260\.1890\.0130\.7640\.4110\.02885\.4FPS0\.9460\.7380\.8440\.1550\.0080\.7310\.4470\.13973\.0500MBaseline0\.9640\.7260\.8490\.2070\.0100\.8060\.4840\.04184\.0

Table 5:Per\-trainer SAE evaluation — Gemma\-2\-2B, ReLU \(64K\)\.Each row is one trainer run\. Missing evaluations shown as “–”\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowUnlearning↑\\uparrowSparsity\(L0\)↓\\downarrow65MRandom0\.9950\.9020\.6480\.1910\.0110\.7750\.6900\.0241700\.5FPS0\.9880\.8940\.6630\.1250\.0080\.7920\.6640\.0131439\.525MRandom0\.9950\.8650\.6690\.1920\.0120\.8110\.6850\.0111932\.7FPS0\.9930\.7580\.6950\.1550\.0090\.8050\.6650\.0091686\.450MRandom0\.9970\.8190\.6730\.1830\.0120\.8130\.6850\.0211987\.0FPS0\.9950\.7290\.7020\.1590\.0090\.8120\.6810\.0511678\.4500MBaseline0\.9970\.7880\.6890\.1840\.0120\.8070\.6830\.0061923\.775MRandom0\.9930\.8810\.6590\.1960\.0120\.7920\.6610\.0191176\.4FPS0\.9820\.8650\.6720\.1460\.0100\.7920\.6460\.017976\.925MRandom0\.9930\.8380\.6860\.1900\.0110\.7960\.6600\.0091340\.8FPS0\.9900\.6950\.6980\.1600\.0090\.8030\.6440\.0081149\.150MRandom0\.9950\.7910\.6920\.1930\.0130\.8120\.6810\.0091381\.5FPS0\.9920\.6730\.7040\.1590\.0100\.7910\.6430\.0451145\.6500MBaseline0\.9950\.7210\.7060\.1800\.0110\.8060\.6420\.0061332\.185MRandom0\.9900\.8380\.7010\.2110\.0130\.7930\.6370\.023685\.9FPS0\.9770\.7640\.7120\.1390\.0070\.7930\.6030\.024553\.825MRandom0\.9920\.7430\.7020\.2010\.0130\.7970\.6330\.041793\.7FPS0\.9850\.6210\.7370\.1570\.0090\.8030\.6190\.019665\.050MRandom0\.9920\.7160\.7280\.1910\.0130\.8130\.6450\.013825\.4FPS0\.9880\.6070\.7300\.1570\.0110\.8030\.6050\.122669\.6500MBaseline0\.9920\.7000\.7230\.1800\.0140\.8120\.6460\.036803\.695MRandom0\.9850\.7070\.7510\.1850\.0130\.7970\.5910\.041304\.1FPS0\.9640\.6680\.7600\.1210\.0060\.7820\.5760\.225239\.025MRandom0\.9880\.6670\.7460\.1890\.0100\.8000\.5970\.028355\.6FPS0\.9770\.5540\.7660\.1380\.0060\.7580\.5820\.118298\.750MRandom0\.9880\.5940\.7550\.1860\.0100\.8150\.6000\.186375\.2FPS0\.9800\.5490\.7820\.1530\.0070\.7620\.6150\.201306\.6500MBaseline0\.9880\.5640\.7660\.1840\.0120\.8190\.5940\.167377\.8105MRandom0\.9790\.6370\.7920\.1850\.0100\.8060\.5730\.135181\.3FPS0\.9570\.5860\.8010\.1290\.0060\.7810\.5580\.086145\.625MRandom0\.9840\.5780\.7920\.1780\.0080\.8090\.5900\.184205\.0FPS0\.9690\.5150\.8100\.1360\.0070\.7670\.5710\.156176\.050MRandom0\.9850\.5470\.8050\.1690\.0090\.8040\.5950\.206216\.3FPS0\.9750\.4940\.7990\.1420\.0060\.7710\.5730\.128179\.1500MBaseline0\.9870\.4970\.7980\.1820\.0100\.8080\.5800\.188219\.9115MRandom0\.9690\.5880\.8300\.1440\.0060\.7920\.5340\.154100\.8FPS0\.9390\.6130\.8410\.0970\.0040\.7550\.5220\.13782\.625MRandom0\.9740\.5100\.8370\.1680\.0060\.7810\.5530\.169103\.4FPS0\.9570\.5020\.8380\.1080\.0050\.7660\.5500\.09289\.950MRandom0\.9750\.5020\.8460\.1560\.0070\.7940\.5670\.137106\.7FPS0\.9620\.5200\.8440\.1100\.0050\.7510\.5470\.08889\.6500MBaseline0\.9790\.4820\.8440\.1460\.0070\.7760\.5730\.030107\.2

Table 6:Per\-trainer SAE evaluation for Gemma\-2\-2B, BatchTopK \(16K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowUnlearning↑\\uparrowSparsity\(L0\)↓\\downarrow05MRandom0\.9790\.9470\.8150\.3520\.0470\.7320\.5550\.04179\.8FPS0\.9510\.9280\.8190\.2430\.0250\.7330\.5170\.06350\.925MRandom0\.9850\.9280\.8330\.3660\.0430\.7870\.5400\.10180\.6FPS0\.9740\.8910\.8450\.2680\.0250\.7240\.5350\.02455\.250MRandom0\.9850\.9500\.8350\.3560\.0500\.7290\.5780\.02479\.7FPS0\.9790\.8440\.8530\.2850\.0320\.7390\.5730\.04557\.6500MBaseline0\.9870\.9070\.8210\.3540\.0390\.7650\.5730\.06580\.215MRandom0\.9850\.9300\.7580\.3450\.0960\.7350\.5870\.032158\.7FPS0\.9640\.9290\.7710\.2790\.0800\.7140\.5540\.028102\.225MRandom0\.9900\.9860\.7790\.3080\.1290\.7570\.6030\.006160\.8FPS0\.9820\.9530\.8150\.2930\.0900\.7650\.5880\.067109\.850MRandom0\.9900\.9810\.7780\.3440\.0870\.7600\.6000\.019160\.0FPS0\.9850\.9470\.8110\.1800\.1650\.7010\.6010\.019115\.7500MBaseline0\.9920\.9750\.8000\.3280\.0870\.7040\.6190\.058159\.9

