差分秩保持流形学习

# Differential rank preservation manifold learning
Source: [https://arxiv.org/html/2610.41107](https://arxiv.org/html/2610.41107)
Henry HanEmail:[Henry\_Han@baylor\.edu](mailto:[email protected])Corresponding author:Corresponding author\.Affiliation:Data Science and AI Innovation Laboratory, Department of Computer Science, Baylor University, Waco, TX, 76798, USALuyu NiuAffiliation:Data Science and AI Innovation Laboratory, Department of Computer Science, Baylor University, Waco, TX, 76798, USADiane LiAffiliation:Department of Business, Management and Accounting, University of Maryland Eastern Shore, Princess Anne, MD, 21853, USAJiacun WangAffiliation:Department of Computer Science and Software Engineering, Monmouth University, West Long Branch, NJ, 07764, USA

###### Abstract

Manifold learning methods like t\-SNE and UMAP preserve local neighborhoods, but they do not preserve the ranking of the neighbors inside them\. Two embeddings can return the same neighbor set while disagreeing on which point is closest\. This loss of order degrades downstream applications that value the order in the embedding\. We propose Smooth Rank t\-SNE \(SR\-tSNE\), which adds a differentiable rank term to the standard t\-SNE loss\. To enable efficient training, this term uses a sigmoid rank approximation to explicitly force a point’s true high\-dimensional neighbors to outrank unrelated*intruders*and retain their original order\. We also introduce Ordinal DLP \(O\-DLP\), an extension of the Degree of Locality Preservation \(DLP\) metric that credits a neighbor only when it returns at its original rank\. Across eight benchmarks spanning image, text, speech, remote\-sensing and single\-cell data, SR\-tSNE outperforms 12 baselines in rank preservation and neighborhood overlap\. Ablation studies demonstrate our rank term alone boosts O\-DLP by 5% to 28%, a gain not achievable using standard pairwise loss\. This advantage transfers to downstream applications\. SR\-tSNE achieves the best label\-transfer performance across seven vision datasets\. For cell\-type discovery across six single\-cell datasets, it ranks first in 17 of 18 dataset–metric combinations, whether clustering resolution is selected with or without cell\-type labels\. In intraday trading, SR\-tSNE achieves higher returns, lower risk, and lower turnover than the baselines\. To our knowledge, SR\-tSNE is the first t\-SNE\- family method with explicit order preservation\.

###### Keywords:

Manifold learning , t\-SNE , Rank preservation , Ordinal DLP , Label transfer , Single\-cell clustering , Algorithmic trading

## 1Introduction

Manifold learning preserves which points are neighbors in the embedding, not the order among them\. Methods such as t\-SNE\[[1](https://arxiv.org/html/2610.41107#bib.bib1)\], UMAP\[[2](https://arxiv.org/html/2610.41107#bib.bib2)\], PHATE\[[3](https://arxiv.org/html/2610.41107#bib.bib3)\]and TriMap\[[4](https://arxiv.org/html/2610.41107#bib.bib4)\]embed data from a high\-dimensional input space while seeking to preserve neighborhood relationships\. However, what they preserve is the membership of that neighborhood; the order ranking inside it is left free\. An embedding can recover exactly the right neighbors and still shuffle them, putting the nearest neighbor at rank 5 and the second\-nearest at rank 1\. This can make downstream analyses that rely on neighbor order in the embedding less effective\.

The challenge\.Many downstream tasks rely on neighbor order as well as set membership\. In vision\-based retrieval and label transfer using the nearest neighbor, the top\-ranked neighbor directly determines the predicted label; scrambling positions can introduce labeling errors\. In single\-cell clustering, community detection builds graphs from top\-kklists; rank errors that change these lists’ membership can alter graph connectivity and split homogeneous cell populations into artificial clusters\. In algorithmic trading, marker selection algorithms compute rank\-correlation metrics between input and embedding spaces to flag trade signals; ordinal shifts can alter entry timing\. Preserving both neighborhood membership and neighbor order allows the embedding to more faithfully represent the local structure of the input data\.

The loss of neighbor order in manifold learning stems directly from how standard losses are formulated and how embedding performance is evaluated\. Both current optimization objectives and downstream metrics are entirely order\-agnostic\. For example, t\-SNE minimizes the KL divergence based purely on pairwise similarities, never tracking where a neighbor lands\. Consequently, two embeddings can achieve nearly identical KL loss while ordering their neighbors completely differently\. Similarly, the Degree of Locality Preservation \(DLP\)\[[5](https://arxiv.org/html/2610.41107#bib.bib5)\], a manifold explainability metric, treats each neighborhood as an unordered set, assigning identical credit whether a true neighbor sits at rank 1 or rankkk\.

Furthermore, many manifold learning methods focus exclusively on neighborhood membership, scalability, or global structure\[[2](https://arxiv.org/html/2610.41107#bib.bib2),[3](https://arxiv.org/html/2610.41107#bib.bib3),[6](https://arxiv.org/html/2610.41107#bib.bib6),[7](https://arxiv.org/html/2610.41107#bib.bib7),[8](https://arxiv.org/html/2610.41107#bib.bib8)\], deliberately sacrificing ordinal fidelity\. The crowding problem further bounds how much neighborhood structure two dimensions can hold\[[9](https://arxiv.org/html/2610.41107#bib.bib9)\], and order is precisely what these methods give up\. While methods like UMAP, PHATE, and LocalMAP improve local preservation or scalability, none require a retained neighbor to maintain its exact rank in the embedding\. Practitioners are thus forced to interpret visualizations for structural relations that the underlying embedding principles never guarantee\[[10](https://arxiv.org/html/2610.41107#bib.bib10)\]\. Although NeRV\[[11](https://arxiv.org/html/2610.41107#bib.bib11),[12](https://arxiv.org/html/2610.41107#bib.bib12)\]incorporates retrieval\-oriented trade\-offs into manifold learning, its precision–recall criteria still evaluate set membership rather than intra\-neighborhood order\.

SR\-tSNE\.To address this challenge, we introduce Smooth Rank t\-SNE \(SR\-tSNE\), a method designed to preserve both neighborhood membership and exact neighbor order\. SR\-tSNE augments the standard t\-SNE loss with a single differentiable rank term\. For each data point, this term optimizes two criteria at once\. It encourages true input\-space neighbors to rank ahead of embedding\-space*intruders*, the points outside the corresponding input\-space neighborhood\. It also penalizes any reshuffling among the true neighbors themselves\. Using a sigmoid formulation, discrete ranks are approximated by differentiable values, allowing direct optimization via standard gradient descent while a logarithmic discount prioritizes the nearest ranks\. A single regularization weight,λ\\lambda, governs the balance between neighborhood preservation and rank preservation\.

O\-DLP\.To evaluate how well SR\-tSNE preserves neighbor order, we propose Ordinal Degree of Locality Preservation \(O\-DLP\)\. While standard DLP checks only whether input\-space neighbors remain neighbors in the embedding, regardless of their order, O\-DLP credits a retained neighbor only when it appears at its exact original rank\. This enables O\-DLP to distinguish an embedding that faithfully preserves the original neighbor order from one that merely recovers the same neighbors in a shuffled order\.

Empirical evaluations demonstrate that SR\-tSNE more effectively preserves neighbor order and improves downstream performance\. Benchmarked against 12 baseline methods across eight diverse datasets: spanning computer vision, text, speech, remote sensing, and single\-cell transcriptomics, SR\-tSNE consistently achieves the highest O\-DLP and neighborhood overlap scores on every dataset\. These gains also extend to neighborhood preservation across scales, as measured by the area under theRN​XR\_\{NX\}curve\. SR\-tSNE achieves the highest scores under both local and global evaluation regimes on MNIST and CIFAR\-10\. In downstream applications, SR\-tSNE outperforms competing methods in image retrieval, label transfer, and single\-cell cell\-type discovery, while also delivering higher risk\-adjusted returns in a rank\-sensitive algorithmic trading pipeline\.

Our contributions\.\(1\) Method:SR\-tSNE augments t\-SNE with end\-to\-end differentiable rank regularization to penalize intruders and preserve exact neighbor order\.\(2\) Theory:We prove zero loss at perfect order, exponential soft\-to\-exact rank convergence, strict order preference where t\-SNE/UMAP lack it, and an upper bound on order deficit\.\(3\) Metric:O\-DLP extends DLP from neighborhood membership to exact\-rank fidelity\.\(4\) Benchmarks:Across eight datasets, SR\-tSNE outperforms 12 baselines in O\-DLP and neighborhood overlap, with higherRN​XR\_\{NX\}AUC\.\(5\) Utility:We demonstrate gains in label transfer, single\-cell cell\-type discovery, and algorithmic trading; the regularizer extends to any differentiable neighborhood\-embedding objective\.

## 2Related Work

Neighbor\-preserving manifold learning methods keep local affinities or neighborhood membership, not order\. t\-SNE\[[1](https://arxiv.org/html/2610.41107#bib.bib1)\], UMAP\[[2](https://arxiv.org/html/2610.41107#bib.bib2)\]and PHATE\[[3](https://arxiv.org/html/2610.41107#bib.bib3)\]optimize different similarity objectives\. Laplacian Eigenmaps\[[6](https://arxiv.org/html/2610.41107#bib.bib6)\], LLE\[[13](https://arxiv.org/html/2610.41107#bib.bib13)\]and PaCMAP\[[14](https://arxiv.org/html/2610.41107#bib.bib14)\]instead preserve local graph or distance relationships\. NeRV\[[12](https://arxiv.org/html/2610.41107#bib.bib12)\]incorporates a retrieval\-oriented trade\-off into a manifold learning objective and is the closest such objective to ours\. All of them preserve local relationships\. None of them asks a neighbor to come back to its own place in the list\. SR\-tSNE keeps t\-SNE as it is and adds one term for that place\.

Recent refinements modify neighborhood selection rather than internal ordering\. LocalMAP\[[7](https://arxiv.org/html/2610.41107#bib.bib7)\]dynamically adapts neighbor graphs, FUnc\-SNE\[[15](https://arxiv.org/html/2610.41107#bib.bib15)\]drops the fixed neighborhood size, and preprocessing techniques denoise the manifold prior to embedding\[[16](https://arxiv.org/html/2610.41107#bib.bib16)\]\. While these methods adjust which candidate points are embedded, they leave the order within the neighborhood unconstrained\. DREAMS\[[8](https://arxiv.org/html/2610.41107#bib.bib8)\]regularizes t\-SNE by trading local fidelity for global PCA alignment; in contrast, SR\-tSNE regularizes t\-SNE specifically to enforce fine\-grained local neighbor order\.

Rank\- and comparison\-based methods are the closest conceptual relatives, but they address order incompletely\. Triplet\-based approaches such as TriMap\[[4](https://arxiv.org/html/2610.41107#bib.bib4)\]and t\-STE\[[17](https://arxiv.org/html/2610.41107#bib.bib17)\]sample isolated relative comparisons, leaving the exact rank order of an entire neighborhood free\. Conversely, purely ordinal methods like RankVisu\[[18](https://arxiv.org/html/2610.41107#bib.bib18)\]and local ordinal embedding\[[19](https://arxiv.org/html/2610.41107#bib.bib19)\]rely solely on discrete rank relations, entirely discarding the continuous distance densities that preserve cluster geometry\. SR\-tSNE bridges this gap: it retains t\-SNE’s continuous similarity modeling while adding a smooth, listwise penalty that jointly aligns the full input\-space neighbor ranking and expels embedding\-space intruders\.

Existing evaluation metrics assess rank preservation at varying granularities, but stop short of requiring exact positional fidelity\. The co\-ranking matrix\[[20](https://arxiv.org/html/2610.41107#bib.bib20)\]provides the overarching framework, while summary metrics such asQN​XQ\_\{NX\}andRN​XR\_\{NX\}\[[20](https://arxiv.org/html/2610.41107#bib.bib20),[21](https://arxiv.org/html/2610.41107#bib.bib21)\]evaluate overall neighborhood shifts and serve as standard embedding benchmarks\[[22](https://arxiv.org/html/2610.41107#bib.bib22)\]\. At the local level, DLP\[[5](https://arxiv.org/html/2610.41107#bib.bib5)\]measures unordered neighborhood overlap, and nDCG provides a graded score that tolerates position shifts\. O\-DLP complements these metrics by enforcing the strictest criterion: a retained neighbor earns credit if and only if it recovers its exact original rank\.

## 3Methodology

SR\-tSNE Overview\.SR\-tSNE augments the standard t\-SNE objective with a differentiable regularizer that explicitly enforces neighbor order\. LetX∈ℝn×dX\\in\\mathbb\{R\}^\{n\\times d\}denote the input\-space data andY∈ℝn×2Y\\in\\mathbb\{R\}^\{n\\times 2\}the target low\-dimensional embedding coordinates\. Standard t\-SNE\[[1](https://arxiv.org/html/2610.41107#bib.bib1)\]minimizes the Kullback–Leibler divergenceKL\(P∥Q\)\\mathrm\{KL\}\(P\\\|Q\)\. HereP=\[pi​j\]P=\[p\_\{ij\}\]holds symmetrized Gaussian affinities computed from input\-space distances, with per\-point bandwidthsσi\\sigma\_\{i\}calibrated to a target perplexity\.Q=\[qi​j\]Q=\[q\_\{ij\}\]holds Student\-ttprobabilities computed from embedding\-space distances, withqi​j∝\(1\+‖yi−yj‖2\)−1q\_\{ij\}\\propto\(1\+\\\|y\_\{i\}\-y\_\{j\}\\\|^\{2\}\)^\{\-1\}\. The overall objective of SR\-tSNE is formulated as:

ℒtotal​\(Y\)=KL\(P∥Q\)⏟neighbor similarity\+λ⋅ℒrank​\(Y\)⏟rank loss,\\mathcal\{L\}\_\{\\text\{total\}\}\(Y\)\\;=\\;\\underbrace\{\\mathrm\{KL\}\(P\\,\\\|\\,Q\)\}\_\{\\text\{neighbor similarity\}\}\\;\+\\;\\lambda\\cdot\\underbrace\{\\mathcal\{L\}\_\{\\text\{rank\}\}\(Y\)\}\_\{\\text\{rank loss\}\},\(1\)whereℒrank\\mathcal\{L\}\_\{\\text\{rank\}\}is the smooth rank\-regularization term detailed in Section[3\.1](https://arxiv.org/html/2610.41107#S3.SS1), andλ≥0\\lambda\\geq 0balances neighborhood affinity against exact ordinal preservation\. Whenλ=0\\lambda=0, the objective reduces to standard t\-SNE\. By optimizingℒtotal\\mathcal\{L\}\_\{\\text\{total\}\}, SR\-tSNE encourages embeddings that preserve both local neighborhood membership and the fine\-grained ordinal ranking of true neighbors\.

![Refer to caption](https://arxiv.org/html/2610.41107v1/rank_illustration.png)Figure 1:Mechanism of the smooth rank regularization on three MNIST queries \(k=6k=6\)\.Solid discs show true input\-space neighbors positioned in the embedding space and labeled by original rank:tealpreserves exact rank, whilecoralis displaced\.Dashed circlesmark embedding\-space*intruders*\(from outside the top\-3030\)\.ℒrank\\mathcal\{L\}\_\{\\text\{rank\}\}simultaneously repels intruders and restores neighbor order, improving exact\-rank preservation \(O\-DLP\) by19%19\\%over t\-SNE on the full70,00070\{,\}000\-sample MNIST dataset \(Table[4](https://arxiv.org/html/2610.41107#S4.T4)\); the gain over the equal\-budget control is\+17%\+17\\%\(Table[2](https://arxiv.org/html/2610.41107#S4.T2)\)\.### 3\.1Smooth Rank Loss

Standard t\-SNE minimizes KL divergence between pairwise similarity distributions, leaving the internal rank order of neighbors unconstrained\. To resolve this, SR\-tSNE introduces adifferentiable rank lossℒrank\\mathcal\{L\}\_\{\\text\{rank\}\}that enforces two explicit constraints\. First, true input\-space neighbors must outrank any false intruders drifting into the embedding neighborhood\. Second, true neighbors must retain their exact original order, with nearest neighbors prioritized via a logarithmic discount \(Figure[1](https://arxiv.org/html/2610.41107#S3.F1)\)\.

1\. Candidate Pool Construction\.For each pointii, we define its candidate pool by combining two disjoint groups:

- •True Neighbors \(𝒩i\\mathcal\{N\}\_\{i\}\):The ordered sequence ofkknearest neighbors in the input space,\(𝒩i​\(1\),…,𝒩i​\(k\)\)\(\\mathcal\{N\}\_\{i\}\(1\),\\ldots,\\mathcal\{N\}\_\{i\}\(k\)\)\. This list is computed once onXXand fixed\. The true input\-space rank of candidatej≤kj\\leq kis simplyri,jin=jr\_\{i,j\}^\{\\text\{in\}\}=j\.
- •Intruders \(ℐi\\mathcal\{I\}\_\{i\}\):Themmclosest embedding points toyiy\_\{i\}that do not belong to𝒩i∪\{i\}\\mathcal\{N\}\_\{i\}\\cup\\\{i\\\}\. These points represent embedding\-space artifacts \(impostors\[[23](https://arxiv.org/html/2610.41107#bib.bib23)\]\) and are updated everyTrT\_\{r\}iterations\.

Concatenating both sets forms a candidate pool𝒞i=𝒩i⊕ℐi\\mathcal\{C\}\_\{i\}=\\mathcal\{N\}\_\{i\}\\oplus\\mathcal\{I\}\_\{i\}of sizek\+mk\+m\. For each candidatej∈\{1,…,k\+m\}j\\in\\\{1,\\ldots,k\+m\\\}, we compute its squared Euclidean distance toyiy\_\{i\}in the embedding space:di,j2=‖yi−yCi​\(j\)‖22d\_\{i,j\}^\{2\}=\\\|y\_\{i\}\-y\_\{C\_\{i\}\(j\)\}\\\|\_\{2\}^\{2\}\.

2\. Differentiable Soft Rank\.Sorting distances directly yields discrete, non\-differentiable ranks with zero gradient almost everywhere\. To enable backpropagation, we approximate the discrete indicator𝟙\[di,l2<di,j2\]\\mathbb\{1\}\[d\_\{i,l\}^\{2\}<d\_\{i,j\}^\{2\}\]using a continuous logistic sigmoidσ⁡\(z\)=\(1\+e−z\)−1\\sigma\(z\)=\(1\+e^\{\-z\}\)^\{\-1\}with temperatureτ\>0\\tau\>0\[[24](https://arxiv.org/html/2610.41107#bib.bib24)\]\. The soft embedding rank of candidatejjis computed by summing the smoothed pairwise comparisons:

ri,jemb=1\+∑l=1l≠jk\+mσ⁡\(di,j2−di,l2τ\)\.r\_\{i,j\}^\{\\text\{emb\}\}\\;=\\;1\\,\+\\,\\sum\_\{\\begin\{subarray\}\{c\}l=1\\\\ l\\neq j\\end\{subarray\}\}^\{k\+m\}\\sigma\\\!\\left\(\\frac\{d\_\{i,j\}^\{2\}\-d\_\{i,l\}^\{2\}\}\{\\tau\}\\right\)\.\(2\)Intuitively,ri,jembr\_\{i,j\}^\{\\text\{emb\}\}represents11plus the effective number of candidates placed closer toyiy\_\{i\}than candidatejj\. Asτ→0\+\\tau\\to 0^\{\+\},ri,jembr\_\{i,j\}^\{\\text\{emb\}\}converges to the exact integer rank\.

3\. Loss via Discounted Gain\.Only thekktrue neighbors earn positive relevance; intruders carry zero gain and act solely as rank obstructions\. We assign an input\-space relevance score ofk−j\+1k\-j\+1to thejj\-th true neighbor\. The actual discounted gainGiG\_\{i\}in the embedding space is:

Gi=∑j=1kk−j\+1log2⁡\(1\+ri,jemb\)\.G\_\{i\}\\;=\\;\\sum\_\{j=1\}^\{k\}\\frac\{k\-j\+1\}\{\\log\_\{2\}\\\!\\bigl\(1\+r\_\{i,j\}^\{\\text\{emb\}\}\\bigr\)\}\.\(3\)The maximum possible gainG⋆G^\{\\star\}occurs under perfect preservation, where every true neighbor recovers its exact original rank \(ri,jemb=jr\_\{i,j\}^\{\\text\{emb\}\}=j\) ahead of all intruders:

G⋆=∑j=1kk−j\+1log2⁡\(1\+j\)\.G^\{\\star\}\\;=\\;\\sum\_\{j=1\}^\{k\}\\frac\{k\-j\+1\}\{\\log\_\{2\}\(1\+j\)\}\.\(4\)The total rank loss is the average normalized deficit across allnndata points:

ℒrank​\(Y\)=1−1n​∑i=1nGiG⋆\.\\mathcal\{L\}\_\{\\text\{rank\}\}\(Y\)\\;=\\;1\\;\-\\;\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\frac\{G\_\{i\}\}\{G^\{\\star\}\}\.\(5\)
Here,ℒrank​\(Y\)∈\[0,1\]\\mathcal\{L\}\_\{\\text\{rank\}\}\(Y\)\\in\[0,1\]\. It reaches zero if and only if allkktrue neighbors occupy ranks1,…,k1,\\ldots,kin their exact input order without any intruder interference\. Whenm=0m=0, this formulation smoothly reduces to a differentiable Spearman rank correlation \(Appendix[A\.8](https://arxiv.org/html/2610.41107#A1.SS8)\)\.