Table 7:Per\-trainer SAE evaluation for Gemma\-2\-2B, BatchTopK \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowUnlearning↑\\uparrowSparsity\(L0\)↓\\downarrow25MRandom0\.9800\.8990\.7850\.3300\.0520\.7930\.5560\.04879\.6FPS0\.9540\.9000\.8480\.2550\.0220\.7690\.5360\.09553\.425MRandom0\.9870\.8960\.8260\.3270\.0380\.7890\.5890\.05680\.7FPS0\.9770\.6660\.8630\.2310\.0280\.7210\.5850\.10257\.450MRandom0\.9870\.8830\.8380\.3390\.0280\.7390\.5660\.10280\.6FPS0\.9840\.6600\.8570\.2420\.0190\.7380\.5760\.09159\.9500MBaseline0\.9900\.7500\.8310\.2610\.0190\.7310\.5840\.16880\.835MRandom0\.9870\.9050\.7570\.3370\.1040\.7470\.5990\.015160\.1FPS0\.9690\.8860\.7800\.2290\.0840\.7390\.5730\.037104\.925MRandom0\.9900\.9330\.7740\.2940\.1530\.7330\.6330\.037161\.0FPS0\.9850\.8020\.7980\.2360\.0910\.7530\.5980\.082114\.250MRandom0\.9920\.9630\.7810\.2430\.1130\.7170\.6160\.041160\.5FPS0\.9880\.8210\.8110\.2390\.0580\.7010\.5950\.088119\.5500MBaseline0\.9920\.9230\.7920\.2020\.0600\.7210\.6190\.084161\.8

Table 8:Per\-trainer SAE evaluation for Gemma\-2\-2B, JumpReLU \(16K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowUnlearning↑\\uparrowSparsity\(L0\)↓\\downarrow05MRandom0\.9770\.9100\.8250\.3650\.0470\.7360\.5230\.01583\.4FPS0\.9490\.9100\.8280\.2320\.0340\.6950\.5110\.09454\.325MRandom0\.9820\.8950\.8110\.3830\.0430\.7150\.5770\.04383\.0FPS0\.9720\.8420\.8440\.2900\.0290\.6910\.5420\.07957\.350MRandom0\.9850\.9140\.8130\.3740\.0530\.7330\.5730\.03082\.7FPS0\.9790\.8020\.8490\.2770\.0310\.7290\.5670\.02660\.3500MBaseline0\.9870\.8650\.8220\.3250\.0450\.7480\.5720\.09083\.015MRandom0\.9820\.9230\.7690\.3440\.1240\.7660\.5700\.034167\.7FPS0\.9620\.9180\.7710\.2420\.0540\.7430\.5340\.009107\.625MRandom0\.9870\.9520\.7660\.3890\.0990\.7360\.6220\.011166\.9FPS0\.9820\.9490\.7830\.2980\.0970\.7330\.5610\.015114\.550MRandom0\.9900\.9580\.7610\.3420\.0830\.7340\.5900\.019166\.6FPS0\.9850\.9010\.7890\.3050\.0460\.7440\.5790\.069120\.4500MBaseline0\.9920\.9520\.7750\.3250\.1080\.7290\.6040\.019166\.7

Table 9:Per\-trainer SAE evaluation for Gemma\-2\-2B, JumpReLU \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowUnlearning↑\\uparrowSparsity\(L0\)↓\\downarrow25MRandom0\.9770\.8730\.8090\.3690\.0440\.7330\.5610\.06983\.5FPS0\.9560\.7570\.8350\.1960\.0140\.6940\.5390\.07155\.525MRandom0\.9870\.8270\.8350\.3280\.0290\.7650\.5670\.08182\.9FPS0\.9750\.7170\.8670\.2460\.0270\.7250\.5810\.09259\.050MRandom0\.9870\.8340\.8250\.3080\.0390\.7370\.5720\.08482\.8FPS0\.9840\.6630\.8520\.2640\.0200\.7410\.5820\.22162\.0500MBaseline0\.9900\.7290\.8130\.2550\.0220\.7180\.5820\.13383\.035MRandom0\.9850\.8950\.7650\.3790\.0540\.7340\.5730\.043167\.5FPS0\.9650\.8980\.7640\.2560\.0470\.7030\.5810\.056110\.825MRandom0\.9900\.9190\.7640\.3270\.0530\.7840\.6170\.047166\.8FPS0\.9850\.7870\.7800\.2980\.0350\.7060\.6050\.109117\.250MRandom0\.9920\.9370\.7760\.3430\.0960\.7280\.6090\.099166\.7FPS0\.9880\.8050\.7990\.2720\.0570\.6990\.5900\.103123\.2500MBaseline0\.9920\.8720\.7550\.2310\.0530\.7090\.6300\.308167\.4

Table 10:Per\-trainer SAE evaluation for Gemma\-2\-2B, Matryoshka \(16K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowUnlearning↑\\uparrowSparsity\(L0\)↓\\downarrow05MRandom0\.9790\.9480\.8450\.4040\.1070\.7550\.5610\.02879\.5FPS0\.9410\.9670\.8630\.2950\.0530\.7240\.5260\.01549\.125MRandom0\.9840\.9660\.8440\.4550\.1040\.7430\.5490\.00680\.0FPS0\.9690\.9530\.8840\.3400\.0920\.7600\.5450\.09054\.450MRandom0\.9840\.9820\.8510\.4600\.1300\.7760\.5620\.04179\.5FPS0\.9750\.9750\.8630\.3730\.1060\.7340\.5690\.05657\.4500MBaseline0\.9850\.9930\.8490\.4590\.1300\.8060\.5600\.03480\.115MRandom0\.9850\.9630\.8300\.3350\.1770\.7760\.5810\.028159\.3FPS0\.9560\.9920\.8380\.2590\.1160\.7190\.5540\.03997\.125MRandom0\.9880\.9880\.8400\.4220\.1870\.7700\.6160\.017160\.1FPS0\.9800\.9980\.8620\.3420\.1420\.7320\.5940\.039106\.850MRandom0\.9880\.9760\.8310\.4590\.2060\.7570\.5940\.024159\.7FPS0\.9850\.9980\.8550\.3700\.1470\.7230\.5700\.075112\.1500MBaseline0\.9900\.9780\.8270\.4420\.2040\.7850\.6100\.011159\.3

Table 11:Per\-trainer SAE evaluation for Gemma\-2\-2B, Matryoshka \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowUnlearning↑\\uparrowSparsity\(L0\)↓\\downarrow25MRandom0\.9750\.9250\.7720\.3990\.0860\.7560\.5430\.02679\.5FPS0\.9490\.9670\.8730\.2660\.0740\.7600\.5400\.08650\.625MRandom0\.9850\.9920\.8320\.3930\.1060\.7830\.6020\.01780\.1FPS0\.9750\.9580\.8850\.3390\.0480\.7560\.6120\.03855\.150MRandom0\.9870\.9740\.8290\.4280\.1010\.7670\.6100\.04379\.9FPS0\.9800\.9610\.8810\.3250\.0860\.7720\.5810\.07957\.9500MBaseline0\.9870\.9800\.8200\.4330\.0910\.7820\.6070\.06279\.635MRandom0\.9840\.9490\.7850\.3680\.1300\.7810\.5910\.047159\.1FPS0\.9670\.9870\.8430\.2680\.1190\.7440\.5540\.03999\.725MRandom0\.9900\.9460\.8130\.3610\.1670\.7750\.6020\.058160\.7FPS0\.9840\.9560\.8630\.3730\.1060\.7430\.6070\.049109\.750MRandom0\.9900\.9750\.8230\.3550\.1610\.7500\.6190\.038159\.7FPS0\.9870\.9730\.8560\.3110\.1120\.7740\.5860\.092116\.7500MBaseline0\.9920\.9860\.8170\.3520\.1510\.7530\.6240\.051160\.2

Table 12:Per\-trainer SAE evaluation for Pythia\-160M, ReLU \(16K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowSparsity\(L0\)↓\\downarrow05MRandom0\.9750\.9440\.6760\.1230\.0350\.7250\.4991208\.2FPS0\.9550\.9420\.6800\.1050\.0210\.7350\.4991000\.025MRandom0\.9800\.9070\.6880\.1420\.0330\.7120\.5001200\.3FPS0\.9600\.9420\.6890\.1200\.0290\.7270\.5011074\.350MRandom0\.9780\.9130\.6970\.1460\.0340\.7190\.5011180\.7FPS0\.9620\.9180\.7020\.1270\.0300\.7240\.5021117\.9500MBaseline0\.9690\.9150\.7130\.1430\.0370\.7160\.4991144\.515MRandom0\.9670\.9160\.6770\.1200\.0310\.7240\.497949\.5FPS0\.9490\.9160\.7000\.1100\.0200\.7330\.497767\.825MRandom0\.9750\.8340\.6870\.1420\.0320\.7320\.498936\.1FPS0\.9540\.8920\.7070\.1160\.0260\.7400\.501825\.050MRandom0\.9720\.9040\.7010\.1440\.0290\.7180\.498918\.9FPS0\.9580\.8930\.7060\.1290\.0260\.7300\.498858\.3500MBaseline0\.9660\.8640\.7150\.1350\.0320\.7100\.498879\.225MRandom0\.9600\.9000\.6790\.1300\.0310\.7290\.496668\.3FPS0\.9330\.8820\.7020\.1100\.0180\.7340\.495525\.025MRandom0\.9650\.8660\.6980\.1430\.0310\.7400\.497652\.6FPS0\.9530\.8420\.7230\.1130\.0240\.7460\.498565\.350MRandom0\.9680\.8280\.7110\.1380\.0300\.7040\.496637\.9FPS0\.9510\.8020\.7210\.1280\.0240\.7270\.498587\.7500MBaseline0\.9600\.7820\.7240\.1340\.0310\.7150\.499602\.935MRandom0\.9490\.8060\.6900\.1240\.0320\.7180\.496387\.4FPS0\.9060\.8110\.7040\.1030\.0180\.7230\.491301\.625MRandom0\.9510\.7940\.7090\.1220\.0330\.7230\.496376\.7FPS0\.9290\.6820\.7240\.1110\.0230\.7450\.494319\.950MRandom0\.9530\.7580\.7130\.1220\.0340\.7240\.494367\.6FPS0\.9420\.6420\.7380\.1140\.0250\.7240\.492332\.0500MBaseline0\.9530\.6740\.7320\.1260\.0310\.7230\.494345\.945MRandom0\.9330\.7930\.7080\.1060\.0400\.7170\.494268\.9FPS0\.8930\.7260\.7110\.0860\.0200\.7170\.489213\.125MRandom0\.9370\.7150\.7160\.1190\.0360\.7200\.491260\.1FPS0\.9140\.6010\.7250\.0960\.0230\.7310\.490221\.550MRandom0\.9360\.6970\.7220\.1210\.0400\.7170\.492255\.5FPS0\.9250\.5600\.7350\.0960\.0240\.7180\.491227\.8500MBaseline0\.9350\.6150\.7410\.1060\.0350\.7380\.494241\.155MRandom0\.9080\.7000\.7030\.0810\.0460\.7420\.491179\.3FPS0\.8530\.7030\.6960\.0750\.0220\.7210\.491148\.825MRandom0\.9130\.6340\.7020\.0830\.0520\.7260\.492172\.6FPS0\.8870\.5500\.7200\.0830\.0350\.7270\.490151\.250MRandom0\.9100\.6290\.7150\.0830\.0500\.7500\.488169\.7FPS0\.8930\.5200\.7140\.0800\.0360\.7140\.489154\.4500MBaseline0\.9100\.5600\.7270\.1020\.0460\.7270\.489162\.8