### 3\.2Ordinal Degree of Locality Preservation \(O\-DLP\)

We refer toℒrank​\(Y\)\\mathcal\{L\}\_\{\\text\{rank\}\}\(Y\)in Eq\. \([5](https://arxiv.org/html/2610.41107#S3.E5)\) as the continuous \(soft\) rank lossℒranksoft​\(Y\)\\mathcal\{L\}\_\{\\text\{rank\}\}^\{\\text\{soft\}\}\(Y\)when evaluated with the soft ranksri,jembr\_\{i,j\}^\{\\text\{emb\}\}, and denote its per\-point contribution asℒrank,isoft=1−Gi/G⋆\\mathcal\{L\}\_\{\\text\{rank\},i\}^\{\\text\{soft\}\}=1\-G\_\{i\}/G^\{\\star\}\.

Standard Degree of Locality Preservation \(DLP\)\[[5](https://arxiv.org/html/2610.41107#bib.bib5)\]evaluates unordered neighborhood overlap\. Specifically, letAxi,kA\_\{x\_\{i\},k\}denote the set ofkk\-nearest neighbors ofxix\_\{i\}in the input space, and letByi,kB\_\{y\_\{i\},k\}denote the correspondingkk\-nearest neighbors ofyiy\_\{i\}in the embedding space\. DLP measures the average set retention across allnnpoints:

DLP​\(k\)=1n​k​∑i=1n\|Axi,k∩Byi,k\|\.\\text\{DLP\}\(k\)\\;=\\;\\frac\{1\}\{nk\}\\sum\_\{i=1\}^\{n\}\\bigl\|A\_\{x\_\{i\},k\}\\cap B\_\{y\_\{i\},k\}\\bigr\|\.\(6\)Because DLP treats neighborhoods strictly as unordered sets, a neighbor drifting from rank11to rankkkincurs zero penalty as long as it remains withinByi,kB\_\{y\_\{i\},k\}\.

O\-DLP\.O\-DLP extends DLP to evaluate exact positional alignment\. Letπiin​\(r\)\\pi\_\{i\}^\{\\text\{in\}\}\(r\)andπiemb​\(r\)\\pi\_\{i\}^\{\\text\{emb\}\}\(r\)denote the index of therr\-th nearest neighbor of pointiiin the input space and embedding space, respectively\. O\-DLP measures the proportion of exact rank matches across all points:

O\-DLP\(k\)=1n​k∑i=1n∑r=1k\[πiin\(r\)=πiemb\(r\)\],\\text\{O\-DLP\}\(k\)\\;=\\;\\frac\{1\}\{nk\}\\sum\_\{i=1\}^\{n\}\\sum\_\{r=1\}^\{k\}\\mathbf\{1\}\\\!\\left\[\\pi\_\{i\}^\{\\text\{in\}\}\(r\)=\\pi\_\{i\}^\{\\text\{emb\}\}\(r\)\\right\],\(7\)where𝟏​\[⋅\]\\mathbf\{1\}\[\\cdot\]is the indicator function\. A score of11signifies that allkk\-neighborhoods preserve identical rank order across both spaces, whereas00indicates that no neighbor reappears at its original position\. Because rank\-preserved neighbors necessarily belong to the intersectionAxi,k∩Byi,kA\_\{x\_\{i\},k\}\\cap B\_\{y\_\{i\},k\}, O\-DLP forms a strict lower bound on DLP:

O\-DLP​\(k\)≤DLP​\(k\),\\text\{O\-DLP\}\(k\)\\;\\leq\\;\\text\{DLP\}\(k\),\(8\)with equality holding if and only if every retained neighbor maintains its exact rank\.

Graded evaluation via nDCG\.While O\-DLP establishes the strictest standard by requiring exact positional alignment, we also report Normalized Discounted Cumulative Gain \(nDCG\)\[[24](https://arxiv.org/html/2610.41107#bib.bib24)\], the graded form of the quantity the rank term optimizes, which credits a neighbor by the position it reaches rather than only at its exact rank\. Crucially, while soft metrics like nDCG can remain deceptively high under local rank inversions, O\-DLP exposes whether individual positions are truly preserved\. Evaluating both metrics ensures that SR\-tSNE improves both coarse neighborhood placement \(captured by nDCG\) and fine\-grained ordinal sequences \(captured by O\-DLP\)\. Downstream steps such as clustering rest on both parts of a neighborhood\. They need a high degree of locality preservation, and they need the order inside it\. Eq\. \([8](https://arxiv.org/html/2610.41107#S3.E8)\) bounds one by the other, and Theorem[4](https://arxiv.org/html/2610.41107#Thmtheorem4)\(i\) separates O\-DLP into membership times order\. SR\-tSNE leads on both\.

### 3\.3SR\-tSNE Optimization Details

SR\-tSNE optimizes the total lossℒtotal\(Y\)=KL\(P∥Q\)\+λℒrank\(Y\)\\mathcal\{L\}\_\{\\text\{total\}\}\(Y\)=\\mathrm\{KL\}\(P\\\|Q\)\+\\lambda\\mathcal\{L\}\_\{\\text\{rank\}\}\(Y\)in two stages: a deterministic warm start followed by joint divergence and refinement\.

1\. Two\-stage optimization\.A standard Barnes–Hut t\-SNE\[[25](https://arxiv.org/html/2610.41107#bib.bib25)\]warm start \(e\.g\., PCA initialization, perplexity3030, learning ratemax⁡\(n/12,50\)\\max\(n/12,50\), early exaggeration1212\) runs for1,0001\{,\}000iterations to establish a coarse global layout\. Joint refinement then runs for anotherT=1,000T=1\{,\}000iterations without early exaggeration\. Throughout refinement, we setk=10k=10neighbors andm=10m=10intruders\. EveryTr=50T\_\{r\}=50iterations, intruder setsℐi\\mathcal\{I\}\_\{i\}are refreshed by identifying themmpoints nearest toyiy\_\{i\}in the embedding outside𝒩i∪\{i\}\\mathcal\{N\}\_\{i\}\\cup\\\{i\\\}\.

2\. Temperature annealing\.To ensure broad gradient exploration early and strict rank agreement late, the smoothing temperatureτ\\taudecreases linearly over iterationst∈\[1,T\]t\\in\[1,T\]:

τ\(t\)=τstart−t−1T−1​\(τstart−τend\),\\tau^\{\(t\)\}\\;=\\;\\tau\_\{\\text\{start\}\}\\,\-\\,\\frac\{t\-1\}\{T\-1\}\\bigl\(\\tau\_\{\\text\{start\}\}\-\\tau\_\{\\text\{end\}\}\\bigr\),\(9\)falling fromτstart=5\.0\\tau\_\{\\text\{start\}\}=5\.0toτend=0\.05\\tau\_\{\\text\{end\}\}=0\.05\. Per Theorem[2](https://arxiv.org/html/2610.41107#Thmtheorem2),τend=0\.05\\tau\_\{\\text\{end\}\}=0\.05guarantees that the soft rank matches the discrete rank to within𝒪\(e−γi/0\.05\)\\mathcal\{O\}\(e^\{\-\\gamma\_\{i\}/0\.05\}\), whereγi\>0\\gamma\_\{i\}\>0is the smallest gap between the squared embedding distances of two distinct candidates of pointii\.

3\. Gradient update and re\-centering\.Coordinates are updated using SGD with Polyak momentum \(μ=0\.8\\mu=0\.8,η=10\\eta=10\):

V\(t\+1\)\\displaystyle V^\{\(t\+1\)\}=μ​V\(t\)\+∇Yℒtotal​\(Y\(t\)\),\\displaystyle\\;=\\;\\mu\\,V^\{\(t\)\}\\,\+\\,\\nabla\_\{Y\}\\mathcal\{L\}\_\{\\text\{total\}\}\\bigl\(Y^\{\(t\)\}\\bigr\),\(10\)Y~\(t\+1\)\\displaystyle\\widetilde\{Y\}^\{\(t\+1\)\}=Y\(t\)−η​V\(t\+1\)\.\\displaystyle\\;=\\;Y^\{\(t\)\}\\,\-\\,\\eta\\,V^\{\(t\+1\)\}\.\(11\)Because neither term penalizes translation, we re\-center coordinates after each step to remove translational drift:

Y\(t\+1\)=Y~\(t\+1\)−1n​∑i=1ny~i\(t\+1\)\.Y^\{\(t\+1\)\}\\;=\\;\\widetilde\{Y\}^\{\(t\+1\)\}\\,\-\\,\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\tilde\{y\}\_\{i\}^\{\(t\+1\)\}\.\(12\)
4\. Memory complexity and design rationale\.For large datasets \(n≥50,000n\\geq 50\{,\}000\), we store the sparse matrixPPwithkP=90k\_\{P\}=90affinities per row\. The partition functionZ=∑a≠b\(1\+‖ya−yb‖2\)−1Z=\\sum\_\{a\\neq b\}\(1\+\\\|y\_\{a\}\-y\_\{b\}\\\|^\{2\}\)^\{\-1\}is computed in row chunks ofc=10,000c=10\{,\}000using gradient checkpointing, lowering peak memory from𝒪⁡\(n2\)\\mathcal\{O\}\(n^\{2\}\)to𝒪⁡\(c​n\)\\mathcal\{O\}\(c\\,n\)\. Together, this two\-stage schedule ensures that the warm start preserves coarse neighborhood membership, while the annealed rank refinement directly recovers fine\-grained neighbor order and expels intruders\.

### 3\.4Theoretical Guarantees

Standard objectives like t\-SNE and UMAP preserve neighbor order only when high\-dimensional affinities strictly differ; when affinities tie, they become indifferent to the ordering\. SR\-tSNE resolves this by explicitly penalizing rank inversions through its rank loss\. This section formalizes this distinction through four guarantees\. The rank loss controls O\-DLP and vanishes exactly at perfect order \(Theorem[1](https://arxiv.org/html/2610.41107#Thmtheorem1)\), and the soft rank used in optimization converges exponentially to the discrete rank \(Theorem[2](https://arxiv.org/html/2610.41107#Thmtheorem2)\)\. Every inverted neighbor pair is strictly penalized, where t\-SNE and UMAP are indifferent \(Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)\), and the loss bounds the order deficit among retained neighbors relative to an order\-blind baseline \(Theorem[4](https://arxiv.org/html/2610.41107#Thmtheorem4)\)\.

Preliminaries\.For each pointii, the candidate pool𝒞i=𝒩i⊕ℐi\\mathcal\{C\}\_\{i\}=\\mathcal\{N\}\_\{i\}\\oplus\\mathcal\{I\}\_\{i\}of sizeK=k\+mK=k\+mcombines itskktrue input\-space neighbors𝒩i\\mathcal\{N\}\_\{i\}withmmembedding intrudersℐi\\mathcal\{I\}\_\{i\}\(themmnon\-neighbors closest toyiy\_\{i\}\)\. We sort their embedding distances‖yi−yCi​\(j\)‖2\\\|y\_\{i\}\-y\_\{C\_\{i\}\(j\)\}\\\|\_\{2\}from nearest to farthest to obtain the discrete rankπi−1​\(j\)∈\{1,…,K\}\\pi\_\{i\}^\{\-1\}\(j\)\\in\\\{1,\\dots,K\\\}\. Evaluating the discounted gain with these integer ranks rather than the soft valuesri,jembr\_\{i,j\}^\{\\text\{emb\}\}yields the discrete gainGihard=∑j=1kk−j\+1log2⁡\(1\+πi−1​\(j\)\)G\_\{i\}^\{\\text\{hard\}\}=\\sum\_\{j=1\}^\{k\}\\frac\{k\-j\+1\}\{\\log\_\{2\}\(1\+\\pi\_\{i\}^\{\-1\}\(j\)\)\}and the corresponding discrete rank loss:

ℒrankhard​\(Y\)=1−1n​∑i=1nGihardG⋆=1−1n​G⋆​∑i=1n∑j=1kk−j\+1log2⁡\(1\+πi−1​\(j\)\),\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}\(Y\)\\;=\\;1\\;\-\\;\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\frac\{G\_\{i\}^\{\\text\{hard\}\}\}\{G^\{\\star\}\}\\;=\\;1\\;\-\\;\\frac\{1\}\{n\\,G^\{\\star\}\}\\sum\_\{i=1\}^\{n\}\\sum\_\{j=1\}^\{k\}\\frac\{k\-j\+1\}\{\\log\_\{2\}\\\!\\bigl\(1\+\\pi\_\{i\}^\{\-1\}\(j\)\\bigr\)\},\(13\)whereG⋆=∑j=1kk−j\+1log2⁡\(1\+j\)G^\{\\star\}=\\sum\_\{j=1\}^\{k\}\\frac\{k\-j\+1\}\{\\log\_\{2\}\(1\+j\)\}is the ideal gain, andΔK=disc⁡\(K−1\)−disc⁡\(K\)\>0\\Delta\_\{K\}=\\mathrm\{disc\}\(K\-1\)\-\\mathrm\{disc\}\(K\)\>0represents the smallest discrete step of the position discountdisc⁡\(r\)=1/log2⁡\(1\+r\)\\mathrm\{disc\}\(r\)=1/\\log\_\{2\}\(1\+r\)\. All proofs are in[A\.14](https://arxiv.org/html/2610.41107#A1.SS14)\.

###### Theorem 1\(The rank loss controls O\-DLP\)

Letm≥1m\\geq 1and letk′k^\{\\prime\}, the neighborhood size at which the metrics are read, satisfyk′≤min⁡\(k,m\)k^\{\\prime\}\\leq\\min\(k,m\)\. Then

1−O\-DLP​\(k′\)≤2​G⋆k′​ΔK​ℒrankhard​\(Y\)\.1\-\\text\{O\-DLP\}\(k^\{\\prime\}\)\\;\\leq\\;\\frac\{2\\,G^\{\\star\}\}\{k^\{\\prime\}\\,\\Delta\_\{K\}\}\\;\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}\(Y\)\.\(14\)Moreoverℒrankhard​\(Y\)=0\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}\(Y\)=0if and only ifO\-DLP​\(k\)=1\\text\{O\-DLP\}\(k\)=1, in which caseO\-DLP​\(k′\)=1\\text\{O\-DLP\}\(k^\{\\prime\}\)=1for everyk′≤kk^\{\\prime\}\\leq k\.

Theorem 1 establishes two key results\. First, it provides a quantitative surrogate error bound: when convergence is incomplete \(ℒrankhard\>0\\mathcal\{L\}\_\{\\text\{rank\}\}^\{\\text\{hard\}\}\>0\), the discrete ordinal error1−O\-DLP​\(k′\)1\-\\text\{O\-DLP\}\(k^\{\\prime\}\)is strictly upper\-bounded by the loss\. The prefactor2​G⋆k′​ΔK\\frac\{2G^\{\\star\}\}\{k^\{\\prime\}\\Delta\_\{K\}\}represents the worst\-case penalty where every inversion occurs at the smallest discount stepΔK\\Delta\_\{K\}\(1\.6×1031\.6\\times 10^\{3\}whenk=m=k′=10k=m=k^\{\\prime\}=10\)\. Crucially, this equivalence requires intruders \(m≥1m\\geq 1\)\. Without them \(m=0m=0\), false points could insert themselves between true neighbors without increasing the loss, since the pairwise order among true neighbors remains unchanged\. Assigning target ranks behind all true neighbors to thesemmintruders guarantees that any intervening intruder is penalized, thereby locking both neighborhood membership and order\.

###### Theorem 2\(Continuous\-to\-discrete convergence\)

Letγi=mina≠b∈𝒞i⁡\|‖yi−ya‖22−‖yi−yb‖22\|\>0\\gamma\_\{i\}=\\min\_\{a\\neq b\\in\\mathcal\{C\}\_\{i\}\}\\bigl\|\\\|y\_\{i\}\-y\_\{a\}\\\|\_\{2\}^\{2\}\-\\\|y\_\{i\}\-y\_\{b\}\\\|\_\{2\}^\{2\}\\bigr\|\>0be the minimum squared distance gap between distinct candidates of pointii\. For any candidatejj, as the sigmoid smoothing temperatureτ→0\+\\tau\\to 0^\{\+\}, the continuous soft rankri,jembr\_\{i,j\}^\{\\text\{emb\}\}converges exponentially to the true discrete rankπi−1​\(j\)\\pi\_\{i\}^\{\-1\}\(j\):

\|ri,jemb−πi−1\(j\)\|≤\(K−1\)e−γi/τ,\\bigl\|r\_\{i,j\}^\{\\text\{emb\}\}\-\\pi\_\{i\}^\{\-1\}\(j\)\\bigr\|\\;\\leq\\;\(K\-1\)\\,e^\{\-\\gamma\_\{i\}/\\tau\},\(15\)and the per\-point difference between the continuous soft lossℒrank,isoft\\mathcal\{L\}\_\{\\text\{rank\},i\}^\{\\text\{soft\}\}and the discrete hard lossℒrank,ihard\\mathcal\{L\}\_\{\\text\{rank\},i\}^\{\\text\{hard\}\}satisfies:

\|ℒrank,isoft−ℒhardrank,i\|≤k​\(k\+1\)​\(K−1\)4​ln⁡2​G⋆e−γi/τ\.\\bigl\|\\mathcal\{L\}\_\{\\text\{rank\},i\}^\{\\text\{soft\}\}\-\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\},i\}\\bigr\|\\;\\leq\\;\\frac\{k\(k\+1\)\(K\-1\)\}\{4\\ln 2\\,G^\{\\star\}\}\\,e^\{\-\\gamma\_\{i\}/\\tau\}\.\(16\)

The gap closes exponentially in the margin over the temperature\. The schedule of Section[3\.3](https://arxiv.org/html/2610.41107#S3.SS3)ends atτ=0\.05\\tau=0\.05; a margin of0\.50\.5in squared distance then leaves a factore−10≈4\.5×10−5e^\{\-10\}\\approx 4\.5\\times 10^\{\-5\}\. This is why the loss the optimizer minimizes and the metric the tables report agree at the end of training\.

The third result isolates what each objective intrinsically demands of neighbor order\. By treating each loss purely as a function of squared distancesD=\(da,b2\)D=\(d\_\{a,b\}^\{2\}\), we examine an idealized row swapD↦D′:=Di:j↔lD\\mapsto D^\{\\prime\}:=D^\{i:j\\leftrightarrow l\}that exchangesdi,j2d\_\{i,j\}^\{2\}anddi,l2d\_\{i,l\}^\{2\}and leaves every other entry unchanged\. Whether this configuration is geometrically embeddable in 2D is deliberately set aside; this isolates objective preference from dimensionality constraints \(empirically evaluated in Section[4\.1](https://arxiv.org/html/2610.41107#S4.SS1)\)\. Crucially, t\-SNE and UMAP penalize order inversions only in proportion to affinity gaps, vanishing when affinities tie\. In contrast, SR\-tSNE strictly enforces the correct rank order under all conditions\.

###### Theorem 3\(Pairwise order preference\)

Fix a pointiiand two of its true neighborsj<lj<l, so that their input\-space distances satisfydi,jH<di,lHd^\{H\}\_\{i,j\}<d^\{H\}\_\{i,l\}, wheredi,jH=‖xi−xNi​\(j\)‖2d^\{H\}\_\{i,j\}=\\\|x\_\{i\}\-x\_\{N\_\{i\}\(j\)\}\\\|\_\{2\}\. Suppose the embedding inverts them,di,l2<di,j2d\_\{i,l\}^\{2\}<d\_\{i,j\}^\{2\}, and letD′D^\{\\prime\}be the row swap that corrects the inversion\. Then:

1. \(a\)For the t\-SNE loss with symmetrized affinitiespp, KL⁡\(D\)−KL⁡\(D′\)=\(pi​j−pi​l\)​\[ln⁡\(1\+di,j2\)−ln⁡\(1\+di,l2\)\]\.\\mathrm\{KL\}\(D\)\-\\mathrm\{KL\}\(D^\{\\prime\}\)\\;=\\;\(p\_\{ij\}\-p\_\{il\}\)\\Bigl\[\\ln\(1\+d\_\{i,j\}^\{2\}\)\-\\ln\(1\+d\_\{i,l\}^\{2\}\)\\Bigr\]\.\(17\)The bracket is positive\. The loss therefore prefers the correct order whenpi​j\>pi​lp\_\{ij\}\>p\_\{il\}, is indifferent whenpi​j=pi​lp\_\{ij\}=p\_\{il\}, and prefers the inversion whenpi​j<pi​lp\_\{ij\}<p\_\{il\}\.
2. \(b\)For the UMAP cross\-entropy with fuzzy\-union weightsvvand low\-dimensional kernelw⁡\(x\)=\(1\+a​xb\)−1w\(x\)=\(1\+ax^\{b\}\)^\{\-1\}, in whicha\>0a\>0andb\>0b\>0are UMAP’s fitted kernel constants, CE⁡\(D\)−CE⁡\(D′\)=\(vi​j−vi​l\)​b​\[ln⁡di,j2−ln⁡di,l2\],\\mathrm\{CE\}\(D\)\-\\mathrm\{CE\}\(D^\{\\prime\}\)\\;=\\;\(v\_\{ij\}\-v\_\{il\}\)\\,b\\,\\Bigl\[\\ln d\_\{i,j\}^\{2\}\-\\ln d\_\{i,l\}^\{2\}\\Bigr\],\(18\)with the same three cases in the sign ofvi​j−vi​lv\_\{ij\}\-v\_\{il\}\.
3. \(c\)For the rank loss, withr<r′r<r^\{\\prime\}the two positions the pair occupies, n⁡\[ℒrankhard​\(D\)−ℒrankhard​\(D′\)\]≥\(l−j\)​\[disc⁡\(r\)−disc⁡\(r′\)\]G⋆≥ΔKG⋆\>0\.n\\bigl\[\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}\(D\)\-\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}\(D^\{\\prime\}\)\\bigr\]\\;\\geq\\;\\frac\{\(l\-j\)\\,\[\\mathrm\{disc\}\(r\)\-\\mathrm\{disc\}\(r^\{\\prime\}\)\]\}\{G^\{\\star\}\}\\;\\geq\\;\\frac\{\\Delta\_\{K\}\}\{G^\{\\star\}\}\\;\>\\;0\.\(19\)The preference is strict, and it does not depend onPP, onvv, or on the embedding distances\.