Table 13:Per\-trainer SAE evaluation for Pythia\-160M, ReLU \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowSparsity\(L0\)↓\\downarrow65MRandom0\.9720\.8920\.6590\.0990\.0120\.7190\.4991861\.4FPS0\.9440\.8780\.6660\.0760\.0080\.7190\.4951371\.225MRandom0\.9760\.8000\.6710\.0880\.0140\.7140\.5002002\.8FPS0\.9640\.7840\.6790\.0900\.0100\.7060\.4991646\.350MRandom0\.9770\.8090\.6710\.0930\.0140\.7300\.4992012\.6FPS0\.9670\.7280\.6740\.0830\.0120\.7180\.4981758\.5500MBaseline0\.9680\.6590\.6830\.1000\.0160\.7350\.5001997\.575MRandom0\.9720\.8660\.6630\.0910\.0140\.7080\.4971472\.9FPS0\.9390\.7980\.6610\.0780\.0090\.6960\.4951076\.925MRandom0\.9740\.7480\.6750\.0940\.0120\.7090\.4971569\.1FPS0\.9600\.6980\.6950\.0840\.0100\.7350\.4981274\.050MRandom0\.9760\.7310\.6730\.0940\.0140\.7210\.4971570\.6FPS0\.9650\.6450\.6860\.0880\.0110\.7110\.4981359\.9500MBaseline0\.9670\.6290\.6720\.0900\.0160\.7300\.4971550\.185MRandom0\.9640\.8100\.6600\.0850\.0140\.7340\.4971079\.1FPS0\.9360\.7400\.6690\.0790\.0090\.7110\.494794\.225MRandom0\.9690\.6620\.6790\.0870\.0120\.7080\.4961126\.1FPS0\.9540\.5870\.7000\.0830\.0120\.7490\.497908\.250MRandom0\.9700\.6320\.6760\.0900\.0140\.7120\.4971123\.6FPS0\.9600\.5060\.6930\.0760\.0120\.7470\.494962\.8500MBaseline0\.9650\.5300\.6860\.0880\.0170\.7450\.4961095\.895MRandom0\.9510\.6760\.6680\.0790\.0130\.7300\.496710\.5FPS0\.9160\.5990\.6750\.0660\.0110\.7140\.493551\.525MRandom0\.9570\.5550\.6910\.0850\.0170\.7290\.494720\.4FPS0\.9390\.3900\.6910\.0780\.0130\.7380\.493590\.750MRandom0\.9570\.5100\.6840\.0830\.0170\.7410\.494713\.1FPS0\.9470\.3810\.6890\.0690\.0150\.7520\.493613\.3500MBaseline0\.9530\.4080\.6990\.0750\.0200\.7610\.494685\.2105MRandom0\.9400\.5850\.6800\.0690\.0190\.7400\.493559\.0FPS0\.9000\.5220\.6810\.0610\.0130\.7240\.492456\.125MRandom0\.9460\.4320\.6780\.0760\.0190\.7250\.494557\.4FPS0\.9240\.3400\.6860\.0650\.0150\.7440\.493475\.450MRandom0\.9460\.4020\.6990\.0750\.0190\.7420\.492551\.4FPS0\.9370\.3020\.6880\.0610\.0180\.7420\.492485\.9500MBaseline0\.9410\.3550\.7050\.0650\.0240\.7470\.491524\.9115MRandom0\.9210\.5170\.7030\.0520\.0240\.7320\.492448\.5FPS0\.8680\.4430\.6920\.0450\.0150\.7000\.491382\.825MRandom0\.9280\.3760\.6930\.0580\.0280\.7390\.489437\.4FPS0\.9040\.3250\.6860\.0480\.0210\.7350\.490392\.950MRandom0\.9300\.3490\.7010\.0480\.0270\.7260\.489429\.8FPS0\.9160\.2890\.7000\.0500\.0290\.7250\.491396\.2500MBaseline0\.9190\.2880\.7170\.0480\.0320\.7310\.491407\.8

Table 14:Per\-trainer SAE evaluation for Pythia\-160M, BatchTopK \(16K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowSparsity\(L0\)↓\\downarrow05MRandom0\.9460\.9020\.7330\.1890\.2450\.7070\.492142\.6FPS0\.9160\.8450\.7440\.2160\.1630\.7070\.490106\.325MRandom0\.9540\.9110\.7550\.1830\.2220\.7040\.492137\.5FPS0\.9400\.8520\.7760\.1960\.1630\.6970\.489108\.250MRandom0\.9530\.9140\.7740\.0270\.2240\.6970\.491135\.3FPS0\.9490\.8400\.7680\.1750\.1830\.6980\.488118\.1500MBaseline0\.9610\.8750\.802\-0\.1840\.2020\.6950\.494125\.015MRandom0\.9500\.8600\.732\-0\.1860\.1930\.7280\.492216\.7FPS0\.9300\.8200\.750\-0\.0710\.2190\.7110\.489167\.825MRandom0\.9580\.9480\.751\-0\.2160\.1940\.7200\.492211\.7FPS0\.9500\.9150\.759\-0\.1540\.1850\.7250\.491167\.150MRandom0\.9660\.9430\.769\-0\.1530\.1810\.7180\.492210\.3FPS0\.9610\.9120\.773\-0\.4090\.1990\.7250\.490174\.5500MBaseline0\.9700\.9110\.807\-0\.3080\.1680\.7220\.490204\.5

Table 15:Per\-trainer SAE evaluation for Pythia\-160M, BatchTopK \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowSparsity\(L0\)↓\\downarrow25MRandom0\.9450\.8860\.6920\.2290\.2480\.7080\.492352\.4FPS0\.9160\.8360\.7030\.1890\.2100\.7040\.488294\.725MRandom0\.9610\.8620\.7170\.2000\.2600\.6970\.492330\.6FPS0\.9460\.7690\.7630\.1960\.2150\.7100\.483276\.450MRandom0\.9610\.8570\.736\-0\.0840\.2350\.7090\.490315\.7FPS0\.9520\.6350\.7460\.1800\.1450\.6910\.488282\.9500MBaseline0\.9610\.7820\.783\-0\.0610\.2240\.6840\.490267\.735MRandom0\.9520\.8520\.6950\.0400\.2130\.7400\.494424\.9FPS0\.9340\.8770\.7050\.0380\.1800\.7310\.491346\.425MRandom0\.9660\.8980\.721\-0\.1740\.1800\.7300\.487393\.6FPS0\.9500\.8110\.760\-0\.0770\.2030\.7380\.487321\.650MRandom0\.9650\.9030\.751\-0\.1550\.1390\.7260\.494373\.7FPS0\.9580\.7520\.767\-0\.2430\.1920\.7470\.481331\.3500MBaseline0\.9700\.8940\.778\-0\.2710\.1560\.7210\.488362\.0

Table 16:Per\-trainer SAE evaluation for Pythia\-160M, JumpReLU \(16K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowSparsity\(L0\)↓\\downarrow05MRandom0\.9460\.8570\.7320\.2400\.2290\.6960\.491139\.5FPS0\.9090\.8090\.7470\.1310\.2100\.6910\.491106\.725MRandom0\.9550\.9070\.7420\.2460\.2130\.6860\.486141\.6FPS0\.9420\.8320\.7630\.1610\.1750\.6860\.490111\.950MRandom0\.9570\.9080\.7530\.2080\.2460\.6950\.492141\.5FPS0\.9480\.8280\.7660\.2140\.1860\.6800\.488115\.6500MBaseline0\.9560\.8890\.7620\.2040\.1990\.7110\.493138\.015MRandom0\.9580\.8700\.7110\.0970\.2400\.7130\.491222\.7FPS0\.9310\.8320\.7130\.1560\.2640\.7160\.490170\.125MRandom0\.9610\.9030\.7250\.2530\.2450\.7030\.494224\.4FPS0\.9560\.8470\.7460\.1160\.2540\.7020\.492177\.050MRandom0\.9630\.9410\.7230\.1010\.2660\.7010\.498224\.9FPS0\.9560\.8870\.751\-0\.0560\.2370\.7110\.493182\.7500MBaseline0\.9600\.9670\.768\-0\.1190\.2490\.7290\.497216\.5

Table 17:Per\-trainer SAE evaluation for Pythia\-160M, JumpReLU \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowSparsity\(L0\)↓\\downarrow25MRandom0\.9500\.8330\.7030\.1470\.2110\.6980\.494297\.4FPS0\.9090\.7390\.7140\.1250\.1650\.6650\.490255\.925MRandom0\.9580\.8180\.7270\.2440\.2380\.6980\.494309\.8FPS0\.9440\.6540\.7330\.1630\.1680\.7220\.487266\.250MRandom0\.9610\.7800\.7250\.2180\.2050\.6760\.492315\.0FPS0\.9570\.6350\.7470\.2070\.1840\.6900\.494272\.5500MBaseline0\.9630\.7370\.7520\.1620\.1910\.7000\.487322\.235MRandom0\.9590\.8190\.6980\.0870\.2320\.7120\.494378\.0FPS0\.9310\.8000\.6910\.1820\.2600\.7090\.490322\.425MRandom0\.9610\.8340\.706\-0\.0850\.2130\.7260\.489397\.8FPS0\.9580\.8320\.723\-0\.0660\.2650\.7300\.494344\.550MRandom0\.9650\.8640\.7140\.0420\.2760\.7240\.489403\.6FPS0\.9610\.8150\.725\-8\.46e\-040\.2380\.7170\.494355\.7500MBaseline0\.9680\.8820\.7390\.0460\.1840\.7190\.487409\.5

Table 18:Per\-trainer SAE evaluation for Pythia\-160M, Matryoshka \(16K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowSparsity\(L0\)↓\\downarrow05MRandom0\.9340\.9020\.7580\.1360\.2440\.7270\.486132\.5FPS0\.9140\.8920\.7550\.1820\.2230\.6930\.490107\.525MRandom0\.9520\.9070\.791\-0\.0400\.2770\.7200\.493126\.6FPS0\.9310\.9380\.795\-0\.0860\.2380\.6990\.493101\.850MRandom0\.9490\.8900\.803\-0\.3020\.2130\.6970\.489120\.6FPS0\.9410\.9520\.8000\.0370\.2320\.7140\.490107\.5500MBaseline0\.9510\.9420\.819\-0\.1870\.2320\.7020\.487117\.615MRandom0\.9420\.7010\.740\-0\.0720\.2100\.7190\.491208\.6FPS0\.9340\.8120\.759\-0\.2210\.1790\.7290\.489161\.325MRandom0\.9610\.7920\.789\-0\.1850\.2270\.6890\.491201\.8FPS0\.9460\.9230\.790\-0\.1350\.1660\.7270\.493161\.350MRandom0\.9610\.8620\.788\-0\.4020\.1920\.7080\.489196\.0FPS0\.9530\.8890\.796\-0\.0670\.2340\.7190\.493158\.9500MBaseline0\.9660\.9040\.802\-0\.1670\.1850\.7070\.493194\.1