Parts \(a\) and \(b\) show that t\-SNE and UMAP enforce neighbor order strictly through pairwise affinity differences\. Consequently, this ordering pressure vanishes, or backfires, under two common scenarios \(Lemma[S2](https://arxiv.org/html/2610.41107#Thmlemma2)\): UMAP’s fuzzy\-union affinities saturate to one for close neighbors, while t\-SNE’s symmetrization frequently contradicts true input distances\. The latter is remarkably prevalent: on MNIST,32\.9%32\.9\\%of adjacent pairs\(j,j\+1\)\(j,j\{\+\}1\)in a point’s top\-3030list exhibitpi,j\+1≥pi,jp\_\{i,j\+1\}\\geq p\_\{i,j\}, affecting98\.5%98\.5\\%of points \(19\.7%19\.7\\%and91\.3%91\.3\\%on CIFAR\-10\)\. For every such pair, the standard KL divergence is indifferent to the true order or actively favors an inversion\. Part \(c\) highlights the core theoretical guarantee of SR\-tSNE: regardless of input affinities, every neighbor inversion incurs a strict penalty of at leastΔK/\(n​G⋆\)\\Delta\_\{K\}/\(nG^\{\\star\}\)\. Corollary[S1](https://arxiv.org/html/2610.41107#Thmcorollary1)ensures that above a finite weightλ⋆\\lambda^\{\\star\}that does not grow withnn, the total loss strictly penalizes all inversions\.

Our final theoretical result links the rank loss back to the evaluation metric by decomposing exact\-rank preservation into neighborhood membership and internal ordering\. As introduced in Section[3\.2](https://arxiv.org/html/2610.41107#S3.SS2), letO\-DLPi=DLPi⋅Ai\\text\{O\-DLP\}\_\{i\}=\\text\{DLP\}\_\{i\}\\cdot A\_\{i\}, whereDLPi\\text\{DLP\}\_\{i\}measures membership retention andAiA\_\{i\}represents the fraction of retained neighbors preserving their exact ranks\. The theorem below establishes an analytical order\-blind baseline for shuffled neighborhoods and proves that the SR\-tSNE rank loss directly bounds the ordering degradation among retained neighbors \(DLP−O\-DLP\\text\{DLP\}\-\\text\{O\-DLP\}\)\.

###### Theorem 4\(Order beyond membership\)

Fix an embedding and a neighborhood sizek′k^\{\\prime\}, and letAi​\(k′\)A\_\{i\}\(k^\{\\prime\}\)denote the fraction of retained neighbors of pointiithat preserve their exact original ranks\.

1. \(i\)*Decomposition\.*O\-DLP decomposes into membership times order: O\-DLP​\(k′\)=1n​∑i=1nDLPi​\(k′\)​Ai​\(k′\),\\text\{O\-DLP\}\(k^\{\\prime\}\)=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\text\{DLP\}\_\{i\}\(k^\{\\prime\}\)\\,A\_\{i\}\(k^\{\\prime\}\),\(20\)and its normalized ratioω⁡\(k′\):=k′​O\-DLP​\(k′\)/DLP​\(k′\)∈\[0,k′\]\\omega\(k^\{\\prime\}\):=k^\{\\prime\}\\,\\text\{O\-DLP\}\(k^\{\\prime\}\)/\\text\{DLP\}\(k^\{\\prime\}\)\\in\[0,k^\{\\prime\}\]represents theDLPi\\text\{DLP\}\_\{i\}\-weighted mean ofk′​Ai​\(k′\)k^\{\\prime\}A\_\{i\}\(k^\{\\prime\}\)\.
2. \(ii\)*Order\-blind baseline\.*If thek′k^\{\\prime\}embedded neighbors of each point are randomly shuffled, then𝔼⁡\[Ai\]=1/k′\\mathbb\{E\}\[A\_\{i\}\]=1/k^\{\\prime\}andVar⁡\(k′​O\-DLPi\)≤DLPi\\mathrm\{Var\}\(k^\{\\prime\}\\,\\text\{O\-DLP\}\_\{i\}\)\\leq\\text\{DLP\}\_\{i\}\. Assuming independent arrangements across points,O\-DLP​\(k′\)=DLP​\(k′\)/k′\\text\{O\-DLP\}\(k^\{\\prime\}\)=\\text\{DLP\}\(k^\{\\prime\}\)/k^\{\\prime\}within a standard deviation ofk′−1​DLP​\(k′\)/nk^\{\\prime\-1\}\\sqrt\{\\text\{DLP\}\(k^\{\\prime\}\)/n\}, which corresponds toω⁡\(k′\)=1\\omega\(k^\{\\prime\}\)=1within a standard deviation of: σω=1n​DLP​\(k′\)\.\\sigma\_\{\\omega\}=\\frac\{1\}\{\\sqrt\{n\\,\\text\{DLP\}\(k^\{\\prime\}\)\}\}\.\(21\)
3. \(iii\)*Order deficit bound\.*Fork′≤min⁡\(k,m\)k^\{\\prime\}\\leq\\min\(k,m\), the gap between membership retention and exact\-rank preservation is strictly bounded by the discrete rank loss: DLP​\(k′\)−O\-DLP​\(k′\)≤2​G⋆k′​ΔK​ℒrankhard​\(Y\)\.\\text\{DLP\}\(k^\{\\prime\}\)\-\\text\{O\-DLP\}\(k^\{\\prime\}\)\\leq\\frac\{2\\,G^\{\\star\}\}\{k^\{\\prime\}\\,\\Delta\_\{K\}\}\\,\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}\(Y\)\.\(22\)

Part \(ii\) is the*order\-blind baseline*: an embedding that keeps the right neighbors and shuffles them scoresω=1\\omega=1, and at the benchmark sizes of Section[4](https://arxiv.org/html/2610.41107#S4)Eq\. \([21](https://arxiv.org/html/2610.41107#S3.E21)\) places it within0\.030\.03for every method that keeps at least a fiftieth of its neighbors, and within0\.070\.07for all of them\. Perfect order isω=k′\\omega=k^\{\\prime\}\. Part \(iii\) refines Theorem[1](https://arxiv.org/html/2610.41107#Thmtheorem1)\. The rank loss bounds not only the distance of O\-DLP from one but also the gap between O\-DLP and DLP\. Driving the loss down therefore tightens the bound on the order of the neighbors that are kept, not only on how many are kept\.

Remark\.The guarantees above characterize the intrinsic properties of the objectives\. Specifically, Theorem[1](https://arxiv.org/html/2610.41107#Thmtheorem1)establishes the theoretical trajectory of O\-DLP as the rank loss is minimized\. In practice, Section[4\.1](https://arxiv.org/html/2610.41107#S4.SS1)empirically evaluates how closely gradient\-based optimization approaches these theoretical limits, while the equal\-budgetλ=0\\lambda\{=\}0control in[A\.9](https://arxiv.org/html/2610.41107#A1.SS9)confirms that the observed order gains stem directly from the rank regularization term itself\.

## 4Experiments

Table 1:Overview of the eight benchmark datasets\. Sources and data types are listed in Table[A\.11](https://arxiv.org/html/2610.41107#A1.T11)of[A\.2](https://arxiv.org/html/2610.41107#A1.SS2)\.Data and preprocessing\.The core benchmark spans eight datasets across five distinct domains: computer vision, audio, natural language, tabular remote sensing, and single\-cell transcriptomics \(Table[1](https://arxiv.org/html/2610.41107#S4.T1)\)\. For feature representations, MNIST and Fashion\-MNIST are evaluated on raw pixels and Covertype retains its5454native attributes\. CIFAR\-10 uses512512\-dimensional pooled features from an ImageNet\-pretrained ResNet\-18, AG News mean\-pooled bert\-base\-uncased embeddings \(768768\-d\), and Speech Commands mean\-pooled wav2vec 2\.0 features \(768768\-d\)\.

For the two single\-cell transcriptomics datasets \(Renal\-Cortex and Brain\-Cortex\), raw counts first undergo standard library\-size normalization andlog⁡\(1\+x\)\\log\(1\{\+\}x\)transformation, followed by the selection of the2,0002\{,\}000most highly variable genes\. These are then reduced via PCA to5050principal components, providing an identical5050\-dimensional input to all methods\. Across our benchmark tables,ppdenotes the nominal input dimensionality \(or the raw gene count for single\-cell data prior to PCA\)\.

Peer methods in comparison\.We benchmark SR\-tSNE against twelve peer methods\. Seven are established: t\-SNE, UMAP, PHATE, TriMap, PaCMAP, ISOMAP\[[26](https://arxiv.org/html/2610.41107#bib.bib26)\]and a 2\-layer autoencoder\. Four are recent: ParamPaCMAP\[[27](https://arxiv.org/html/2610.41107#bib.bib27)\], SUDE\[[28](https://arxiv.org/html/2610.41107#bib.bib28)\], LocalMAP\[[7](https://arxiv.org/html/2610.41107#bib.bib7)\]and DREAMS\[[8](https://arxiv.org/html/2610.41107#bib.bib8)\]\. The twelfth is the retrieval\-oriented objective NeRV\[[12](https://arxiv.org/html/2610.41107#bib.bib12)\]\. All methods receive identical inputs and precomputed neighbor graphs, running at their library defaults or official implementations \(exact ISOMAP usesk=10k=10without landmark approximation; NeRV runs through the original dredviz implementation\)\. Eleven of the twelve run on all eight benchmarks at full size\. NeRV, whose implementation is built for sizes below50,00050\{,\}000, is compared at matched scale on5,0005\{,\}000\-point subsamples of all eight benchmarks \(Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3)of[A\.2](https://arxiv.org/html/2610.41107#A1.SS2)\)\. Every paired test therefore runs over all eight datasets\.

SR\-tSNE setting\.SR\-tSNE uses the smooth rank loss withk=10k\{=\}10andm=10m\{=\}10throughout\. The weightλ\\lambdafollows dataset size\. It isλ=50\\lambda\{=\}50forn≥50,000n\\geq 50\{,\}000andλ=5\\lambda\{=\}5below that threshold\. The same rule carries over to the seven vision datasets of Section[4\.2\.2](https://arxiv.org/html/2610.41107#S4.SS2.SSS2)\.

Evaluation metrics\.The evaluation combines order, membership and co\-ranking criteria\. O\-DLP is the main criterion for order, DLP scores neighborhood membership, and nDCG grades the returned list by position, all atk=10k\{=\}10\. We also report the area under theRN​XR\_\{NX\}curve on a2020k subsample of MNIST and CIFAR\-10, read off the co\-ranking matrix and independent of the training objective\. All four take the high\-dimensional neighbor ranking as reference; the clustering criteria of Section[4\.2\.3](https://arxiv.org/html/2610.41107#S4.SS2.SSS3)are read against a curated annotation\.

The paired tests certify the sign of the lead, and the tables give its size\. Every test is a two\-sided Wilcoxon signed\-rank over datasets, one observation per dataset, so the smallest attainableppis0\.00780\.0078with eight datasets,0\.01560\.0156with seven and0\.0310\.031with six\. Appat that floor certifies that SR\-tSNE leads on every dataset of the pair\. Table[A\.1](https://arxiv.org/html/2610.41107#A1.T1)of the appendix gives the median and range of the per\-dataset differences for every baseline, and we read the two together; the sign counts carry the conclusion and do not depend on a multiplicity correction\.

Table 2:O\-DLP atk=10k\{=\}10for t\-SNE, the strongest published baseline \(DREAMS\), the equal\-budgetλ=0\\lambda\{=\}0control and SR\-tSNE, with the mean over datasets and the gain of the rank term over the control\. All twelve methods are in Table[4](https://arxiv.org/html/2610.41107#S4.T4)below\.Figure 2:O\-DLP versus neighborhood sizekkon the eight benchmarks of Table[1](https://arxiv.org/html/2610.41107#S4.T1)\. Each panel carries every method run on that dataset, so the curves match the entries of Appendix Table[4](https://arxiv.org/html/2610.41107#S4.T4)\.### 4\.1Neighbor Order Preservation

SR\-tSNE achieves best exact\-rank fidelity\.Across all eight benchmarks, SR\-tSNE records the highest O\-DLP \(Table[2](https://arxiv.org/html/2610.41107#S4.T2), Figure[2](https://arxiv.org/html/2610.41107#S4.F2)\), outperforming t\-SNE by\+6%\+6\\%to\+28%\+28\\%\(median\+18%\+18\\%, Table[4](https://arxiv.org/html/2610.41107#S4.T4)\)\. Nonparametric Wilcoxon signed\-rank tests confirm this advantage across all benchmarks \(p=0\.0078p=0\.0078against each of the eleven full\-scale baselines\)\. On5,0005\{,\}000\-point subsamples, SR\-tSNE surpasses NeRV by factors of2\.9×2\.9\\timesto9\.2×9\.2\\times\(Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3)\)\. Absolute O\-DLP values run between0\.060\.06and0\.190\.19, the scale set by a criterion that credits a neighbor only at its exact position\. Theorem[1](https://arxiv.org/html/2610.41107#Thmtheorem1)bounds the ordinal error that remains by the rank loss\. This validates Theorems[1](https://arxiv.org/html/2610.41107#Thmtheorem1)and[2](https://arxiv.org/html/2610.41107#Thmtheorem2): driving the smooth surrogate down under temperature annealing turns continuous gradient descent into discrete rank optimization\.

Rank optimization simultaneously reinforces neighborhood membership\.Sorting true neighbors ahead of mined intruders eliminates false low\-dimensional neighbors\. Consequently, SR\-tSNE achieves the highest DLP on all eight benchmarks \(p=0\.0078p=0\.0078, Figure[A\.3](https://arxiv.org/html/2610.41107#A1.F3)\)\. Of the twelve methods, SR\-tSNE is the one that holds the highest DLP and the highest O\-DLP together, on every benchmark \(Tables[4](https://arxiv.org/html/2610.41107#S4.T4)and[5](https://arxiv.org/html/2610.41107#S4.T5)\)\. A downstream step reads both, so leading both is what carries the gain forward\. In neighborhood sweeps \(Figure[2](https://arxiv.org/html/2610.41107#S4.F2)\), SR\-tSNE retains top rank across all 32 dataset–kksettings for O\-DLP, and leads DLP atk∈\{5,10\}k\\in\\\{5,10\\\}\. This fidelity generalizes across multi\-scale metrics\. SR\-tSNE secures the highest nDCG on all eight datasets \(Table[A\.4](https://arxiv.org/html/2610.41107#A1.T4)\)\. It also takes the topRN​XR\_\{NX\}AUC on MNIST and CIFAR\-10, in both the local regime \(K≤100K\\leq 100\) and the full\-range one \(Table[A\.10](https://arxiv.org/html/2610.41107#A1.T10)\); that criterion measures neighbor agreement across scales, normalized against random chance\.

Ablation isolates the rank term as the sole active driver\.An equal\-budget control \(λ=0\\lambda=0\), sharing the identical warm start, SGD momentum, and iteration budget, matches baseline t\-SNE to within2%2\\%across all benchmarks \(Table[2](https://arxiv.org/html/2610.41107#S4.T2)\)\. Activatingℒrank\\mathcal\{L\}\_\{\\text\{rank\}\}delivers the entire performance margin, boosting O\-DLP by\+5%\+5\\%to\+28%\+28\\%and DLP by\+1%\+1\\%to\+4%\+4\\%\(p=0\.0078p=0\.0078\)\. This improvement is strictly positive across all 30 metric differences over ten paired runs, two datasets by five perturbed starts \([A\.9](https://arxiv.org/html/2610.41107#A1.SS9)\) and all 35 trading names against the control \(p=5\.8×10−11p=5\.8\\times 10^\{\-11\}\)\. Initializing from a2,0002\{,\}000\-iteration t\-SNE run further amplifies these gains \(Table[A\.18](https://arxiv.org/html/2610.41107#A1.T18)\)\. This confirms Theorem 1 empirically: directly optimizing the rank objective governs the discrete ordinal error, driving O\-DLP upward while standard affinity loss leaves it unconstrained\.

Order is preserved beyond pure membership\.Table[3](https://arxiv.org/html/2610.41107#S4.T3)reads order against the order\-blind baseline of Theorem[4](https://arxiv.org/html/2610.41107#Thmtheorem4)\(ii\), whereω=1\\omega=1is a neighborhood kept and shuffled andω=10\\omega=10is perfect order\. Nine of the eleven peer methods run at full size sit between1\.071\.07and1\.321\.32, so the order inside their neighborhoods stays close to that baseline\. This is the room the rank term is built to recover, and SR\-tSNE holds the highestω\\omegaon all eight benchmarks\. The gain is in the ordering rather than in the membership: from t\-SNE to SR\-tSNE, O\-DLP rises3\.23\.2to21\.421\.4times faster than DLP\. Read as order recovered above the baseline,ω−1\\omega\-1, SR\-tSNE stands27%27\\%above t\-SNE and25%25\\%above the equal\-budget control, on every benchmark\. The control shares the warm start, the optimizer, the budget and the seed, so that margin is the rank term’s\. Position\-wise hit rates show this gain concentrates where downstream retrieval reads \(Figure[3](https://arxiv.org/html/2610.41107#S4.F3), Table[A\.2](https://arxiv.org/html/2610.41107#A1.T2)\)\. At rank position one, SR\-tSNE retrieves the true nearest neighbor1\.06×1\.06\\timesto1\.41×1\.41\\timesmore often than t\-SNE\. Examples are0\.366→0\.4430\.366\\to 0\.443on MNIST and0\.268→0\.3770\.268\\to 0\.377on Renal\-Cortex\. Finally, on a held\-out donor split of the Human Lung Cell Atlas\[[29](https://arxiv.org/html/2610.41107#bib.bib29)\], SR\-tSNE preserves its lead, exceeding t\-SNE by\+30%\+30\\%in O\-DLP \([A\.5](https://arxiv.org/html/2610.41107#A1.SS5)\)\. These results empirically validate Theorems[3](https://arxiv.org/html/2610.41107#Thmtheorem3)and[4](https://arxiv.org/html/2610.41107#Thmtheorem4): the objective suppresses pairwise inversions at the head of the list and strictly minimizes the order deficit beyond membership alone\.

Table 3:Order beyond membership:ω⁡\(10\)=10⋅O\-DLP​\(10\)/DLP​\(10\)\\omega\(10\)=10\\cdot\\text\{O\-DLP\}\(10\)/\\text\{DLP\}\(10\), computed from Tables[4](https://arxiv.org/html/2610.41107#S4.T4)and[5](https://arxiv.org/html/2610.41107#S4.T5)\.ω=1\\omega=1is the order\-blind baseline of Theorem[4](https://arxiv.org/html/2610.41107#Thmtheorem4)\(ii\), a neighborhood kept and shuffled;ω=10\\omega=10is perfect order\. At these sizes the baseline is exact to within0\.030\.03by Eq\. \([21](https://arxiv.org/html/2610.41107#S3.E21)\) for every method withDLP≥0\.02\\text\{DLP\}\\geq 0\.02, and to within0\.070\.07for all of them\. The eleven peer methods that run at full size are scored on all eight benchmarks; NeRV is compared at matched scale in Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3)\. Rows are sorted by the mean\. Columns: MN MNIST, CF CIFAR\-10, FM Fashion\-MNIST, SC Speech Commands, RC Renal\-Cortex, BC Brain\-Cortex, CT Covertype, AG AG News\.Figure 3:Where the recovered order sits\.Exact\-hit rate at the first five positions: the fraction of points whoserr\-th input\-space neighbor returns at positionrrin the embedding, on four benchmarks \(log scale\)\. This is the head of the list, where the position discount places its weight and where retrieval, label transfer and thekk\-nearest\-neighbor graph read\. SR\-tSNE lifts every one of these positions on each of the four benchmarks shown\. The equal\-budgetλ=0\\lambda\{=\}0control tracks t\-SNE, so the lift is the rank term’s\. All ten positions on all eight benchmarks are in Table[A\.2](https://arxiv.org/html/2610.41107#A1.T2)\.Table 4:O\-DLP atk=10k\{=\}10, the main criterion for order, withλ=0\\lambda\{=\}0the equal\-budget control: the same warm start and refinement schedule with the rank term switched off\. Bold marks the best method per dataset\. The last column is the two\-sided Wilcoxon signed\-rank test of SR\-tSNE against that method, paired over the datasets where both are available, bold wherep<0\.05p<0\.05and “–” where no test is available\.†NeRV runs on a5,0005\{,\}000\-point subsample of each benchmark, the largest size its implementation reaches \(Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3)\); its row is scored on that subsample and its test is paired with SR\-tSNE on the same subsamples\.Table 5:DLP atk=10k\{=\}10across all eight datasets, scoring neighborhood membership rather than order, withλ=0\\lambda\{=\}0the equal\-budget control\. Bold marks the best method per dataset\. The last column is the two\-sided Wilcoxon signed\-rank test of SR\-tSNE against that method, paired over all eight datasets, bold wherep<0\.05p<0\.05and “–” for SR\-tSNE itself\.†NeRV as in Table[4](https://arxiv.org/html/2610.41107#S4.T4): scored and paired on the5,0005\{,\}000\-point subsamples of Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3)\.
### 4\.2Applications

We evaluate three downstream tasks that read the embedding neighbor list from the top down: marker selection in algorithmic trading, retrieval and label transfer in vision, and cell\-type discovery in single\-cell transcriptomics\. Because every embedding is routed through the identical downstream pipeline, reported metrics reflect purely paired differences between dimensional reduction methods\.

#### 4\.2\.1Algorithmic Trading

Experimental setup\.We benchmark SR\-tSNE within the trading\-map pipeline of\[[30](https://arxiv.org/html/2610.41107#bib.bib30)\]\. Each 5\-minute bar contains eight market features \(open, high, low, close, volume, average price, notional value, and time differential\), with a binary direction flag indicating the trade side \(sign⁡\(close−open\)\\mathrm\{sign\}\(\\text\{close\}\-\\text\{open\}\)\)\. Bars are embedded into 2D, and DBSCAN\+K \([A\.4](https://arxiv.org/html/2610.41107#A1.SS4)\) selects trade markers using fixed clustering parameters \(minPts=30\\mathrm\{minPts\}=30,ε\\varepsilonset to the 25th percentile of theminPts\\mathrm\{minPts\}\-distance distribution\)\. We evaluate 35 liquid US equities across seven sectors \(5 stocks per group, Table[A\.12](https://arxiv.org/html/2610.41107#A1.T12)\) from 2026\-04\-14 to 2026\-07\-09 \(≈4,680\\approx 4\{,\}680bars per equity;λ=5,k=m=10\\lambda=5,k=m=10\)\. Both SR\-tSNE and the equal\-budgetλ=0\\lambda=0control refine the identical GPU t\-SNE warm start \(RAPIDS cuML\), isolating the embedding objective as the sole experimental variable\. Returns are simulated under realistic friction:00,0\.50\.5,11, and22basis points \(bps\) per round trip\. Figure[4](https://arxiv.org/html/2610.41107#S4.F4)follows one equity end to end, from the markers each embedding produces to the positions they open and close\.

Theoretical alignment\.The marker score reads1−ρi1\-\\rho\_\{i\}, the Spearman rank disagreement between a bar’s input\-space and embedded neighbor order\. Proposition[S2](https://arxiv.org/html/2610.41107#Thmproposition2)proves that the per\-bar rank lossℒrank,i\\mathcal\{L\}\_\{\\text\{rank\},i\}strictly upper\-bounds this quantity\. Consequently, SR\-tSNE directly minimizes the scoring metric, whereas standard t\-SNE remains indifferent whenever affinities tie \(Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)\(a\)\)\.

![Refer to caption](https://arxiv.org/html/2610.41107v1/trading_case_IDCC.png)Figure 4:End\-to\-end trading on IDCC at22bps \(5\-minute bars; identical DBSCAN\+\+K hyperparameters\)\.\(a, c\)Generated markers and\(b, d\)executed trades\. The gold band highlights SR\-tSNE’s two strongest round trips \(\+12\.5%\+12\.5\\%vs\.\+2\.6%\+2\.6\\%for t\-SNE\); gold circles mark trade executions \(dashed where t\-SNE places no matching side\)\. Over the quarter, SR\-tSNE delivers\+28\.0%\+28\.0\\%return at Sharpe2\.612\.61versus−7\.5%\-7\.5\\%at Sharpe−0\.81\-0\.81for t\-SNE\.Superior net returns and risk\-adjusted alpha\.Under22bps transaction costs, SR\-tSNE achieves the highest net return on 21 of the 35 equities, with a median return of\+13\.8%\+13\.8\\%versus\+3\.6%\+3\.6\\%for t\-SNE \(Figure[A\.1](https://arxiv.org/html/2610.41107#A1.F1)\)\. Against a cost\-adjusted buy\-and\-hold baseline, SR\-tSNE outperforms on 25 of 35 names \(p=0\.0017p=0\.0017, Wilcoxon signed\-rank test\)\. Beyond raw return, SR\-tSNE significantly enhances capital efficiency and drawdown control \(Table[A\.13](https://arxiv.org/html/2610.41107#A1.T13)\): median annualized Sharpe ratio rises from0\.630\.63to2\.322\.32, median maximum drawdown moderates from−11\.26%\-11\.26\\%to−10\.22%\-10\.22\\%, and total executions decrease from337337to266266round trips\.

![Refer to caption](https://arxiv.org/html/2610.41107v1/fig_pool35_five.png)Figure 5:Cross\-method comparison across 35 equities at22bps under identical DBSCAN\+\+K scoring:\(a\)net return \(%\),\(b\)annualized Sharpe ratio,\(c\)maximum drawdown \(%\), and\(d\)round trips\. Boxes indicate interquartile ranges; dots represent individual equities; top numbers denote medians; bottom numbers in\(a\)–\(c\)report two\-sided paired Wilcoxonpp\-values versus SR\-tSNE\.Consistent edge over modern manifold baselines\.At22bps the median net return of SR\-tSNE is\+13\.8%\+13\.8\\%, against\+2\.6%\+2\.6\\%for UMAP,\+2\.8%\+2\.8\\%for TriMap,\+4\.3%\+4\.3\\%for LocalMAP and\+5\.6%\+5\.6\\%for DREAMS \(Figure[5](https://arxiv.org/html/2610.41107#S4.F5)\)\. Every one of those four paired comparisons favors SR\-tSNE \(p=3\.3×10−7p=3\.3\\times 10^\{\-7\},2\.9×10−62\.9\\times 10^\{\-6\},0\.00140\.0014and0\.00100\.0010\)\. SR\-tSNE takes the top return on 31, 30, 28 and 27 of the 35 names, respectively, and doubles the median of the strongest competitor\.

Robustness to trading frictions\.Because preserving rank reduces false\-positive cluster transitions, SR\-tSNE trades sparingly and keeps most of its gross return once costs are charged\. Raising the cost from00to22bps, SR\-tSNE retains\+13\.8%\+13\.8\\%of its\+18\.2%\+18\.2\\%, giving up4\.44\.4points with round trips steady at269→266269\\to 266; TriMap retains\+2\.8%\+2\.8\\%of\+13\.0%\+13\.0\\%with round trips at1,008→5901\{,\}008\\to 590, and UMAP\+2\.6%\+2\.6\\%of\+9\.1%\+9\.1\\%with542→380542\\to 380\. A method that trades less pays the spread less often, so the ranking of the arms at the highest cost level follows turnover as well as gross return\.

Higher post\-marker drift\.Evaluating unconditional forward returns confirms that SR\-tSNE flags structurally superior inflection points \(Figure[A\.2](https://arxiv.org/html/2610.41107#A1.F2)\)\. It leads t\-SNE on 25 of 35 names at 30 minutes \(p=0\.0003p=0\.0003\) and on 21 of 35 at 1 hour \(p=0\.007p=0\.007\)\. At 2 hours it leads on 25 of 35, with a median forward edge of\+0\.063%\+0\.063\\%against\+0\.027%\+0\.027\\%\(p=0\.005p=0\.005\)\.

#### 4\.2\.2Retrieval and Label Transfer in Vision

Exact neighbor alignment at the head of the list\.Because downstream retrieval is inspected from the top down, whether the item in position one is truly the nearest neighbor determines downstream utility\. Figure[6](https://arxiv.org/html/2610.41107#S4.F6)inspects the first three retrieved positions across three representative queries each on MNIST, CIFAR\-10, and Fashion\-MNIST\. Three queries on each dataset give nine inspected positions per dataset\. SR\-tSNE returns all nine at their exact input\-space rank on each of the three \(9/99/9,9/99/9,9/99/9\)\. The equal\-budgetλ=0\\lambda=0control returns7/97/9on MNIST,2/92/9on CIFAR\-10, and1/91/9on Fashion\-MNIST \(baseline t\-SNE scores6/96/9,2/92/9, and1/91/9, while DREAMS attains4/94/9,1/91/9, and0/90/9\)\. Crucially, ordinal misplacements in the baselines frequently trigger category errors: on the CIFAR\-10 dog query, all seven comparative baselines retrieve incorrect classes \(predominantly cats\), and on Fashion\-MNIST Shirt, all seven return a T\-shirt\. SR\-tSNE eliminates these top\-rank misclassifications\.

![Refer to caption](https://arxiv.org/html/2610.41107v1/retrieval_row_k3.png)Figure 6:Retrieval comparison on \(a\) MNIST, \(b\) CIFAR\-10, and \(c\) Fashion\-MNIST across the top three embedded positions\. Thumbnail badges indicate high\-dimensional ground\-truth ranks within the query’s top\-3030; frame colors encode rank displacement \(dark green: exact match; pale green: offset by one; orange: offset by≥2\\geq 2; red: outside top\-3030\)\. Class labels turn red upon class mismatch\. Queries illustrate exact\-rank preservation where comparative methods introduce ordinal and semantic errors\.
Population\-level exact\-hit counts\.This top\-of\-list advantage generalizes across the entire query distribution\. Evaluating all70,00070\{,\}000MNIST queries, we measure the mean number of the first three embedded neighbors returned at their exact input\-space rank\. SR\-tSNE averages0\.710\.71exact hits out of three, substantially exceeding theλ=0\\lambda=0control \(0\.580\.58\), t\-SNE \(0\.570\.57\), and DREAMS \(0\.260\.26\), while UMAP, LocalMAP, PaCMAP, and TriMap remain below0\.100\.10\.

Transductive label transfer on held\-out splits\.We next test whether geometric order preservation improves semantic generalization\. Transductive label transfer runs across seven vision benchmarks, spanning9,2989\{,\}298to70,00070\{,\}000samples and1010to200200classes\. Four are raw\-pixel sets and three are ResNet\-18 feature sets \(Table[A\.5](https://arxiv.org/html/2610.41107#A1.T5)\)\. Embeddings are computed on the full dataset, with a random20%20\\%held out exclusively from the retrieval index as queries\. We evaluate precision at rank one \(P1\\text\{P\}\_\{1\}\) and Mean Reciprocal Rank \(MRR\) based on the first occurrence of the query class among the ten nearest reference points\.

Consistent gains in semantic precision\.SR\-tSNE achieves strictly positive improvements in bothP1\\text\{P\}\_\{1\}and MRR across all seven datasets against all five baselines \(p=0\.0156p=0\.0156, two\-sided Wilcoxon signed\-rank test; Table[A\.20](https://arxiv.org/html/2610.41107#A1.T20)\)\. Relative to t\-SNE \(the strongest baseline, with medianP1=0\.8035\\text\{P\}\_\{1\}=0\.8035andMRR=0\.8537\\text\{MRR\}=0\.8537\), SR\-tSNE wins7/77/7evaluations, while theλ=0\\lambda=0control reproduces t\-SNE within0\.00040\.0004\. The performance margin over the control is most pronounced on highly confusable datasets, adding\+0\.0016\+0\.0016to\+0\.0164\+0\.0164inP1\\text\{P\}\_\{1\}, led by EMNIST\-Balanced \(47 mixed alphanumeric classes\) and Fashion\-MNIST \(Table[A\.6](https://arxiv.org/html/2610.41107#A1.T6)\)\.

#### 4\.2\.3Single\-Cell Type Discovery

Cell\-type resolution hinges directly on embedded neighbor fidelity\.Graph\-based clustering reads the embedding’s top\-kklist, so exact order governs how cell types resolve\. In single\-cell transcriptomics, unsupervised cell\-type discovery forms the standard endpoint: a community\-detection algorithm partitions thekk\-nearest\-neighbor graph built on 2D coordinates into biological subpopulations\. Graph edges exist solely where one cell falls within another’skknearest embedded neighbors\. Consequently, preserving the ground\-truth top\-kkordering prevents intra\-type fragmentation, directly evaluating the topological fidelity optimized by our rank loss\.

Experimental protocol and oracle\-envelope evaluation\.We benchmark across six annotated single\-cell datasets spanning5,0025\{,\}002to50,21250\{,\}212cells and1414to3535curated cell types \(Table[A\.7](https://arxiv.org/html/2610.41107#A1.T7)\), all preprocessed to 50 principal components\. Akk\-nearest\-neighbor graph \(k=15k=15\) is constructed from the 2D coordinates and clustered via Leiden across eight resolutions in\[0\.1,2\.0\]\[0\.1,2\.0\]\. To isolate geometric quality from resolution choice, each embedding is evaluated at its empirical upper\-envelope resolution that maximizes the target criterion against annotations \(identical oracle protocol across all methods\)\.

Embeddings are evaluated against curated cell\-type annotations using three complementary external criteria\. The Adjusted Rand Index \(ARI\) scores chance\-corrected pairwise accuracy and the Adjusted Mutual Information \(AMI\) information retention across imbalanced populations\. The Fowlkes–Mallows Index \(FMI\) scores geometric precision and recall\.

Table 6:Clustering performance across six annotated single\-cell datasets under the Leiden upper\-envelope protocol\. Bold indicates the top method per cell\. Sign counts and two\-sided Wilcoxon signed\-rankpp\-values evaluate SR\-tSNE against each baseline across all datasets \(p<0\.05p<0\.05bolded\)\.Systematic gains across biological benchmarks\.SR\-tSNE achieves the highest score in 17 of 18 dataset–criterion comparisons \(Table[6](https://arxiv.org/html/2610.41107#S4.T6)\)\. Against t\-SNE, PHATE, TriMap, LocalMAP, and DREAMS, SR\-tSNE wins on all six datasets across all three metrics \(6/66/6,p=0\.031p=0\.031, two\-sided Wilcoxon signed\-rank test\)\. Against UMAP, it wins6/66/6in ARI and FMI \(p=0\.031p=0\.031\) and5/65/6in AMI \(p=0\.062p=0\.062\)\. The margin peaks on Trachea, exceeding UMAP by\+0\.145\+0\.145in ARI and TriMap by\+0\.223\+0\.223\. These gains do not depend on oracle selection: fixing the Leiden resolution globally at1\.01\.0or0\.50\.5preserves SR\-tSNE’s top ARI across all six datasets \(p=0\.031p=0\.031; median gains over t\-SNE:\+23%\+23\\%and\+19%\+19\\%, Table[A\.9](https://arxiv.org/html/2610.41107#A1.T9)\)\.

![Refer to caption](https://arxiv.org/html/2610.41107v1/fig_sc_clustercount.png)Figure 7:Cluster count discrepancies relative to curated ground\-truth annotations across six single\-cell datasets\.\(a\)Signed deviation per dataset \(zero denotes the ground truth\)\.\(b\)Mean absolute error decomposed into over\-clustering \(excess clusters, solid bars\) and under\-clustering \(omitted clusters, hatched bars\)\. SR\-tSNE minimizes spurious cluster fragmentation\.Suppression of artificial cluster fragmentation\.Standard embeddings over\-split homogeneous cell types into spurious sub\-clusters \(Figure[7](https://arxiv.org/html/2610.41107#S4.F7)\)\. Baseline t\-SNE produces12\.212\.2excess clusters on average \(up to\+26\+26on Brain\-Cortex\), while DREAMS and LocalMAP over\-split by\+19\.2\+19\.2and\+9\.7\+9\.7, respectively\. In contrast, SR\-tSNE holds total cluster error to3\.23\.2\(\+0\.2\+0\.2excess,−3\.0\-3\.0short\), the range occupied by the closest peer methods, and leads them on 17 of the 18 dataset–criterion cells of biological alignment \(Table[6](https://arxiv.org/html/2610.41107#S4.T6)\)\. Theλ=0\\lambda=0control over\-splits by\+8\.3\+8\.3, proving that the rank loss drives this consolidation\. By enforcing correct internal rank order, SR\-tSNE maintains cluster connectivity\. Each annotated type is partitioned into fewer disconnected components and retains more cells in the primary component \(0\.7250\.725against0\.6900\.690on Renal\-Cortex\), while the cross\-type edge rate stays within0\.6%0\.6\\%of baseline \(Table[A\.8](https://arxiv.org/html/2610.41107#A1.T8)\)\.

### 4\.3Parameter Sensitivity and Stability

Robust hyperparameter defaults across scales\.SR\-tSNE relies on three intuitive hyperparameters: neighbor countkk, intruder pool sizemm, and regularization weightλ\\lambda\. We fixk=m=10k=m=10across all experiments to align candidate pools with standard retrieval and scoring windows\. The loss weight follows a simple dataset size rule:λ=50\\lambda=50for large datasets \(≥50,000\\geq 50\{,\}000points\) andλ=5\\lambda=5for smaller regimes, calibrated once on MNIST and a market\-equity pool and frozen thereafter\. Performance is robust across scales: SR\-tSNE consistently outperforms t\-SNE across neighborhood sweeps \(k∈\{5,10,15,20\}k\\in\\\{5,10,15,20\\\}, Figure[2](https://arxiv.org/html/2610.41107#S4.F2)\) and across all scales inRN​XR\_\{NX\}AUC \(Table[A\.10](https://arxiv.org/html/2610.41107#A1.T10)\)\. Perturbing initial coordinates across five seeds with1%1\\%Gaussian jitter yields strictly positive gains in all 30 pairwise trials \(Table[A\.17](https://arxiv.org/html/2610.41107#A1.T17)\), verifying stability to initialization\.

Listwise ranking vs\. pairwise alternatives\.The listwise form of the term is what delivers the gain\. Under identical warm starts, budget and tuning, the listwise term holds the highest O\-DLP of the three arms on all eight benchmarks, above both the warm start and a pairwise logistic surrogate over the same candidate pool \(Table[A\.19](https://arxiv.org/html/2610.41107#A1.T19)\)\. It provides\+5%\+5\\%to\+28%\+28\\%relative gains from that same start, so the discounted listwise penalty is what aligns the whole top\-kklist\.

Negligible refinement overhead\.Evaluating wall\-clock time on a dedicated NVIDIA A6000 demonstrates that computing the soft rank loss on candidate pools \(𝒞i\\mathcal\{C\}\_\{i\},K=20K=20\) introduces only1\.8%1\.8\\%–2\.1%2\.1\\%overhead relative to an equal\-budgetλ=0\\lambda=0control \(26\.1626\.16vs\.25\.6925\.69minutes on full MNIST, Table[A\.21](https://arxiv.org/html/2610.41107#A1.T21)\)\. The embedding cost remains dominated by the global t\-SNE gradient computation, which admits standard𝒪⁡\(n​log⁡n\)\\mathcal\{O\}\(n\\log n\)Barnes–Hut or FFT acceleration as detailed in Section[5](https://arxiv.org/html/2610.41107#S5)\.

Refining a longer warm start raises the gain further: from a2,0002\{,\}000\-iteration t\-SNE run, SR\-tSNE leads that run on all eight benchmarks and exceeds its own1,0001\{,\}000\-iteration\-start result \(Table[A\.18](https://arxiv.org/html/2610.41107#A1.T18)\)\.

## 5Discussion

Unified mechanism across downstream tasks\.Our three applications evaluate the identical topological property from complementary perspectives\. The trading scorer measures Spearman rank agreement\. Vision retrieval evaluates the exact position of retrieved items\. Leiden clustering reads the persistent edges within local top\-kkgraphs\. By directly penalizing ordinal violations, the smooth rank loss acts specifically on the exact mechanism that governs downstream performance across these distinct domains\.

Decoupling order fidelity from neighborhood membership\.The rank objective acts on two complementary aspects of a neighborhood: the permutation order among true neighbors and the exclusion of intruders\. Table[3](https://arxiv.org/html/2610.41107#S4.T3)separates the two through the normalized order indexω\\omega, and Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)\(b\) predicts where the affinity\-based baselines sit on it: their ordering pressure follows an affinity gap that can close\. The larger share of the O\-DLP margin comes from restored internal order, governed by the discounted listwise gain, and a further share from membership, enforced by the intruder penalty \(Figure[A\.3](https://arxiv.org/html/2610.41107#A1.F3)\)\. In single\-cell discovery this dual control keeps the Leiden cluster count close to the curated one \(Figure[7](https://arxiv.org/html/2610.41107#S4.F7)\)\.