Table 19:Per\-trainer SAE evaluation for Pythia\-160M, Matryoshka \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowRAVELDisent\.↑\\uparrowSparsity\(L0\)↓\\downarrow25MRandom0\.9480\.8910\.6870\.1200\.2140\.7170\.490340\.3FPS0\.9230\.8770\.6900\.2090\.2140\.7170\.490314\.725MRandom0\.9540\.9390\.712\-0\.2590\.2510\.7140\.492330\.7FPS0\.9450\.9410\.734\-0\.0350\.1920\.7040\.489283\.350MRandom0\.9540\.9670\.744\-0\.0850\.2190\.6940\.489300\.2FPS0\.9510\.9420\.7470\.0110\.1730\.7100\.489270\.7500MBaseline0\.9580\.9500\.763\-0\.1540\.2280\.7060\.493279\.135MRandom0\.9530\.8430\.6940\.0460\.1750\.7250\.496404\.8FPS0\.9380\.8380\.712\-0\.3380\.1860\.7280\.492343\.225MRandom0\.9650\.8810\.720\-0\.1790\.1900\.7360\.488389\.4FPS0\.9540\.9600\.738\-0\.0260\.1930\.7240\.494325\.750MRandom0\.9630\.8590\.731\-0\.4780\.1810\.7230\.493378\.3FPS0\.9590\.9500\.740\-0\.1070\.1930\.7290\.490334\.1500MBaseline0\.9660\.9240\.740\-0\.3860\.1800\.7220\.490379\.4

Table 20:Per\-trainer SAE evaluation for Qwen3\-8B, ReLU \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowSparsity\(L0\)↓\\downarrow05MRandom0\.9930\.9590\.8230\.1780\.1550\.7321337\.0FPS0\.9840\.9700\.8080\.1160\.0570\.7321199\.025MRandom0\.9930\.8940\.8330\.1470\.1010\.7261835\.8FPS0\.9910\.8600\.8120\.1370\.0650\.7321638\.750MRandom0\.9950\.8150\.8150\.1510\.0900\.7801944\.7FPS0\.9910\.7550\.7930\.1250\.0540\.7371703\.2500MBaseline0\.9950\.7340\.7770\.1460\.0660\.6662003\.915MRandom0\.9870\.9490\.8170\.1880\.1480\.700878\.7FPS0\.9780\.9400\.8090\.1150\.0650\.739778\.525MRandom0\.9910\.8590\.8320\.1490\.1330\.7161225\.5FPS0\.9870\.7370\.8130\.1270\.0730\.7741065\.550MRandom0\.9930\.7780\.8300\.1570\.1290\.7691306\.0FPS0\.9890\.6930\.8090\.1320\.0840\.7591122\.0500MBaseline0\.9930\.6990\.7970\.1550\.1010\.6731355\.025MRandom0\.9840\.8920\.8180\.1640\.1830\.698502\.3FPS0\.9730\.8690\.8120\.1280\.0890\.733428\.825MRandom0\.9870\.7670\.8390\.1430\.1680\.720693\.9FPS0\.9840\.6280\.8250\.1320\.0960\.728587\.650MRandom0\.9870\.7040\.8290\.1550\.1470\.768746\.8FPS0\.9840\.5610\.8330\.1420\.1060\.742628\.2500MBaseline0\.9890\.5910\.8190\.1420\.1090\.687788\.235MRandom0\.9800\.7670\.8330\.1610\.1710\.695243\.9FPS0\.9620\.6770\.8360\.1210\.0880\.745198\.025MRandom0\.9820\.6450\.8430\.1470\.1910\.687314\.6FPS0\.9770\.4530\.8430\.1130\.1110\.749260\.050MRandom0\.9840\.5750\.8380\.1550\.2110\.703338\.1FPS0\.9780\.4330\.8370\.1240\.1250\.742282\.3500MBaseline0\.9860\.4940\.8370\.1380\.1780\.706360\.545MRandom0\.9770\.6680\.8500\.1510\.1680\.708160\.4FPS0\.9530\.5640\.8460\.1120\.0830\.730127\.225MRandom0\.9800\.5490\.8540\.1400\.1690\.699192\.4FPS0\.9730\.3930\.8540\.1210\.1310\.706160\.350MRandom0\.9800\.4840\.8470\.1490\.1970\.749202\.9FPS0\.9750\.3920\.8660\.1220\.1460\.733172\.1500MBaseline0\.9840\.4220\.8540\.1340\.1970\.736216\.055MRandom0\.9690\.5990\.8670\.1240\.135–97\.1FPS0\.9370\.5190\.8690\.1080\.0790\.71574\.025MRandom0\.9750\.4380\.8720\.1470\.1670\.707105\.3FPS0\.9620\.3730\.8690\.0930\.1550\.70987\.450MRandom0\.9750\.4100\.8680\.1000\.2160\.720108\.3FPS0\.9680\.3460\.8710\.0820\.1890\.71491\.7500MBaseline0\.9770\.3940\.8700\.0860\.2110\.711111\.5