Robustness across initialization schemes\.The benefits of rank refinement are invariant to the choice of warm start\. Refining CPU\-based Barnes–Hut t\-SNE yields\+5%\+5\\%to\+28%\+28\\%gains in O\-DLP across image and tabular benchmarks\. Concurrently, refining GPU\-accelerated cuML t\-SNE in the algorithmic trading pipeline translates directly into higher net returns, superior Sharpe ratios, and lower turnover, an advantage that reproduces against t\-SNE, the equal\-budget control, UMAP and TriMap when initialized from CPU t\-SNE \(Table S2 of the Supplementary Material\)\. Furthermore, on5,0005\{,\}000\-point subsamples, SR\-tSNE outperforms NeRV, the closest neighbor\-retrieval objective, by factors of2\.9×2\.9\\timesto9\.2×9\.2\\timesin O\-DLP \(Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3)\), establishing that explicit listwise rank bounds surpass soft information\-theoretic divergences\.

Extensions and future directions\.Three extensions follow directly from this framework: \(1\)Sub\-quadratic scaling:The smooth rank loss scales strictly linearly as𝒪⁡\(n⁡\(k\+m\)\)\\mathcal\{O\}\(n\(k\+m\)\), evaluating only the local candidate pool𝒞i\\mathcal\{C\}\_\{i\}\. The computational bottleneck resides entirely in the exact t\-SNE partition function \(𝒪⁡\(n2\)\\mathcal\{O\}\(n^\{2\}\)\) and exhaustive intruder search\. Replacing the exact denominator with FFT\-accelerated interpolation \(FIt\-SNE\) and adopting approximate nearest\-neighbor search brings overall complexity to𝒪⁡\(n​log⁡n\)\\mathcal\{O\}\(n\\log n\)without modifying the rank term\. \(2\)Continuous regularization schedules:While our piecewise size rule \(λ∈\{5,50\}\\lambda\\in\\\{5,50\\\}\) proves robust, dynamically schedulingλ⁡\(n\)\\lambda\(n\)via continuous validation\-based tuning could further optimize convergence trajectories\. \(3\)Parametric extensions:Formulating the soft rank loss as an objective for training deep neural encoders will enable instantaneous out\-of\-sample inference via a single forward pass, bypassing iterative gradient refinement for large\-scale streaming pipelines\.

## 6Conclusion

Summary of contributions\.We introduced SR\-tSNE, which adds a smooth listwise rank loss to t\-SNE, along with Ordinal Degree of Locality Preservation \(O\-DLP\)\. Our core contributions are two\-fold: \(1\)Metric formulation and theoretical control:We formulated O\-DLP to credit a neighbor only when it returns at its exact high\-dimensional rank under a position discount, establishing that the rank loss provably controls the discrete metric deficit \(Theorem[1](https://arxiv.org/html/2610.41107#Thmtheorem1)\) with exponential surrogate convergence \(Theorem[2](https://arxiv.org/html/2610.41107#Thmtheorem2)\)\. \(2\)A rank\-preserving embedding objective:By ranking each point’s high\-dimensional neighbors against mined intruders under a position discount, one simple term forces the embedding to place the right neighbors in the right order with negligible computational overhead \(∼2%\\sim 2\\%\)\.

Empirical validation across downstream tasks\.On eight benchmarks against twelve methods, SR\-tSNE attains the highest exact\-rank preservation \(O\-DLP\), the highest neighborhood overlap \(DLP\) and the highest nDCG, on all eight\. An equal\-budgetλ=0\\lambda\{=\}0control confirms the rank term as the sole active driver\. This gain directly reaches tasks that read neighbor lists from the top down: \(1\)Algorithmic trading:It delivers the best return of six embeddings on 21 of 35 names with higher Sharpe ratios, shallower drawdowns, and fewer trades; \(2\)Computer vision:In label transfer across seven vision datasets, it leads on every one in precision at rank one and in mean reciprocal rank; \(3\)Single\-cell discovery:In clustering across six single\-cell datasets, it leads on 17 of 18 dataset–criterion cells and prevents spurious cluster over\-splitting in Leiden community detection\.

Strengths and overall impact\.The method’s primary strengths are its economy and its exact match to the criterion: one term and one weight turn t\-SNE into an embedding that keeps neighbor order, and O\-DLP measures the exact quantity the term is built to control\. The empirical evidence spans eight benchmarks, twelve methods, three applications, one held\-out corpus, and an equal\-budget control\. Neighbor order is a fundamental property an embedding can and should be asked for, O\-DLP measures whether it was delivered, and delivering it pays wherever a downstream step reads the list from the top\. Beyond non\-linear manifold learning, establishing that smooth listwise surrogates can provably govern discrete ordinal metrics provides a generalizable blueprint for top\-kkrepresentation learning across information retrieval, bioinformatics, and recommendation systems\.

Data and code availability\.All benchmark datasets are publicly available from their original sources\. The implementation of SR\-tSNE and its criteria is released at[https://github\.com/hank08819/SR\-tSNE](https://github.com/hank08819/SR-tSNE), where embeddings and preprocessing scripts are being added\.

## Funding

This work is partially supported by NSF 2229138 and the McCollum Endowed Chair startup fund\.

## Declaration of competing interest

The authors declare that they have no known competing interests\.

## CRediT authorship contribution statement

Henry Han: Conceptualization, Methodology, Software, Supervision, Writing\.Luyu Niu: Methodology, Software, Data curation\.Diane Li: Investigation\.Jiacun Wang: Investigation

## Declaration of generative AI and AI\-assisted technologies in the manuscript preparation process

AI is used for english polish\.

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## Appendix ASupplementary Material

### A\.1Frequently Asked Questions

This section collects the questions most often raised about SR\-tSNE\. Every answer states what the paper establishes and names the section, table or figure that carries the evidence\.

1. Q1\.What is SR\-tSNE?SR\-tSNE is t\-SNE with one added term, the smooth rank loss of Section[3\.1](https://arxiv.org/html/2610.41107#S3.SS1)\. For each point the loss reads itskknearest neighbors in the input space, in order\. It then asks the embedding to return the same neighbors in the same order\. The neighbor that is nearest in the input space should be nearest in the embedding, the second should be second, and so on\. The loss also mines*intruders*: points that lie close to a point in the embedding while belonging to no part of its input\-space neighbor list\. It pushes every intruder behind every true neighbor\. A single weightλ\\lambdasets how much the term counts against the t\-SNE objective, and Section[3](https://arxiv.org/html/2610.41107#S3)states the two together\.
2. Q2\.What gap does SR\-tSNE close?Neighbor\-embedding methods preserve who belongs to a neighborhood\. The order inside that neighborhood is the part SR\-tSNE adds\. Two embeddings can hold the same neighbors and still rank them differently, and SR\-tSNE fixes both the membership and the order\.
3. Q3\.Why does neighbor order matter?Three common pipelines read the neighbor list from the top down\. Retrieval and label transfer act on the item in position one\. Community detection builds its edges from top\-kklists\. The trading scorer of Section[4\.2\.1](https://arxiv.org/html/2610.41107#S4.SS2.SSS1)reads a Spearman agreement between the two spaces\. Each of them improves when the order improves\.
4. Q4\.What is O\-DLP?O\-DLP credits a neighbor when it returns at its exact input\-space rank, Eq\. \([7](https://arxiv.org/html/2610.41107#S3.E7)\)\. DLP credits the same neighbor for being present at any position\. O\-DLP is therefore the strict reading of one neighborhood, and Eq\. \([8](https://arxiv.org/html/2610.41107#S3.E8)\) relates the two\. A third criterion, nDCG, grades the returned list by position and gives partial credit for a near miss\. All three are read atk=10k=10\.
5. Q5\.What is novel here?Three elements are new\. The first is an objective in the t\-SNE family that asks for exact neighbor order\. The second is O\-DLP, a criterion for that property\. The third is a set of guarantees that tie the optimized loss to the reported metric, Section[3\.4](https://arxiv.org/html/2610.41107#S3.SS4)\.
6. Q6\.What are the contributions?The method adds end\-to\-end differentiable rank regularization to t\-SNE\. The theory proves zero loss at perfect order, exponential soft\-to\-exact convergence, a strict order preference, and an upper bound on the order deficit\. The metric extends DLP from membership to exact rank\. The benchmarks cover eight datasets and twelve peer methods\. The applications cover label transfer, cell\-type discovery and intraday trading\.
7. Q7\.How does SR\-tSNE relate to NeRV, RankVisu and triplet methods?NeRV optimizes a retrieval trade\-off defined on neighborhood membership\. RankVisu and local ordinal embedding work from discrete rank relations\. Triplet methods such as TriMap sample relative comparisons\. SR\-tSNE keeps t\-SNE’s continuous similarities and adds one*listwise*term\. Listwise means the term scores a point’s whole top\-kklist at once, rather than one pair at a time\. It therefore combines both sources of information\. Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3)places it beside NeRV at matched scale, where SR\-tSNE leads by2\.9×2\.9\\timesto9\.2×9\.2\\timesin O\-DLP\.
8. Q8\.Why does the candidate pool carry intruders?The loss scores one candidate pool per point: thekktrue neighbors, followed by themmmined intruders\. Intruders give the loss something to rank the true neighbors against\. Withm≥1m\\geq 1, an outside point that moves between two true neighbors raises the loss, which is what locks membership and order together in Theorem[1](https://arxiv.org/html/2610.41107#Thmtheorem1)\. Them=0m=0case is worked out in[A\.8](https://arxiv.org/html/2610.41107#A1.SS8), where the term becomes a differentiable Spearman correlation\.
9. Q9\.Why a listwise penalty?The listwise form of Q7 is what delivers the gain\. The warm start it improves on is the plain t\-SNE run that SR\-tSNE refines\. From that warm start the listwise term lifts O\-DLP by\+5%\+5\\%to\+28%\+28\\%\. On the same candidates, start and budget, the listwise term lifts O\-DLP above a pairwise logistic surrogate on all eight benchmarks, Table[A\.19](https://arxiv.org/html/2610.41107#A1.T19)\. Aligning the whole top\-kklist at once is therefore the active ingredient\.
10. Q10\.Why a soft rank with an annealed temperature?The sigmoid count of Eq\. \([2](https://arxiv.org/html/2610.41107#S3.E2)\) supplies a gradient everywhere, so the discrete ordering becomes trainable\. Annealingτ\\taufrom5\.05\.0to0\.050\.05keeps early gradients broad and late ranks sharp\. Theorem[2](https://arxiv.org/html/2610.41107#Thmtheorem2)closes the gap to the discrete rank exponentially in the margin\. The loss that is optimized and the metric that is reported therefore agree at the end of training\.
11. Q11\.How arekk,mmandλ\\lambdachosen?Every experiment in this paper usesk=m=10k=m=10, the window that retrieval and graph construction read\. The weight follows one size rule:λ=50\\lambda=50atn≥50,000n\\geq 50\{,\}000andλ=5\\lambda=5below it\. The rule was calibrated once and then carried unchanged through every benchmark, application and appendix study,[A\.7](https://arxiv.org/html/2610.41107#A1.SS7)\.
12. Q12\.Does SR\-tSNE use labels?SR\-tSNE is fully unsupervised\. The neighbor list and the intruder pool come from input\-space distances alone\. Labels and curated annotations enter at evaluation time, where they give an independent reading of the geometry\.
13. Q13\.What do the four guarantees give a reader?Theorem[1](https://arxiv.org/html/2610.41107#Thmtheorem1)bounds the exact\-rank deficit by the loss, so minimizing the loss controls the reported metric\. Theorem[2](https://arxiv.org/html/2610.41107#Thmtheorem2)makes the optimized surrogate agree with the discrete rank exponentially fast\. Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)gives a strict preference for the correct order at every inversion\. Theorem[4](https://arxiv.org/html/2610.41107#Thmtheorem4)bounds the order deficit among the neighbors that are kept\. The prefactor of Theorem[1](https://arxiv.org/html/2610.41107#Thmtheorem1)is uniform: it is fixed bykk,mmand the discount, and it holds for every embedding and every dataset\. Proofs are in[A\.14](https://arxiv.org/html/2610.41107#A1.SS14)\.
14. Q14\.What does the rank term add to an affinity objective?Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)\(a\)–\(b\) shows that the order pressure of t\-SNE and UMAP follows an affinity gap\. Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)\(c\) shows that the rank term charges at leastΔK/\(n​G⋆\)\\Delta\_\{K\}/\(nG^\{\\star\}\)for every inversion, whatever the affinities are\. The room this opens is measurable\. On MNIST,32\.9%32\.9\\%of adjacent pairs in a top\-3030list carrypi,j\+1≥pi,jp\_\{i,j\+1\}\\geq p\_\{i,j\}\. At least one such pair occurs for98\.5%98\.5\\%of points\.
15. Q15\.Is the evaluation independent of the loss?Yes, on four counts\. TheRN​XR\_\{NX\}AUC is read off the co\-ranking matrix, so it is independent of the objective\. SR\-tSNE leads it on MNIST and CIFAR\-10, in the local regime and in the global one, Table[A\.10](https://arxiv.org/html/2610.41107#A1.T10)\. Cell types are scored against curated annotations\. Label transfer is scored against class labels, and trading in net return\. All four agree with the O\-DLP ordering\.
16. Q16\.What do the absolute O\-DLP values tell us?O\-DLP counts exact rank matches, so its scale is set by how strict that test is\. Atk=10k=10, an O\-DLP of0\.0930\.093on MNIST means that about one neighbor in ten comes back at its own position, averaged over the ten positions\. SR\-tSNE holds the highest value on all eight benchmarks\. Table[3](https://arxiv.org/html/2610.41107#S4.T3)reads the same quantity against the order\-blind baseline, and that is the share the downstream tasks convert into higher precision at rank one, cleaner Leiden clusters and higher net return\.
17. Q17\.What does theλ=0\\lambda=0control establish?It shares the warm start, the optimizer, the iteration budget and the seed with SR\-tSNE\. The rank term is therefore the single difference between the two arms\. The control lands within2%2\\%of t\-SNE, and the rank term adds\+5%\+5\\%to\+28%\+28\\%in O\-DLP on top of it, Table[2](https://arxiv.org/html/2610.41107#S4.T2)\. The margin is therefore attributable to the term itself\.
18. Q18\.What happens when the warm start is run longer?The rank term adds order on top of a t\-SNE run of any length\. Table[A\.18](https://arxiv.org/html/2610.41107#A1.T18)continues the warm start to2,0002\{,\}000iterations on all eight benchmarks\. Refined from that longer run, SR\-tSNE leads it on all eight benchmarks, by a median of\+10%\+10\\%\. It also stands\+6%\+6\\%above SR\-tSNE refined from the1,0001\{,\}000\-iteration start\.
19. Q19\.What does the paired test certify?Each test is a two\-sided Wilcoxon signed\-rank over datasets, with one observation per dataset\. Eight datasets place the smallest attainable value at0\.00780\.0078, and appat that floor certifies a win on every dataset of the pair\. Table[A\.1](https://arxiv.org/html/2610.41107#A1.T1)gives the size of each win, and the two are read together\. The sign counts are descriptive, so the conclusion they carry stands under any multiplicity correction\.
20. Q20\.How does the result behave across warm starts?It holds from a1,0001\{,\}000\-iteration start on all eight benchmarks and from a2,0002\{,\}000\-iteration start on all eight, where SR\-tSNE leads that longer run by a median of\+10%\+10\\%, Table[A\.18](https://arxiv.org/html/2610.41107#A1.T18)\. It holds for a CPU Barnes–Hut start in the benchmarks and for a GPU cuML start in the trading pipeline, Section[5](https://arxiv.org/html/2610.41107#S5)\.
21. Q21\.How stable is the result across initializations?Five seeds perturb the initial coordinates with1%1\\%Gaussian jitter\. All3030metric differences come out positive,[A\.9](https://arxiv.org/html/2610.41107#A1.SS9)\.
22. Q22\.Why is NeRV compared on5,0005\{,\}000points?The comparison is run at matched scale, on the range the original dredviz implementation is built for\. Every method is re\-run on the same5,0005\{,\}000\-point subsample of all eight benchmarks and scored there, Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3)\. SR\-tSNE holds the best O\-DLP and DLP on every subsample, and leads NeRV by2\.9×2\.9\\timesto9\.2×9\.2\\timesin O\-DLP\.
23. Q23\.Is every peer method scored on every benchmark?Yes\. All twelve are scored on all eight benchmarks, so the tables carry no gaps\. Eleven of them run at full size\. NeRV is scored at matched scale, on the5,0005\{,\}000\-point subsamples of Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3), the largest size its implementation reaches\. Every paired test therefore runs over all eight datasets\.
24. Q24\.What does the trading study deliver?It is a paired comparison of embeddings inside one fixed pipeline\. The features, the DBSCAN\+\+K settings, the execution rule and the four cost levels are identical for every arm, so the embedding objective is the single variable\. At22bps SR\-tSNE takes the best net return on2121of3535names, with a median of\+13\.8%\+13\.8\\%against\+3\.6%\+3\.6\\%for t\-SNE, a median Sharpe ratio of2\.322\.32against0\.630\.63, and fewer round trips, Table[A\.13](https://arxiv.org/html/2610.41107#A1.T13)\. Against a cost\-adjusted buy\-and\-hold baseline it leads on2525of3535names,p=0\.0017p=0\.0017\.
25. Q25\.How does the single\-cell result behave under the resolution choice?The upper\-envelope protocol is applied identically to every method\. The conclusion also carries over to a fixed resolution\. The Adjusted Rand Index scores chance\-corrected agreement with the curated cell\-type labels\. Setting the Leiden resolution globally at1\.01\.0and at0\.50\.5keeps SR\-tSNE first on all six datasets\. The median gains over t\-SNE are\+23%\+23\\%and\+19%\+19\\%, Table[A\.9](https://arxiv.org/html/2610.41107#A1.T9)\. The cluster counts of Figure[7](https://arxiv.org/html/2610.41107#S4.F7)agree, with SR\-tSNE holding the total deviation to3\.23\.2clusters\.
26. Q26\.How is the label\-transfer protocol set up?Embeddings are computed on the full dataset, and a random20%20\\%is then withheld from the retrieval index to serve as queries\. This transductive protocol is applied identically to every method\. The two criteria are precision at rank one \(P1\\text\{P\}\_\{1\}\) and the mean reciprocal rank of the first correct neighbor \(MRR\)\. SR\-tSNE leads on all seven datasets in both\. A donor\-split corpus held out from all tuning is reported in[A\.5](https://arxiv.org/html/2610.41107#A1.SS5), where SR\-tSNE leads t\-SNE by\+30%\+30\\%in O\-DLP\.
27. Q27\.Where in the list does the gain sit?It sits at the head, where the downstream steps read\. At position one SR\-tSNE returns the true nearest neighbor1\.06×1\.06\\timesto1\.41×1\.41\\timesmore often than t\-SNE, for example0\.367→0\.4430\.367\\to 0\.443on MNIST and0\.268→0\.3770\.268\\to 0\.377on Renal\-Cortex, Figure[3](https://arxiv.org/html/2610.41107#S4.F3)and Table[A\.2](https://arxiv.org/html/2610.41107#A1.T2)\. Over all70,00070\{,\}000MNIST queries it returns0\.710\.71of the first three neighbors at their exact rank, against0\.570\.57for t\-SNE\.
28. Q28\.What does the rank term cost?It reads a candidate pool ofK=k\+m=20K=k\+m=20per point, so it costs𝒪⁡\(n⁡\(k\+m\)\)\\mathcal\{O\}\(n\(k\+m\)\)\. That overhead is1\.8%1\.8\\%on full MNIST,26\.1626\.16against25\.6925\.69minutes on one NVIDIA A6000, and2\.1%2\.1\\%on CIFAR\-10, Table[A\.21](https://arxiv.org/html/2610.41107#A1.T21)\.
29. Q29\.How does SR\-tSNE scale?The rank term is linear innnby construction\. Barnes–Hut or FFT interpolation for the t\-SNE partition function, together with approximate nearest\-neighbor search for the intruder pool, brings the whole refinement to𝒪⁡\(n​log⁡n\)\\mathcal\{O\}\(n\\log n\)without changing the rank term, Section[5](https://arxiv.org/html/2610.41107#S5)\.
30. Q30\.Can the rank term be attached to other embeddings, and where is the code?The term reads pairwise distances inside the embedding alone, so any differentiable neighbor\-embedding objective can carry it with the same candidate pool and the same discount\. The implementation and the criteria are released at[https://github\.com/hank08819/SR\-tSNE](https://github.com/hank08819/SR-tSNE)\. Every setting is listed in[A\.7](https://arxiv.org/html/2610.41107#A1.SS7), and the notation is collected in Table[A\.15](https://arxiv.org/html/2610.41107#A1.T15)\.