Table 21:Per\-trainer SAE evaluation for Qwen3\-8B, BatchTopK \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowSparsity\(L0\)↓\\downarrow25MRandom0\.9840\.9740\.8520\.2820\.3030\.70385\.6FPS0\.9680\.9620\.8780\.1890\.2580\.67963\.425MRandom0\.9860\.9510\.8670\.2640\.3030\.69286\.1FPS0\.9770\.9830\.8820\.2020\.2960\.67166\.950MRandom0\.9860\.9440\.8670\.2580\.2890\.69285\.8FPS0\.9820\.9430\.8770\.2360\.3130\.69869\.6500MBaseline0\.9890\.8800\.8660\.1800\.2470\.66987\.235MRandom0\.9870\.9470\.8540\.2730\.3410\.738170\.0FPS0\.9770\.9990\.8400\.1950\.2780\.700127\.725MRandom0\.9890\.9570\.8690\.2100\.3150\.692172\.1FPS0\.9860\.9730\.8600\.2020\.2620\.673132\.250MRandom0\.9910\.9790\.8540\.2120\.2900\.683171\.5FPS0\.9870\.9860\.8590\.2040\.2770\.663138\.8500MBaseline0\.9910\.8710\.838\-0\.0120\.2040\.646173\.8

Table 22:Per\-trainer SAE evaluation for Qwen3\-8B, JumpReLU \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowSparsity\(L0\)↓\\downarrow25MRandom0\.9800\.9900\.8570\.3360\.2260\.65287\.4FPS0\.9680\.9430\.8600\.1920\.1060\.70068\.525MRandom0\.9840\.9320\.8690\.2700\.2290\.66487\.3FPS0\.9820\.9650\.8840\.2120\.0980\.67767\.750MRandom0\.9860\.9620\.8760\.2570\.1980\.67187\.4FPS0\.9820\.9090\.8680\.2420\.1510\.66270\.1500MBaseline0\.9870\.8330\.8670\.2060\.1320\.67288\.335MRandom0\.9860\.9710\.8460\.3010\.3010\.723174\.7FPS0\.9800\.9990\.8350\.2240\.2000\.707134\.125MRandom0\.9890\.9460\.8490\.2790\.2240\.691176\.0FPS0\.9860\.9820\.8460\.2500\.1920\.709135\.450MRandom0\.9890\.9620\.8420\.2310\.2890\.671177\.3FPS0\.9870\.9740\.8540\.2450\.175–140\.7500MBaseline0\.9910\.9410\.8230\.1580\.1160\.624179\.2

Table 23:Per\-trainer SAE evaluation for Qwen3\-8B, Matryoshka \(64K\)\.TrainerBudgetSelectionLossRecovered↑\\uparrow1−1\-Abs\.Fraction↑\\uparrowAutoInterp↑\\uparrowSCRTop\-20↑\\uparrowTPPTop\-20↑\\uparrowSparse ProbeTop\-1↑\\uparrowSparsity\(L0\)↓\\downarrow25MRandom0\.9800\.9820\.8480\.2900\.3420\.66285\.3FPS0\.9620\.9980\.8920\.2380\.2860\.72658\.325MRandom0\.9840\.9900\.8630\.2680\.3250\.70885\.5FPS0\.9780\.9860\.8940\.2960\.2980\.67262\.650MRandom0\.9860\.9860\.8570\.3650\.3110\.69185\.9FPS0\.9800\.9780\.8940\.3120\.2990\.70465\.7500MBaseline0\.9840\.9740\.8500\.3720\.3140\.70586\.135MRandom0\.9860\.9050\.8350\.3340\.3330\.716169\.6FPS0\.9750\.9700\.8760\.2360\.3070\.699111\.225MRandom0\.9890\.9640\.8540\.3160\.3290\.694170\.9FPS0\.9840\.9960\.8850\.2270\.3100\.667122\.350MRandom0\.9890\.9570\.8420\.3450\.3520\.717171\.0FPS0\.9860\.9780\.8680\.3440\.3270\.748128\.2500MBaseline0\.9890\.9790\.8220\.3230\.3180\.689171\.4

## Appendix BProofs: When Should Feature matchings Agree?

### B\.1Separate encoder and decoder feature alignment[1](https://arxiv.org/html/2610.20680#Thmtheorem1)

###### Proof\.

Define

ar,N=supθ\|R^r,N​\(θ\)−R⁡\(θ\)\|\.a\_\{r,N\}=\\sup\_\{\\theta\}\\left\|\\widehat\{R\}\_\{r,N\}\(\\theta\)\-R\(\\theta\)\\right\|\.Statistical consistency \(assumption \(ii\)\) impliesa1,N→𝑝0a\_\{1,N\}\\xrightarrow\{p\}0andaq,N→𝑝0a\_\{q,N\}\\xrightarrow\{p\}0\.

Letθ⋆\\theta^\{\\star\}be a population minimizer\. Approximate empirical optimality gives

R⁡\(θ^r,N\)\\displaystyle R\(\\widehat\{\\theta\}\_\{r,N\}\)≤R^r,N​\(θ^r,N\)\+ar,N\\displaystyle\\leq\\widehat\{R\}\_\{r,N\}\(\\widehat\{\\theta\}\_\{r,N\}\)\+a\_\{r,N\}\(9\)≤R^r,N​\(θ⋆\)\+ηr,N\+ar,N\\displaystyle\\leq\\widehat\{R\}\_\{r,N\}\(\\theta^\{\\star\}\)\+\\eta\_\{r,N\}\+a\_\{r,N\}\(10\)≤R⋆\+2​ar,N\+ηr,N\.\\displaystyle\\leq R^\{\\star\}\+2a\_\{r,N\}\+\\eta\_\{r,N\}\.\(11\)Hereηr,N≥0\\eta\_\{r,N\}\\geq 0denotes the optimization error, i\.e\.,R^r,N​\(θ^r,N\)≤infθR^r,N​\(θ\)\+ηr,N\\widehat\{R\}\_\{r,N\}\(\\widehat\{\\theta\}\_\{r,N\}\)\\leq\\inf\_\{\\theta\}\\widehat\{R\}\_\{r,N\}\(\\theta\)\+\\eta\_\{r,N\}, andηr,N→𝑝0\\eta\_\{r,N\}\\xrightarrow\{p\}0by optimization consistency \(assumption \(iv\)\)\.

Hence

R⁡\(θ^r,N\)−R⋆→𝑝0\.R\(\\widehat\{\\theta\}\_\{r,N\}\)\-R^\{\\star\}\\xrightarrow\{p\}0\.
By encoder identifiability and separation, any sequence whose encoder features remain at leastϵ\\epsilonaway from the population encoder orbit must incur population excess risk at leastΔE​\(ϵ\)\>0\\Delta\_\{E\}\(\\epsilon\)\>0\. Therefore

Pr\[dE\(θ^r,N,θ⋆\)≥ϵ\]≤Pr\[2ar,N\+ηr,N≥ΔE\(ϵ\)\]→0\.\\Pr\\left\[d\_\{E\}\(\\widehat\{\\theta\}\_\{r,N\},\\theta^\{\\star\}\)\\geq\\epsilon\\right\]\\leq\\Pr\\left\[2a\_\{r,N\}\+\\eta\_\{r,N\}\\geq\\Delta\_\{E\}\(\\epsilon\)\\right\]\\rightarrow 0\.
The same argument using decoder separation givesdD​\(θ^r,N,θ⋆\)→𝑝0\.d\_\{D\}\(\\widehat\{\\theta\}\_\{r,N\},\\theta^\{\\star\}\)\\xrightarrow\{p\}0\.Applying the triangle inequality separately in encoder and decoder feature spaces gives

dE​\(θ^q,N,θ^1,N\)→𝑝0d\_\{E\}\(\\widehat\{\\theta\}\_\{q,N\},\\widehat\{\\theta\}\_\{1,N\}\)\\xrightarrow\{p\}0and

dD​\(θ^q,N,θ^1,N\)→𝑝0\.d\_\{D\}\(\\widehat\{\\theta\}\_\{q,N\},\\widehat\{\\theta\}\_\{1,N\}\)\\xrightarrow\{p\}0\.∎

### B\.2Proof of Agreement of encoder and decoder Hungarian matchings\)[2](https://arxiv.org/html/2610.20680#Thmcorollary2)

###### Proof\.

By Theorem[1](https://arxiv.org/html/2610.20680#Thmtheorem1), both normalized encoder and decoder feature systems converge to their respective population feature systems after one\-to\-one alignment\.

By assumption \(iii\), each run’s encoder and decoder features approach the population features under a single labeling shared by encoder and decoder\. LetPN⋆P^\{\\star\}\_\{N\}denote the permutation that aligns the two runs’ labelings\. Because distinct population features are separated by a positive margin, small perturbations cannot change the optimal Hungarian assignment, so with probability tending to onePE\(N\)=PN⋆P\_\{E\}^\{\(N\)\}=P^\{\\star\}\_\{N\}, and likewisePD\(N\)=PN⋆P\_\{D\}^\{\(N\)\}=P^\{\\star\}\_\{N\}\.

Hence,

Pr⁡\(PE\(N\)=PD\(N\)\)→1\.\\Pr\\left\(P\_\{E\}^\{\(N\)\}=P\_\{D\}^\{\(N\)\}\\right\)\\rightarrow 1\.∎

Table 24:FPS distribution shift for Qwen3\-8B and Pythia\-160M\.\*Metrics are defined as in Table[1](https://arxiv.org/html/2610.20680#S5.T1)\. The Gemma\-2\-2B pattern holds across model families: random selection is distribution\-neutral at every budget, while FPS shifts the selected subset toward higher\-norm activations, with divergence growing as the subset shrinks\.\*Same protocol as Table[1](https://arxiv.org/html/2610.20680#S5.T1): all selectors are applied to identical activation pools; entries are mean±\\pmSD across pool refreshes \(50 for norm statistics, 15 for direction statistics\)\. Entries shown as±0\.000\\pm 0\.000have SD below10−410^\{\-4\}\.

Table 25:Feature correspondence under increasingly strict alignment criteria\.*Encoder\-decoder agreement*is the fraction of features for which independently computed encoder and decoder Hungarian matchings select the same reference latent\.*Shared features*additionally require both encoder and decoder cosine similarities to exceed the specified thresholdτ\\tau\. Results are reported for Qwen3\-8B, and Pythia\-160M\.Table 26:Paired statistical tests for Gemma\-2\-2B ReLU and BatchTopK\.Panel A compares the500500M reference with Random subsets \(Δ=500​M−Random\\Delta=\\mathrm\{500M\}\-\\mathrm\{Random\}\); Panel B compares FPS with Random at matched budgets \(Δ=FPS−Random\\Delta=\\mathrm\{FPS\}\-\\mathrm\{Random\}\)\. Each entry reportsΔ/p\\Delta/pfrom a paired two\-sidedtt\-test across matched SAE configurations\.Observation\.Reduced\-data SAEs become increasingly comparable to the500500M reference as the budget grows, with several metrics already showing parity at small budgets\. Selection effects are metric\-dependent: Random generally favors reconstruction and SCR, whereas FPS frequently improves AutoInterp\.∗∗∗p<0\.001\{\}^\{\*\*\*\}p<0\.001,p∗⁣∗<0\.01\{\}^\{\*\*\}p<0\.01,∗p<0\.05\{\}^\{\*\}p<0\.05\.