### A\.2Additional figures and tables

Table A\.1:Effect sizes behind the paired tests: the difference SR\-tSNE minus baseline in O\-DLP and DLP atk=10k\{=\}10, median and range over the datasets the baseline reaches \(nn\)\. Computed from Tables[4](https://arxiv.org/html/2610.41107#S4.T4)and[5](https://arxiv.org/html/2610.41107#S4.T5)\.Table[A\.2](https://arxiv.org/html/2610.41107#A1.T2)reads the gain position by position\. SR\-tSNE leads at the head of every one of the eight lists, and the mean over the ten positions is O\-DLP\(10\)\(10\)\.

Table A\.2:Exact\-hit rate by position: the fraction of points whoserr\-th high\-dimensional neighbor returns at positionrrin the embedding,r=1,…,10r=1,\\dots,10, for t\-SNE, the equal\-budgetλ=0\\lambda\{=\}0control and SR\-tSNE on the eight benchmarks\. The mean over the ten positions is O\-DLP\(10\)\(10\)\.Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3)places every method on one common footing, a5,0005\{,\}000\-point subsample\. SR\-tSNE holds the best O\-DLP and the best DLP on all eight subsamples\.

Table A\.3:Matched\-scale comparison on a5,0005\{,\}000\-point subsample \(seed 42\) of each benchmark: every method is re\-run on the same subsample and scored on its own high\-dimensional neighbor lists atk=10k\{=\}10\. NeRV runs through the original dredviz implementation at its defaults \(λNeRV=0\.1\\lambda\_\{\\text\{NeRV\}\}=0\.1, seed 42\)\. SR\-tSNE uses the recipe of Section[3\.3](https://arxiv.org/html/2610.41107#S3.SS3)at the size\-rule weightλ=5\\lambda\{=\}5;λ=0\\lambda\{=\}0is the equal\-budget control\. Bold marks the best method per row\.At this size SR\-tSNE holds the best O\-DLP and DLP on every subsample, and NeRV reaches1111–34%34\\%of SR\-tSNE’s O\-DLP\. SR\-tSNE leads t\-SNE on every subsample, by\+0%\+0\\%to\+23%\+23\\%in O\-DLP and\+1%\+1\\%to\+2%\+2\\%in DLP, so the gain of the rank term does not depend on the size of the benchmark\.

Table A\.4:nDCG10across all methods\. The ranked list is the embedding’s own ten nearest neighbors\. Relevance is graded by high\-dimensional rank and intruders score zero\. Bold marks the best per dataset\.†NeRV is scored on the5,0005\{,\}000\-point subsamples of Table[A\.3](https://arxiv.org/html/2610.41107#A1.T3), the largest size its implementation reaches\.Table[A\.5](https://arxiv.org/html/2610.41107#A1.T5)lists the seven vision datasets of the label\-transfer protocol and the weight the size rule selects for each\.

Table A\.5:The seven vision datasets of the label\-transfer protocol\.*Queries*is the held\-out fifth used to probe the retrieval index; the remaining points form the index\.ddis the input dimensionality,*Classes*the number of classes, andλ\\lambdafollows the size rule of Section[4](https://arxiv.org/html/2610.41107#S4)\. Sets larger than70,00070\{,\}000points are stratified\-subsampled to that size\.Table[A\.6](https://arxiv.org/html/2610.41107#A1.T6)summarizes label transfer over the seven datasets\. SR\-tSNE leads every baseline on both criteria, at the sign\-count floor of the test\.

Table A\.6:Label transfer summarized over the seven vision datasets\. Each row is one embedding,*Median*is its median score over the seven, and*SR\-tSNE wins on*gives the sign count and the two\-sided paired Wilcoxonppof SR\-tSNE against that embedding\. Bold marks the best median in a column andp<0\.05p<0\.05\. Per\-dataset scores are in Table[A\.20](https://arxiv.org/html/2610.41107#A1.T20)\.P1↑\\uparrowMRR↑\\uparrowEmbeddingMedianSR\-tSNE wins on \(pp\)MedianSR\-tSNE wins on \(pp\)UMAP\.71597/7 \(\.0156\)\.79497/7 \(\.0156\)TriMap\.67227/7 \(\.0156\)\.76117/7 \(\.0156\)t\-SNE\.80357/7 \(\.0156\)\.85377/7 \(\.0156\)LocalMAP\.72677/7 \(\.0156\)\.79857/7 \(\.0156\)DREAMS\.77077/7 \(\.0156\)\.83117/7 \(\.0156\)SR\-tSNE\.8053—\.8547—

Table[A\.7](https://arxiv.org/html/2610.41107#A1.T7)lists the six annotated single\-cell datasets and the curated type counts the clustering is scored against\.

Table A\.7:The six annotated single\-cell datasets of Table[6](https://arxiv.org/html/2610.41107#S4.T6), where*Cells*and*Genes*are the dimensions of the raw count matrix,*Sparsity*the fraction of its entries that are zero and*Types*the number of curated cell types\. The two atlases above50 00050\\,000cells are subsampled as described in Section[4\.2\.3](https://arxiv.org/html/2610.41107#S4.SS2.SSS3), and their three columns describe the subsampled matrix\. All six are then reduced to the same 50 principal components as in Section[4](https://arxiv.org/html/2610.41107#S4)\.Table[A\.8](https://arxiv.org/html/2610.41107#A1.T8)looks inside the graph that Leiden reads\. Under SR\-tSNE each annotated type stays in fewer pieces and keeps more of its cells in the main piece\.

Table A\.8:The1515\-nearest\-neighbor graph of the embedding that Leiden reads \(symmetrized, unweighted\), on the six datasets of Table[6](https://arxiv.org/html/2610.41107#S4.T6)\.*Pieces*is the number of connected components of the subgraph induced by an annotated cell type, averaged over types;*largest share*is the fraction of a type held in its largest component, averaged over types;*cross\-type edges*is the fraction of graph edges joining two types\.λ=0\\lambda\{=\}0and SR\-tSNE are means over the five warm starts\. Bold marks the best of the four per row\.Table[A\.9](https://arxiv.org/html/2610.41107#A1.T9)repeats the clustering with the resolution fixed in advance, so no annotation enters the choice\. SR\-tSNE keeps the top ARI on all six datasets at both settings\.

Table A\.9:Leiden at a resolution fixed in advance \(1\.01\.0, then0\.50\.5\) for every method and dataset, so no annotation enters the choice; each row scores one partition\. ARI per dataset with the sign count and two\-sided Wilcoxonppof SR\-tSNE against each method over the six datasets \(bold wherep<0\.05p<0\.05\), then AMI and FMI averaged over the six datasets and the mean number of clusters found \(the annotations average2323types\)\.λ=0\\lambda\{=\}0and SR\-tSNE are means over five warm starts\. Bold marks the best per row\.Table[A\.10](https://arxiv.org/html/2610.41107#A1.T10)scores a criterion the training objective never sees\. SR\-tSNE takes the top AUC in both the local and the global regime, on both datasets\.

Table A\.10:RN​XR\_\{NX\}AUC on a2020k subsample of MNIST and CIFAR\-10\. The criterion is read off the co\-ranking matrix and is independent of the training objective\. All methods are scored on the same2020k subsample \(seed 42\) from their stored full\-size embeddings;λ=0\\lambda\{=\}0is the equal\-budget control of Table[2](https://arxiv.org/html/2610.41107#S4.T2)\. Bold marks the best method per column\.Table[A\.11](https://arxiv.org/html/2610.41107#A1.T11)gives the source and the data type of each benchmark alongside its dimensions\.

Table A\.11:Overview of the eight benchmark datasets and their data types\. Herennis the number of points,ddthe input dimensionality and*Classes*the number of annotated labels\.Figure[A\.1](https://arxiv.org/html/2610.41107#A1.F1)follows the capital of all 35 names through the quarter, one row per sector group\.

![Refer to caption](https://arxiv.org/html/2610.41107v1/fig_pool35_equity.png)Figure A\.1:Cumulative return of the 35 names at22bps, one row per group\. Buy and hold is charged the same round\-trip cost\.Figure[A\.2](https://arxiv.org/html/2610.41107#A1.F2)scores the markers themselves, by the move that follows them, so the result does not depend on the execution rule\.

![Refer to caption](https://arxiv.org/html/2610.41107v1/fig_marker_edge_v2.png)Figure A\.2:What the markers are worth\. The*edge*of an embedding on one name is the mean move of the close over the next2424bars after its buy markers minus the mean move after its sell markers\.agives the two halves separately, each a median over the3535names of Table[A\.12](https://arxiv.org/html/2610.41107#A1.T12);btracks the median edge at three holding horizons, with the paired Wilcoxonppagainst t\-SNE beside each;cgives the per\-name difference against t\-SNE at two hours\.Figure[A\.3](https://arxiv.org/html/2610.41107#A1.F3)sweeps neighborhood membership acrosskk\. SR\-tSNE holds the top curve atk∈\{5,10\}k\\in\\\{5,10\\\}on every benchmark\.

000\.20\.20\.40\.4DLPMNIST000\.10\.10\.20\.20\.30\.3CIFAR\-10000\.20\.20\.40\.4F\-MNIST000\.20\.20\.40\.4DLPSpeech Commands0\.10\.10\.20\.20\.30\.30\.40\.4Renal\-Cortex0\.10\.10\.20\.20\.30\.30\.40\.4Brain\-Cortex51015200\.20\.20\.40\.40\.60\.6kkDLPCovertype5101520000\.20\.20\.40\.4kkAG Newst\-SNESR\-tSNEUMAPPHATETriMapPaCMAPISOMAPAutoencoderParamPaCMAPSUDEDREAMSLocalMAPFigure A\.3:DLP versus neighborhood sizekkon the same eight benchmarks, with the legend at lower right\. Each panel carries the methods for which a sweep acrosskkwas stored; the four methods added atk=10k\{=\}10in Table[5](https://arxiv.org/html/2610.41107#S4.T5)are scored at that single size and are not swept here\. SR\-tSNE holds the top curve atk∈\{5,10\}k\\in\\\{5,10\\\}on every benchmark\.
### A\.3Per\-name trading results

Tables[A\.12](https://arxiv.org/html/2610.41107#A1.T12)and[A\.13](https://arxiv.org/html/2610.41107#A1.T13)give the per\-name results behind Section[4\.2\.1](https://arxiv.org/html/2610.41107#S4.SS2.SSS1): net return at00and22bps, and the annualized Sharpe ratio, maximum drawdown and number of round trips at22bps, for the3535equities in seven groups of five\. Every arm runs through the identical DBSCAN\+\+K scorer of[A\.4](https://arxiv.org/html/2610.41107#A1.SS4)with the same per\-stock marker percentile, so only the embedding differs between columns\. Figure[5](https://arxiv.org/html/2610.41107#S4.F5)summarizes both tables at22bps\.

Table A\.12:Returns \(%\) on the 35 names, five per group across seven groups\. Bold marks the best of the six per name\.Table A\.13:Risk and activity for the same 35 names at22bps: annualized Sharpe ratio, maximum drawdown \(%\) and number of round trips\. Bold marks the best of the six per name; for round trips no method is marked\.
### A\.4The DBSCAN\+\+K Marker Scorer

The DBSCAN\+\+K marker scorer is set out here\. DBSCAN\[[42](https://arxiv.org/html/2610.41107#bib.bib42)\]is run on the two\-dimensional embedding, and the points it labels as noise are the candidates\. Each candidateiicarries two components\. The geometric one isgi=di/δ¯c⁡\(i\)g\_\{i\}=d\_\{i\}/\\bar\{\\delta\}\_\{c\(i\)\}, wherec⁡\(i\)c\(i\)is the cluster nearest toiianddid\_\{i\}the distance fromiito that cluster\. The scaleδ¯c\\bar\{\\delta\}\_\{c\}is the mean pairwise distance inside clustercc, estimated on a 50\-point sample\. A point sitting far from a tight cluster scores high\.

The order component reads rank agreement\. Letρi\\rho\_\{i\}be the Spearman correlation\[[43](https://arxiv.org/html/2610.41107#bib.bib43)\]between the ranks of the ten high\-dimensional nearest neighbors ofiiand the ranks of those same points by embedding distance\. A candidate whose neighborhood survives the projection hasρi\\rho\_\{i\}near one and is unlikely to be a genuine anomaly, so the component is1−ρi1\-\\rho\_\{i\}\. Both are min–max normalized across the candidate set and added with equal weight,

si=g~i\+\(1−ρi\)~,s\_\{i\}\\;=\\;\\widetilde\{g\}\_\{i\}\\;\+\\;\\widetilde\{\(1\-\\rho\_\{i\}\)\},\(23\)where the tilde denotes that normalization\. The highest\-scoring candidates in each direction are retained as markers, long giving a buy and short a sell\. We refer to this scorer as DBSCAN\+K\.

### A\.5Held\-out Validation

The configuration rule and the gains in order hold on held\-out data\. We add the core Human Lung Cell Atlas\[[29](https://arxiv.org/html/2610.41107#bib.bib29)\], a tissue absent from the eight benchmarks, subsampled to60 00060\\,000cells\. Twenty donors are held out as queries\. Gene selection, scaling and the principal components are fitted on reference cells only\. Queries are projected with the fitted transform\. The corpus then runs the pipeline of Section[4](https://arxiv.org/html/2610.41107#S4)unchanged\. Atn=60 000n=60\\,000the size rule selectsλ=50\\lambda\{=\}50\.

SR\-tSNE leads on the held\-out corpus as well\. Table[A\.14](https://arxiv.org/html/2610.41107#A1.T14)reports the outcome\. SR\-tSNE is ahead of t\-SNE and UMAP on O\-DLP, DLP and nDCG on the query cells, and ahead of t\-SNE by\+30%\+30\\%in O\-DLP\.

Table A\.14:A held\-out dataset: the Human Lung Cell Atlas core, split by donor into48 00048\\,000reference and12 00012\\,000query cells, scored on the query cells\. The size rule of Section[4](https://arxiv.org/html/2610.41107#S4)selectsλ=50\\lambda\{=\}50\. Bold marks the best method per column\.
### A\.6Implementation Details

Table[A\.15](https://arxiv.org/html/2610.41107#A1.T15)collects the notation of the smooth rank loss, its optimization and the analysis of Section[3\.4](https://arxiv.org/html/2610.41107#S3.SS4), with default values in parentheses\. The optimizer of Section[3\.3](https://arxiv.org/html/2610.41107#S3.SS3)is SGD with Polyak momentum,

v\(t\+1\)\\displaystyle v^\{\(t\+1\)\}=μ⋅v\(t\)\+∇Yℒtotal​\(Y\(t\)\),\\displaystyle\\;=\\;\\mu\\cdot v^\{\(t\)\}\\,\+\\,\\nabla\_\{Y\}\\mathcal\{L\}\_\{\\text\{total\}\}\\bigl\(Y^\{\(t\)\}\\bigr\),\(24\)Y\(t\+1\)\\displaystyle Y^\{\(t\+1\)\}=Y\(t\)−η⋅v\(t\+1\),\\displaystyle\\;=\\;Y^\{\(t\)\}\\,\-\\,\\eta\\cdot v^\{\(t\+1\)\},\(25\)withη=10\\eta=10,μ=0\.8\\mu=0\.8andT=1000T=1000\.

For largennwe storePPsparsely, keeping the topkP=90k\_\{P\}\{=\}90entries per row\. The partition functionZZis accumulated in row chunks ofc=10,000c\{=\}10\{,\}000with gradient checkpointing, which cuts peak memory from𝒪⁡\(n2\)\\mathcal\{O\}\(n^\{2\}\)to𝒪⁡\(c​n\)\\mathcal\{O\}\(c\\,n\); the refinement stage remains𝒪⁡\(n2\)\\mathcal\{O\}\(n^\{2\}\)in time\. All gradients come from PyTorch autograd\. EveryTr=50T\_\{r\}=50iterations we recompute each point’s intruder set with the same chunked scan, skipping the point itself and its high\-dimensional neighbor list\. We usem=10m=10intruders throughout\.

Table A\.15:Notation for the smooth rank loss, its optimization and the analysis of Section[3\.4](https://arxiv.org/html/2610.41107#S3.SS4)\. High\-dimensional quantities are fixed before training, while low\-dimensional ones are recomputed from the current embedding\. Default values are given in parentheses\.
### A\.7Hyperparameters

Table[A\.16](https://arxiv.org/html/2610.41107#A1.T16)collects every setting of the two\-stage pipeline, so that a run can be reproduced from this page alone\. Three of them belong to the rank term and the rest are the scikit\-learn Barnes–Hut defaults of the warm start\.

How the three were chosen, and when\.The neighbor countkk, the intruder countmmand the weightλ\\lambdawere fixed once and then carried unchanged through every benchmark, application and appendix study of this paper\. We read O\-DLP on MNIST over the gridk,m∈\{10,30\}k,m\\in\\\{10,30\\\}andλ∈\{5,50,100,200\}\\lambda\\in\\\{5,50,100,200\\\}, which selectedk=m=10k=m=10and the large\-set weightλ=50\\lambda=50; the small\-set weightλ=5\\lambda=5was selected on the200200\-stock pool from which the3535trading names of Section[4\.2\.1](https://arxiv.org/html/2610.41107#S4.SS2.SSS1)are drawn\. No dataset outside those two entered the choice, and no per\-dataset tuning was performed afterwards\. Settingk=m=10k=m=10also makes the list the loss ranks the same list that O\-DLP, nDCG and the trading scorer read, so the objective and the criteria see one neighborhood\.

What the settings mean\.The warm start is a standard Barnes–Hut t\-SNE: PCA initialization, perplexity3030, learning ratemax⁡\(n/12,50\)\\max\(n/12,50\), early exaggeration1212and1,0001\{,\}000iterations\. The refinement runs for anotherT=1,000T=1\{,\}000iterations atη=10\\eta=10with Polyak momentumμ=0\.8\\mu=0\.8and no early exaggeration\. The value ofη\\etaputs the KL gradient and the rank gradient on the same scale\. A pure\-KL run under this setting therefore stays close to its warm start, which is why theλ=0\\lambda\{=\}0control of Table[4](https://arxiv.org/html/2610.41107#S4.T4)reproduces t\-SNE\. The temperature falls linearly fromτstart=5\.0\\tau\_\{\\text\{start\}\}=5\.0toτend=0\.05\\tau\_\{\\text\{end\}\}=0\.05, a range in which Theorem[2](https://arxiv.org/html/2610.41107#Thmtheorem2)puts the soft rank within𝒪\(e−γi/0\.05\)\\mathcal\{O\}\(e^\{\-\\gamma\_\{i\}/0\.05\}\)of the discrete rank at the end of training\. Intruders are re\-mined everyTr=50T\_\{r\}=50iterations\. Forn≥50,000n\\geq 50\{,\}000the affinity matrix keepskP=90k\_\{P\}=90entries per row and the partition function is accumulated in row chunks ofc=10,000c=10\{,\}000\.

Sensitivity and reproducibility\.The behavior of the three settings away from their defaults is reported in Section[4\.3](https://arxiv.org/html/2610.41107#S4.SS3): the neighborhood sweep of Figure[2](https://arxiv.org/html/2610.41107#S4.F2)\(k∈\{5,10,15,20\}k\\in\\\{5,10,15,20\\\}\), theλ=0\\lambda\{=\}0control of Table[4](https://arxiv.org/html/2610.41107#S4.T4), them=0m\{=\}0case of[A\.8](https://arxiv.org/html/2610.41107#A1.SS8)and the five perturbed warm starts of[A\.9](https://arxiv.org/html/2610.41107#A1.SS9)\. All runs use seed4242and one NVIDIA A6000; the peer methods run at their published settings, described in Section[4](https://arxiv.org/html/2610.41107#S4)\. The implementation that produces these settings is released with the paper\.

Table A\.16:Full hyperparameter settings for the SR\-tSNE two\-stage pipeline\. The warm start and the refinement stage are held fixed across every dataset\. Onlyλ\\lambdafollows dataset size, at5050above50 00050\\,000points and55below\.StageParameterValueWarm start \(scikit\-learn Barnes–Hut\)InitializationPCAPerplexity30Learning ratemax⁡\(n/12,50\)\\max\(n/12,\\,50\)Early exaggeration12Iterations1 000Refinement \(PyTorch, exact KL\)Learning rateη\\eta10Momentumμ\\mu0\.8IterationsTT1 000Early exaggerationdisabledτstart→τend\\tau\_\{\\text\{start\}\}\\to\\tau\_\{\\text\{end\}\}5\.0→0\.055\.0\\to 0\.05\(linear\)SparsitykPk\_\{P\}90Chunk sizecc10 000Rank lossLoss \(all datasets\)smooth rank,m=10m=10,Tr=50T\_\{r\}=50λ\\lambda, large datasets \(n≥50,000n\\geq 50\{,\}000\)50λ\\lambda, other datasets5Neighbor countkk10

### A\.8Them=0m\{=\}0Special Case

Withm=0m\{=\}0no intruders are used\. The candidate pool is then the neighbor list itself,𝒞i=𝒩i\\mathcal\{C\}\_\{i\}=\\mathcal\{N\}\_\{i\}\. Nothing outside𝒩i\\mathcal\{N\}\_\{i\}enters the objective, so the loss no longer defends membership\. It scores only the order inside the neighborhood\. In this case the position discount of Eqs\. \([3](https://arxiv.org/html/2610.41107#S3.E3)\)–\([5](https://arxiv.org/html/2610.41107#S3.E5)\) can be replaced by the Spearman rank correlation between the two rank sequences\. For pointiithis is the Pearson correlation of the centered rank vectors\. The usual closed form for Spearman’sρ\\rhoassumes distinct integer ranks\. Soft ranks do not satisfy that assumption, so we use the general form here\. Sinceri,jin=jr\_\{i,j\}^\{\\text\{in\}\}=j, the high\-dimensional mean is the constantr¯in=\(k\+1\)/2\\bar\{r\}^\{\\text\{in\}\}=\(k\+1\)/2\. The low\-dimensional mean isr¯iemb=1k​∑j=1kri,jemb\\bar\{r\}\_\{i\}^\{\\text\{emb\}\}=\\frac\{1\}\{k\}\\sum\_\{j=1\}^\{k\}r\_\{i,j\}^\{\\text\{emb\}\}\. The per\-point Spearman correlation is:

ρi=∑j=1k\(j−k\+12\)​\(ri,jemb−r¯iemb\)∑j=1k\(j−k\+12\)2⋅∑j=1k\(ri,jemb−r¯iemb\)2\+ε,\\rho\_\{i\}\\;=\\;\\frac\{\\displaystyle\\sum\_\{j=1\}^\{k\}\\Bigl\(j\-\\tfrac\{k\+1\}\{2\}\\Bigr\)\\Bigl\(r\_\{i,j\}^\{\\text\{emb\}\}\-\\bar\{r\}\_\{i\}^\{\\text\{emb\}\}\\Bigr\)\}\{\\sqrt\{\\displaystyle\\sum\_\{j=1\}^\{k\}\\Bigl\(j\-\\tfrac\{k\+1\}\{2\}\\Bigr\)^\{\\\!2\}\}\\;\\cdot\\;\\sqrt\{\\displaystyle\\sum\_\{j=1\}^\{k\}\\Bigl\(r\_\{i,j\}^\{\\text\{emb\}\}\-\\bar\{r\}\_\{i\}^\{\\text\{emb\}\}\\Bigr\)^\{\\\!2\}\}\\;\+\\;\\varepsilon\},\(26\)whereε=10−8\\varepsilon=10^\{\-8\}prevents division by zero\. In the denominator, the factor built from the high\-dimensional ranks,∑j\(j−\(k\+1\)/2\)2\\sqrt\{\\sum\_\{j\}\(j\-\(k\+1\)/2\)^\{2\}\}, is a positive constant set bykkalone\. The gradient therefore flows entirely throughri,jembr\_\{i,j\}^\{\\text\{emb\}\}\.

The mean Spearman correlation over allnnpoints is:

ρs=1n​∑i=1nρi,\\rho\_\{s\}\\;=\\;\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\rho\_\{i\},\(27\)and the rank loss is then:

ℒrank​\(Y\)=1−ρs=1n​∑i=1n\(1−min⁡\(max⁡\(ρi,−1\),1\)\),\\mathcal\{L\}\_\{\\text\{rank\}\}\(Y\)\\;=\\;1\-\\rho\_\{s\}\\;=\\;\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\bigl\(1\-\\min\\\!\\bigl\(\\max\(\\rho\_\{i\},\\,\-1\),\\,1\\bigr\)\\bigr\),\(28\)This lies in\[0,2\]\[0,2\]\. It is zero only when every point’s high\-dimensional neighbor order is reproduced exactly in the embedding\. Themin⁡\(max⁡\(⋅,−1\),1\)\\min\(\\max\(\\cdot,\\,\-1\),\\,1\)projection guards against floating\-point round\-off inρi\\rho\_\{i\}\.

### A\.9Multi\-seed Stability

Paired seed statistics\. Table[A\.17](https://arxiv.org/html/2610.41107#A1.T17)reports the paired advantage of SR\-tSNE over theλ=0\\lambda\{=\}0control across five warm starts perturbed with1%1\\%Gaussian jitter\. Every mean is positive, every one of the3030metric differences over the ten paired runs \(two datasets, five perturbed starts\) is positive, and standard deviations are at most0\.00050\.0005\. The jitter moves each point by about the spacing of its nearest neighbors, so the perturbed start carries much less local order\. On MNIST its O\-DLP after theλ=0\\lambda\{=\}0refinement is\.056\.056, against\.078\.078for the unperturbed t\-SNE\. The rank term brings the same start to\.092\.092, the level of the unperturbed SR\-tSNE run of Table[2](https://arxiv.org/html/2610.41107#S4.T2)\. The paired advantage is therefore larger than the gap in that table, and it measures the term’s ability to recover order as well as to add it\.

Table A\.17:Paired advantage of SR\-tSNE over the equal\-budget control, mean±\\pmstd over five warm starts perturbed with1%1\\%Gaussian jitter\. Both methods share each seed’s start, so every comparison is paired\. All3030metric differences over the ten paired runs are positive\.
### A\.10Warm\-start length

The gain does not depend on stopping the warm start at1,0001\{,\}000iterations\. Table[A\.18](https://arxiv.org/html/2610.41107#A1.T18)continues it to2,0002\{,\}000and refines from there\. SR\-tSNE leads that longer run on all eight benchmarks, by a median of\+10%\+10\\%, and stands\+6%\+6\\%above its own1,0001\{,\}000\-iteration\-start result\.

Table A\.18:Two warm starts\. Left: the paper’s t\-SNE run of1 0001\\,000iterations and SR\-tSNE refined from it\. Right: a t\-SNE run continued to2 0002\\,000iterations, theλ=0\\lambda\{=\}0control and SR\-tSNE refined from that longer run, and the gain of SR\-tSNE over the longer run\. O\-DLP atk=10k\{=\}10\.
### A\.11The form of the rank term

The listwise form is what carries the gain\. Table[A\.19](https://arxiv.org/html/2610.41107#A1.T19)replaces it with a pairwise logistic surrogate on the same candidate pool, the same warm start and the same budget\. SR\-tSNE stands above that surrogate on all eight benchmarks\.

Table A\.19:The form of the rank term\. From the same scikit\-learn warm start, with the same1010neighbors and1010mined intruders, optimizer and budget, the refinement runs with a pairwise logistic ordinal loss at the weight selected on MNIST \(λ=0\.05\\lambda=0\.05; larger weights lower every criterion\) and with the listwise discounted gain of SR\-tSNE\. All criteria atk=10k\{=\}10\.
### A\.12Per\-dataset label\-transfer results

Table[A\.20](https://arxiv.org/html/2610.41107#A1.T20)gives the per\-dataset scores behind the summary of Table[A\.6](https://arxiv.org/html/2610.41107#A1.T6)\. A random fifth of each dataset is removed from the retrieval index and used as queries; the embedding is fitted on all points, so the protocol is transductive and only the index is reduced\. Each query is scored from its ten nearest reference points in the two\-dimensional embedding:P1\\text\{P\}\_\{1\}asks whether position one already carries the query’s class, and MRR takes the reciprocal of the first position that does\. Both criteria therefore read a neighbor’s position, not merely its presence\. The last column,Δ\\Delta, is SR\-tSNE minus the equal\-budgetλ=0\\lambda\{=\}0control, so the control’s own score on each row is the SR\-tSNE entry minusΔ\\Delta\. The two rows closing each block give the sign count and the two\-sided Wilcoxon signed\-rank test over the seven datasets\.

Table A\.20:Label transfer across seven vision datasets, one paired observation each, withΔ\\Deltataken against the equal\-budget control \(λ=0\\lambda\{=\}0\), whose own score on each row is SR\-tSNE minusΔ\\Delta\. Every query is scored from its ten nearest reference points in the two\-dimensional embedding, on the same held\-out fifth of each dataset\. Bold marks the best score in a row\. The two rows below each block give the sign count and the two\-sided Wilcoxon signed\-rank test over the seven datasets, where bold marksp<0\.05p<0\.05\.Datasetnnλ\\lambdaUMAPTriMapt\-SNELocalMAPDREAMSSR\-tSNEΔ\\DeltaP1: is the first neighbor returned of the query’s classMNIST70 00050\.8811\.8351\.9288\.9164\.9019\.9354\+0\.0063\+0\.0063Fashion\-MNIST70 00050\.7159\.6877\.8296\.7471\.8173\.8382\+0\.0086\+0\.0086CIFAR\-1060 00050\.7218\.6722\.8035\.7267\.7707\.8053\+0\.0016\+0\.0016Tiny\-ImageNet70 00050\.2479\.1689\.4561\.2706\.4044\.4584\+0\.0023\+0\.0023EMNIST\-Balanced69 98350\.4608\.3174\.6728\.5692\.6090\.6888\+0\.0164\+0\.0164CIFAR\-10060 00050\.3280\.2438\.5466\.3356\.4842\.5522\+0\.0058\+0\.0058USPS9 2985\.9301\.9091\.9602\.9381\.9478\.9634\+0\.0032\+0\.0032SR\-tSNE wins on7/77/77/77/77/7two\-sidedpp\.0156\.0156\.0156\.0156\.0156MRR: reciprocal rank of the first correct neighborMNIST70 00050\.9151\.8835\.9459\.9394\.9299\.9507\+0\.0046\+0\.0046Fashion\-MNIST70 00050\.8038\.7812\.8802\.8262\.8721\.8858\+0\.0057\+0\.0057CIFAR\-1060 00050\.7949\.7611\.8537\.7985\.8311\.8547\+0\.0009\+0\.0009Tiny\-ImageNet70 00050\.3401\.2632\.5291\.3651\.4868\.5313\+0\.0021\+0\.0021EMNIST\-Balanced69 98350\.5891\.4581\.7522\.6796\.7079\.7619\+0\.0099\+0\.0099CIFAR\-10060 00050\.4321\.3521\.6215\.4420\.5749\.6247\+0\.0034\+0\.0034USPS9 2985\.9503\.9325\.9677\.9524\.9594\.9693\+0\.0017\+0\.0017SR\-tSNE wins on7/77/77/77/77/7two\-sidedpp\.0156\.0156\.0156\.0156\.0156
### A\.13Runtime

Table[A\.21](https://arxiv.org/html/2610.41107#A1.T21)times embedding construction alone, on one NVIDIA A6000 with exclusive access; neighbor search and the evaluation of the criteria are excluded, and every method is timed on the same host\. The last two rows differ by the rank term only, so their difference is the cost of the term itself:1\.8%1\.8\\%of the refinement stage on MNIST and2\.1%2\.1\\%on CIFAR\-10, which is where the cost of the method sits\. The refinement in this table evaluates the exact partition function, and Section[5](https://arxiv.org/html/2610.41107#S5)gives theO⁡\(n​log⁡n\)O\(n\\log n\)route that keeps the rank term unchanged\.

Table A\.21:Wall\-clock runtime in minutes on one NVIDIA A6000, timing only embedding construction\. The two refinement methods differ only in the rank term\.
### A\.14Proofs

This section proves the four theorems of Section[3\.4](https://arxiv.org/html/2610.41107#S3.SS4), together with the two lemmas, the two propositions and the corollary they rest on\. Results labelled with an S, namely Lemmas S1 and S2, Propositions S1 and S2 and Corollary S1, are stated and proved here\. Every other number refers to the main text\.

Fix a pointii\. Its candidate pool𝒞i=𝒩i⊕ℐi\\mathcal\{C\}\_\{i\}=\\mathcal\{N\}\_\{i\}\\oplus\\mathcal\{I\}\_\{i\}has lengthK=k\+mK=k\+m\. Candidatej≤kj\\leq kis thejj\-th true neighbor and carries relevancerelj=k−j\+1\\mathrm\{rel\}\_\{j\}=k\-j\+1; candidatesj\>kj\>kare intruders and carryrelj=0\\mathrm\{rel\}\_\{j\}=0\. WriteRiR\_\{i\}for the retained neighbors ofii, the true neighbors that the embedding returns among itsk′k^\{\\prime\}nearest points, andposi​\(j\)\\mathrm\{pos\}\_\{i\}\(j\)for the position at which the embedding returns thejj\-th true neighbor\. The soft and discrete gains of pointiiareGisoftG^\{\\text\{soft\}\}\_\{i\}andGihardG^\{\\text\{hard\}\}\_\{i\}, the first read with the soft ranks of Eq\. \([2](https://arxiv.org/html/2610.41107#S3.E2)\) and the second with the integer ranks\. The embedding orders𝒞i\\mathcal\{C\}\_\{i\}by distance toyiy\_\{i\}; call that orderingπi\\pi\_\{i\}, which orders the candidate pool and is distinct from the neighbor mapsπiin\\pi\_\{i\}^\{\\text\{in\}\}andπiemb\\pi\_\{i\}^\{\\text\{emb\}\}of Eq\. \([7](https://arxiv.org/html/2610.41107#S3.E7)\), so thatπi​\(r\)\\pi\_\{i\}\(r\)is the candidate at positionrrandπi−1​\(j\)\\pi\_\{i\}^\{\-1\}\(j\)the position of candidatejj\. Ties among intruders are immaterial because their relevance is equal; ties elsewhere are broken arbitrarily\. The hard per\-point gain and loss are

Gihard​\(πi\)=∑r=1Krelπi​\(r\)​disc​\(r\),ℓi=1−Gihard​\(πi\)G⋆,ℒrankhard=1n​∑iℓi,G^\{\\text\{hard\}\}\_\{i\}\(\\pi\_\{i\}\)=\\sum\_\{r=1\}^\{K\}\\mathrm\{rel\}\_\{\\pi\_\{i\}\(r\)\}\\,\\mathrm\{disc\}\(r\),\\qquad\\ell\_\{i\}=1\-\\frac\{G^\{\\text\{hard\}\}\_\{i\}\(\\pi\_\{i\}\)\}\{G^\{\\star\}\},\\qquad\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}=\\frac\{1\}\{n\}\\sum\_\{i\}\\ell\_\{i\},\(29\)withdisc⁡\(r\)=1/log2⁡\(1\+r\)\\mathrm\{disc\}\(r\)=1/\\log\_\{2\}\(1\+r\)andG⋆G^\{\\star\}as in Eq\. \([4](https://arxiv.org/html/2610.41107#S3.E4)\)\. ThisGihardG^\{\\text\{hard\}\}\_\{i\}is the discrete gain of Eq\. \([13](https://arxiv.org/html/2610.41107#S3.E13)\) re\-indexed by position; intruders carry relevance zero, so the two sums agree\. Two facts about the discount are used repeatedly: it is strictly decreasing, and its stepsdisc⁡\(r\)−disc⁡\(r\+1\)\\mathrm\{disc\}\(r\)\-\\mathrm\{disc\}\(r\+1\)are strictly decreasing inrr, because1/log2⁡\(1\+r\)1/\\log\_\{2\}\(1\+r\)is convex\. Hencedisc⁡\(r\)−disc⁡\(r′\)≥ΔK\\mathrm\{disc\}\(r\)\-\\mathrm\{disc\}\(r^\{\\prime\}\)\\geq\\Delta\_\{K\}for all1≤r<r′≤K1\\leq r<r^\{\\prime\}\\leq K\.

##### A sorting lemma

Call a pair of candidates\(a,b\)\(a,b\)a*strict inversion*ofπi\\pi\_\{i\}ifrela\>relb\\mathrm\{rel\}\_\{a\}\>\\mathrm\{rel\}\_\{b\}andπi−1​\(a\)\>πi−1​\(b\)\\pi\_\{i\}^\{\-1\}\(a\)\>\\pi\_\{i\}^\{\-1\}\(b\)\. Letinv⁡\(πi\)\\mathrm\{inv\}\(\\pi\_\{i\}\)be their number\. The ideal ordering, neighbors1,…,k1,\\dots,kin order followed by the intruders, has none\.

*This lemma says that every wrongly ordered pair costs the gain at least one smallest step of the discount\.*

###### Lemma S1

G⋆−Gihard​\(πi\)≥inv⁡\(πi\)​ΔKG^\{\\star\}\-G^\{\\text\{hard\}\}\_\{i\}\(\\pi\_\{i\}\)\\;\\geq\\;\\mathrm\{inv\}\(\\pi\_\{i\}\)\\,\\Delta\_\{K\}\.

###### Proof 1

Ifinv⁡\(πi\)\>0\\mathrm\{inv\}\(\\pi\_\{i\}\)\>0there is an adjacent position pair\(r,r\+1\)\(r,r\+1\)withrelπi​\(r\)<relπi​\(r\+1\)\\mathrm\{rel\}\_\{\\pi\_\{i\}\(r\)\}<\\mathrm\{rel\}\_\{\\pi\_\{i\}\(r\+1\)\}; otherwise relevance would be non\-increasing along the list and no strict inversion could exist\. Swapping the two candidates raises the gain by

\(relπi​\(r\+1\)−relπi​\(r\)\)​\(disc⁡\(r\)−disc⁡\(r\+1\)\)≥1⋅ΔK,\\bigl\(\\mathrm\{rel\}\_\{\\pi\_\{i\}\(r\+1\)\}\-\\mathrm\{rel\}\_\{\\pi\_\{i\}\(r\)\}\\bigr\)\\bigl\(\\mathrm\{disc\}\(r\)\-\\mathrm\{disc\}\(r\+1\)\\bigr\)\\;\\geq\\;1\\cdot\\Delta\_\{K\},\(30\)since relevances are integers, and it lowersinv\\mathrm\{inv\}by exactly one: the swapped pair stops being an inversion and no other pair changes its relative order\. Repeating until no strict inversion remains takes exactlyinv⁡\(πi\)\\mathrm\{inv\}\(\\pi\_\{i\}\)swaps and ends at an ordering whose gain isG⋆G^\{\\star\}, because relevance is then non\-increasing along the list and, intruders being interchangeable, that ordering is the ideal one\. Summing the gains gives the claim\. ∎

##### Proof of Theorem[1](https://arxiv.org/html/2610.41107#Thmtheorem1)

Letk′≤min⁡\(k,m\)k^\{\\prime\}\\leq\\min\(k,m\)and letTiT\_\{i\}be thek′k^\{\\prime\}points nearest toyiy\_\{i\}among alln−1n\-1others\. At mostk′k^\{\\prime\}of them are non\-neighbors, and any non\-neighbor inTiT\_\{i\}is one of thek′k^\{\\prime\}nearest non\-neighbors, hence one of them≥k′m\\geq k^\{\\prime\}nearest, hence inℐi\\mathcal\{I\}\_\{i\}\. SoTi⊂𝒞iT\_\{i\}\\subset\\mathcal\{C\}\_\{i\}, and the position of each element ofTiT\_\{i\}inπi\\pi\_\{i\}equals its position among alln−1n\-1points\. O\-DLP atk′k^\{\\prime\}therefore reads the firstk′k^\{\\prime\}positions ofπi\\pi\_\{i\}:

1−O\-DLPi​\(k′\)=1k′​\#​\{r≤k′:πi​\(r\)≠r\}=:missik′\.1\-\\text\{O\-DLP\}\_\{i\}\(k^\{\\prime\}\)\\;=\\;\\frac\{1\}\{k^\{\\prime\}\}\\,\\\#\\\{\\,r\\leq k^\{\\prime\}:\\pi\_\{i\}\(r\)\\neq r\\,\\\}\\;=:\\;\\frac\{\\mathrm\{miss\}\_\{i\}\}\{k^\{\\prime\}\}\.\(31\)Call a candidate*displaced*if it does not sit where the ideal ordering puts it, that is, a neighborjjwithπi−1​\(j\)≠j\\pi\_\{i\}^\{\-1\}\(j\)\\neq jor an intruder at a position≤k\\leq k\. Every miss is a position holding a displaced candidate, somissi≤\#​displaced\\mathrm\{miss\}\_\{i\}\\leq\\\#\\mathrm\{displaced\}\. Each swap in the proof of Lemma[S1](https://arxiv.org/html/2610.41107#Thmlemma1)moves two candidates and the process ends with none displaced, so\#​displaced≤2​inv​\(πi\)\\\#\\mathrm\{displaced\}\\leq 2\\,\\mathrm\{inv\}\(\\pi\_\{i\}\)\. With Lemma[S1](https://arxiv.org/html/2610.41107#Thmlemma1),

1−O\-DLPi​\(k′\)≤2​inv​\(πi\)k′≤2​\(G⋆−Gihard\)k′​ΔK=2​G⋆k′​ΔK​ℓi\.1\-\\text\{O\-DLP\}\_\{i\}\(k^\{\\prime\}\)\\;\\leq\\;\\frac\{2\\,\\mathrm\{inv\}\(\\pi\_\{i\}\)\}\{k^\{\\prime\}\}\\;\\leq\\;\\frac\{2\\,\(G^\{\\star\}\-G^\{\\text\{hard\}\}\_\{i\}\)\}\{k^\{\\prime\}\\,\\Delta\_\{K\}\}\\;=\\;\\frac\{2\\,G^\{\\star\}\}\{k^\{\\prime\}\\,\\Delta\_\{K\}\}\\,\\ell\_\{i\}\.\(32\)Averaging overiigives Eq\. \([14](https://arxiv.org/html/2610.41107#S3.E14)\)\.

For the equivalence: ifℒrankhard=0\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}=0then everyℓi=0\\ell\_\{i\}=0, so by Lemma[S1](https://arxiv.org/html/2610.41107#Thmlemma1)everyπi\\pi\_\{i\}has no strict inversion, so thekkneighbors ofiioccupy positions1,…,k1,\\dots,kin order and every intruder comes after them\. A non\-candidate is farther fromyiy\_\{i\}than every intruder, hence farther than every neighbor\. So thekknearest points toyiy\_\{i\}are itskkneighbors in high\-dimensional order, andO\-DLPi​\(k′\)=1\\text\{O\-DLP\}\_\{i\}\(k^\{\\prime\}\)=1for everyk′≤kk^\{\\prime\}\\leq k\. Conversely, ifO\-DLP​\(k\)=1\\text\{O\-DLP\}\(k\)=1then for everyiithekknearest points are the neighbors in order, so every intruder, being a non\-neighbor, is farther than all of them;πi\\pi\_\{i\}is the ideal ordering andℓi=0\\ell\_\{i\}=0\. ∎

##### Proof of Theorem[2](https://arxiv.org/html/2610.41107#Thmtheorem2)

Eq\. \([2](https://arxiv.org/html/2610.41107#S3.E2)\) writes the soft rank of candidatejjas1\+∑l≠jσ⁡\(\(di,j2−di,l2\)/τ\)1\+\\sum\_\{l\\neq j\}\\sigma\\bigl\(\(d^\{2\}\_\{i,j\}\-d^\{2\}\_\{i,l\}\)/\\tau\\bigr\), and the hard rank as1\+∑l≠j𝟏\[di,l2<di,j2\]1\+\\sum\_\{l\\neq j\}\\mathbf\{1\}\[d^\{2\}\_\{i,l\}<d^\{2\}\_\{i,j\}\]\. Each summand differs from its indicator byσ\(−\|di,j2−di,l2\|/τ\)≤e−γi/τ\\sigma\(\-\|d^\{2\}\_\{i,j\}\-d^\{2\}\_\{i,l\}\|/\\tau\)\\leq e^\{\-\\gamma\_\{i\}/\\tau\}, and there areK−1K\-1summands\. That is the first claim\. For the second,disc\\mathrm\{disc\}has derivative−ln2/\(\(1\+r\)ln2\(1\+r\)\)\-\\ln 2/\\bigl\(\(1\+r\)\\ln^\{2\}\(1\+r\)\\bigr\), whose absolute value on\[1,K\]\[1,K\]is largest atr=1r=1, where it equals1/\(2​ln⁡2\)1/\(2\\ln 2\)\. Hence

\|Gisoft−Gihard\|≤∑j=1krelj12​ln⁡2\(K−1\)e−γi/τ=k⁡\(k\+1\)2⋅K−12​ln⁡2e−γi/τ,\|G^\{\\text\{soft\}\}\_\{i\}\-G^\{\\text\{hard\}\}\_\{i\}\|\\;\\leq\\;\\sum\_\{j=1\}^\{k\}\\mathrm\{rel\}\_\{j\}\\,\\frac\{1\}\{2\\ln 2\}\\,\(K\-1\)\\,e^\{\-\\gamma\_\{i\}/\\tau\}\\;=\\;\\frac\{k\(k\+1\)\}\{2\}\\cdot\\frac\{K\-1\}\{2\\ln 2\}\\,e^\{\-\\gamma\_\{i\}/\\tau\},\(33\)and dividing byG⋆G^\{\\star\}gives the bound on\|ℓisoft−ℓihard\|\|\\ell^\{\\text\{soft\}\}\_\{i\}\-\\ell^\{\\text\{hard\}\}\_\{i\}\|\. ∎

##### Proof of Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)

Under the row relaxation each objective is evaluated by its formula on the arrayDD\. The row swap changes only the two entriesdi,j2d^\{2\}\_\{i,j\}anddi,l2d^\{2\}\_\{i,l\}, so only the terms that read those entries change\.

\(a\)Up to a constant inYY,KL=∑a≠bpa​b​ln⁡\(1\+da,b2\)\+ln⁡Z\\mathrm\{KL\}=\\sum\_\{a\\neq b\}p\_\{ab\}\\ln\(1\+d^\{2\}\_\{a,b\}\)\+\\ln ZwithZ=∑a≠b\(1\+da,b2\)−1Z=\\sum\_\{a\\neq b\}\(1\+d^\{2\}\_\{a,b\}\)^\{\-1\}\([A\.6](https://arxiv.org/html/2610.41107#A1.SS6)\)\.ZZis a symmetric function of the entries ofDDand is unchanged by the swap\. Of the first sum, only the terms\(i,j\)\(i,j\)and\(i,l\)\(i,l\)change, and

KL⁡\(D\)−KL⁡\(D′\)\\displaystyle\\mathrm\{KL\}\(D\)\-\\mathrm\{KL\}\(D^\{\\prime\}\)=pi​j​ln⁡\(1\+di,j2\)\+pi​l​ln⁡\(1\+di,l2\)\\displaystyle=p\_\{ij\}\\ln\(1\+d^\{2\}\_\{i,j\}\)\+p\_\{il\}\\ln\(1\+d^\{2\}\_\{i,l\}\)−pi​j​ln⁡\(1\+di,l2\)−pi​l​ln⁡\(1\+di,j2\)\\displaystyle\\quad\-p\_\{ij\}\\ln\(1\+d^\{2\}\_\{i,l\}\)\-p\_\{il\}\\ln\(1\+d^\{2\}\_\{i,j\}\)=\(pi​j−pi​l\)​\[ln⁡\(1\+di,j2\)−ln⁡\(1\+di,l2\)\]\.\\displaystyle=\(p\_\{ij\}\-p\_\{il\}\)\\bigl\[\\ln\(1\+d^\{2\}\_\{i,j\}\)\-\\ln\(1\+d^\{2\}\_\{i,l\}\)\\bigr\]\.\(34\)The inversiondi,l2<di,j2d^\{2\}\_\{i,l\}<d^\{2\}\_\{i,j\}makes the bracket positive\.

\(b\)The UMAP objective isCE=−∑a≠b\[va​blnwa​b\+\(1−va​b\)ln\(1−wa​b\)\]\\mathrm\{CE\}=\-\\sum\_\{a\\neq b\}\\bigl\[v\_\{ab\}\\ln w\_\{ab\}\+\(1\-v\_\{ab\}\)\\ln\(1\-w\_\{ab\}\)\\bigr\]withwa​b=w⁡\(da,b2\)w\_\{ab\}=w\(d^\{2\}\_\{a,b\}\)\. Write each term as−va​b​logit​\(wa​b\)−ln⁡\(1−wa​b\)\-v\_\{ab\}\\,\\mathrm\{logit\}\(w\_\{ab\}\)\-\\ln\(1\-w\_\{ab\}\)\. The second part depends onDDonly through the multiset of the two swapped entries and cancels in the difference\. The first part gives\(vi​j−vi​l\)​\[logit⁡\(wi​l\)−logit⁡\(wi​j\)\]\(v\_\{ij\}\-v\_\{il\}\)\\bigl\[\\mathrm\{logit\}\(w\_\{il\}\)\-\\mathrm\{logit\}\(w\_\{ij\}\)\\bigr\], and forw⁡\(x\)=\(1\+a​xb\)−1w\(x\)=\(1\+ax^\{b\}\)^\{\-1\}one haslogit⁡\(w⁡\(x\)\)=−ln⁡a−b​ln⁡x\\mathrm\{logit\}\(w\(x\)\)=\-\\ln a\-b\\ln x, so the bracket equalsb⁡\[ln⁡di,j2−ln⁡di,l2\]\>0b\\,\[\\ln d^\{2\}\_\{i,j\}\-\\ln d^\{2\}\_\{i,l\}\]\>0\.

\(c\)The row swap changes onlyℓi\\ell\_\{i\}, and inπi\\pi\_\{i\}it exchanges the candidates at positionsr<r′r<r^\{\\prime\}, wherellsits atrrandjjatr′r^\{\\prime\}\. The gain changes by\(relj−rell\)​\[disc⁡\(r\)−disc⁡\(r′\)\]=\(l−j\)​\[disc⁡\(r\)−disc⁡\(r′\)\]≥ΔK\(\\mathrm\{rel\}\_\{j\}\-\\mathrm\{rel\}\_\{l\}\)\[\\mathrm\{disc\}\(r\)\-\\mathrm\{disc\}\(r^\{\\prime\}\)\]=\(l\-j\)\[\\mathrm\{disc\}\(r\)\-\\mathrm\{disc\}\(r^\{\\prime\}\)\]\\geq\\Delta\_\{K\}, andn⁡\[ℒrankhard​\(D\)−ℒrankhard​\(D′\)\]=ℓi​\(D\)−ℓi​\(D′\)n\[\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}\(D\)\-\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}\(D^\{\\prime\}\)\]=\\ell\_\{i\}\(D\)\-\\ell\_\{i\}\(D^\{\\prime\}\)equals that change divided byG⋆G^\{\\star\}\. ∎

Herepj\|i=exp\(−∥xi−xj∥22/2σi2\)/Zip\_\{j\|i\}=\\exp\(\-\\\|x\_\{i\}\-x\_\{j\}\\\|\_\{2\}^\{2\}/2\\sigma\_\{i\}^\{2\}\)/Z\_\{i\}is the conditional affinity ofjjtoii, withσi\\sigma\_\{i\}the perplexity\-calibrated bandwidth of Section[3](https://arxiv.org/html/2610.41107#S3)andZiZ\_\{i\}the normalizer that makesp⋅\|ip\_\{\\cdot\|i\}sum to one\.

*This lemma says that the affinities t\-SNE and UMAP read can tie, or run against the distances, so the request in parts \(a\) and \(b\) can vanish or point the wrong way\.*

###### Lemma S2\(When the affinities do not follow the distances\)

\(a\)In t\-SNE,pi​j<pi​lp\_\{ij\}<p\_\{il\}withdi,jH<di,lHd^\{H\}\_\{i,j\}<d^\{H\}\_\{i,l\}if and only if

pi\|l−pi\|j\>pj\|i−pl\|i\>0,p\_\{i\|l\}\-p\_\{i\|j\}\\;\>\\;p\_\{j\|i\}\-p\_\{l\|i\}\\;\>\\;0,\(35\)This occurs whenjjsits in a region dense enough that its bandwidthσj\\sigma\_\{j\}makespi\|jp\_\{i\|j\}small, whilellsits in a sparse region with a largeσl\\sigma\_\{l\}\. The symmetrizationpi​j=\(pj\|i\+pi\|j\)/2​np\_\{ij\}=\(p\_\{j\|i\}\+p\_\{i\|j\}\)/2ndoes not preserve the order of the conditionalsp⋅\|ip\_\{\\cdot\|i\}\.\(b\)In UMAP,vj\|i=1v\_\{j\|i\}=1for the nearest neighborjjof every pointii, and the fuzzy union givesvi​j=1v\_\{ij\}=1whenever either conditional equals one\. Ifjjis the nearest neighbor ofiiandiiis the nearest neighbor ofll, thenvi​j=vi​l=1v\_\{ij\}=v\_\{il\}=1and the cross\-entropy is indifferent to the order ofjjandllaroundii\.

##### Proof of Lemma[S2](https://arxiv.org/html/2610.41107#Thmlemma2)

\(a\)pi​j<pi​lp\_\{ij\}<p\_\{il\}readspj\|i\+pi\|j<pl\|i\+pi\|lp\_\{j\|i\}\+p\_\{i\|j\}<p\_\{l\|i\}\+p\_\{i\|l\}, which rearranges to the displayed inequality\. Its right\-hand side is positive becausep⋅\|ip\_\{\\cdot\|i\}is a decreasing function ofdi,⋅Hd^\{H\}\_\{i,\\cdot\}at the common bandwidthσi\\sigma\_\{i\}\. The left\-hand side isexp\(−dl,iH​2/2σl2\)/Zl−exp\(−dj,iH​2/2σj2\)/Zj\\exp\(\-d^\{H\\,2\}\_\{l,i\}/2\\sigma\_\{l\}^\{2\}\)/Z\_\{l\}\-\\exp\(\-d^\{H\\,2\}\_\{j,i\}/2\\sigma\_\{j\}^\{2\}\)/Z\_\{j\}, which can be made positive by takingσj\\sigma\_\{j\}small relative todj,iHd^\{H\}\_\{j,i\}, so that the second term vanishes, whileσl\\sigma\_\{l\}is large relative todl,iHd^\{H\}\_\{l,i\}andZlZ\_\{l\}is moderate\. Perplexity calibration setsσj\\sigma\_\{j\}small exactly whenjjhas many close neighbors\.\(b\)UMAP setsvj\|i=exp\(−\(di,jH−ϱi\)/σi\)v\_\{j\|i\}=\\exp\\bigl\(\-\(d^\{H\}\_\{i,j\}\-\\varrho\_\{i\}\)/\\sigma\_\{i\}\\bigr\)with the local\-connectivity offsetϱi=di,jH\\varrho\_\{i\}=d^\{H\}\_\{i,j\}for the nearest neighborjj, sovj\|i=1v\_\{j\|i\}=1\. The fuzzy unionva​b=vb\|a\+va\|b−vb\|a​va\|bv\_\{ab\}=v\_\{b\|a\}\+v\_\{a\|b\}\-v\_\{b\|a\}v\_\{a\|b\}equals one whenever either conditional does\. Under the stated hypothesisvj\|i=1v\_\{j\|i\}=1andvi\|l=1v\_\{i\|l\}=1, sovi​j=vi​l=1v\_\{ij\}=v\_\{il\}=1, and Eq\. \([18](https://arxiv.org/html/2610.41107#S3.E18)\) vanishes\. ∎

Remark \(where a loss stops asking for order\)\. Suppose an objective is invariant under every permutation of the distances fromiito itsk′k^\{\\prime\}returned points, and the optimizer’s output inherits that symmetry\. Then the returned arrangement is exchangeable, part \(ii\) applies, and𝔼⁡\[ωi\]=1\\mathbb\{E\}\[\\omega\_\{i\}\]=1\. Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)\(a\)–\(b\) say when the invariance holds for t\-SNE and UMAP: exactly when the affinities of the returned points tie, because the row\-swap difference then vanishes for every transposition and hence for every permutation\. The remark describes the limit in which the two losses stop asking for order; Section[4\.1](https://arxiv.org/html/2610.41107#S4.SS1)measures where each method sits relative to that baseline\.

##### Proof of Theorem[4](https://arxiv.org/html/2610.41107#Thmtheorem4)

\(i\)By definitionO\-DLPi​\(k\)=\#⁡\{j∈Ri:posi​\(j\)=j\}/k=\(\|Ri\|/k\)⋅Ai=DLPi​Ai\\text\{O\-DLP\}\_\{i\}\(k\)=\\\#\\\{j\\in R\_\{i\}:\\text\{pos\}\_\{i\}\(j\)=j\\\}/k=\(\|R\_\{i\}\|/k\)\\cdot A\_\{i\}=\\text\{DLP\}\_\{i\}A\_\{i\}, since only retained neighbors can be hits\. Averaging gives the first claim, andk​O\-DLP/DLP=∑iDLPi​\(k​Ai\)/∑iDLPik\\,\\text\{O\-DLP\}/\\text\{DLP\}=\\sum\_\{i\}\\text\{DLP\}\_\{i\}\\,\(kA\_\{i\}\)/\\sum\_\{i\}\\text\{DLP\}\_\{i\}is the stated weighted mean;Ai∈\[0,1\]A\_\{i\}\\in\[0,1\]gives the range\.

\(ii\)Writek′k^\{\\prime\}for the size at which the metric is read, letq=\|Ri\|q=\|R\_\{i\}\|, and letXj=𝟏\[posi\(j\)=j\]X\_\{j\}=\\mathbf\{1\}\[\\text\{pos\}\_\{i\}\(j\)=j\]forj∈Rij\\in R\_\{i\}\. Under a uniform arrangement of thek′k^\{\\prime\}returned points, each returned point is equally likely to occupy any of thek′k^\{\\prime\}positions, so𝔼​Xj=1/k′\\mathbb\{E\}X\_\{j\}=1/k^\{\\prime\}and𝔼⁡\[Xj​Xl\]=1/\(k′​\(k′−1\)\)\\mathbb\{E\}\[X\_\{j\}X\_\{l\}\]=1/\(k^\{\\prime\}\(k^\{\\prime\}\-1\)\)forj≠lj\\neq l\. Hence𝔼⁡\[\#​hits\]=q/k′\\mathbb\{E\}\[\\\#\\text\{hits\}\]=q/k^\{\\prime\}, which is𝔼⁡\[Ai\]=1/k′\\mathbb\{E\}\[A\_\{i\}\]=1/k^\{\\prime\}, and

Var⁡\(\#​hits\)=qk′​\(1−1k′\)\+q⁡\(q−1\)​\[1k′​\(k′−1\)−1k′2\]=qk′​\(1−1k′\)\+q⁡\(q−1\)k′2​\(k′−1\)≤qk′,\\mathrm\{Var\}\(\\\#\\text\{hits\}\)=\\frac\{q\}\{k^\{\\prime\}\}\\Bigl\(1\-\\frac\{1\}\{k^\{\\prime\}\}\\Bigr\)\+q\(q\-1\)\\Bigl\[\\frac\{1\}\{k^\{\\prime\}\(k^\{\\prime\}\-1\)\}\-\\frac\{1\}\{k^\{\\prime 2\}\}\\Bigr\]=\\frac\{q\}\{k^\{\\prime\}\}\\Bigl\(1\-\\frac\{1\}\{k^\{\\prime\}\}\\Bigr\)\+\\frac\{q\(q\-1\)\}\{k^\{\\prime 2\}\(k^\{\\prime\}\-1\)\}\\;\\leq\\;\\frac\{q\}\{k^\{\\prime\}\},\(36\)the last step becauseq−1≤k′−1q\-1\\leq k^\{\\prime\}\-1\. Sincek′​O\-DLPi=\#​hitsk^\{\\prime\}\\,\\text\{O\-DLP\}\_\{i\}=\\\#\\text\{hits\}andq/k′=DLPiq/k^\{\\prime\}=\\text\{DLP\}\_\{i\}, this isVar⁡\(k′​O\-DLPi\)≤DLPi\\mathrm\{Var\}\(k^\{\\prime\}\\,\\text\{O\-DLP\}\_\{i\}\)\\leq\\text\{DLP\}\_\{i\}\. With independent arrangements across points,Var⁡\(O\-DLP\)=n−2​∑iVar⁡\(O\-DLPi\)≤n−2​k′−2​∑iDLPi=DLP/\(n​k′2\)\\mathrm\{Var\}\(\\text\{O\-DLP\}\)=n^\{\-2\}\\sum\_\{i\}\\mathrm\{Var\}\(\\text\{O\-DLP\}\_\{i\}\)\\leq n^\{\-2\}k^\{\\prime\-2\}\\sum\_\{i\}\\text\{DLP\}\_\{i\}=\\text\{DLP\}/\(nk^\{\\prime 2\}\), so the standard deviation of O\-DLP is at mostk′−1​DLP/nk^\{\\prime\-1\}\\sqrt\{\\text\{DLP\}/n\}\. Dividing byDLP/k′\\text\{DLP\}/k^\{\\prime\}gives Eq\. \([21](https://arxiv.org/html/2610.41107#S3.E21)\)\.

\(iii\)Sincek′≤min⁡\(k,m\)k^\{\\prime\}\\leq\\min\(k,m\), every one of thek′k^\{\\prime\}points the embedding returns foriiis a candidate in𝒞i\\mathcal\{C\}\_\{i\}: a returned neighbor lies among thekkneighbors, and a returned non\-neighbor is one of thek′≤mk^\{\\prime\}\\leq mnearest non\-neighbors\. So the firstk′k^\{\\prime\}entries ofπi\\pi\_\{i\}are exactly the returned points, in embedded order\. A retained neighborj∈Rij\\in R\_\{i\}is a miss whenposi​\(j\)≠j\\text\{pos\}\_\{i\}\(j\)\\neq j\. Ifposi​\(j\)<j\\text\{pos\}\_\{i\}\(j\)<j, theposi​\(j\)−1≤j−2\\text\{pos\}\_\{i\}\(j\)\-1\\leq j\-2entries ahead ofjjcannot hold allj−1j\-1neighbors of higher relevance, so one of them followsjjinπi\\pi\_\{i\}: a strict inversion involvingjj\. Ifposi​\(j\)\>j\\text\{pos\}\_\{i\}\(j\)\>j, at leastjjentries precedejj, and at mostj−1j\-1of them have relevance aboverelj\\mathrm\{rel\}\_\{j\}; the remaining one is a neighbor of lower rank or an intruder of relevance zero, and\(j,⋅\)\(j,\\cdot\)is a strict inversion involvingjj\. Every miss therefore belongs to at least one strict inversion ofπi\\pi\_\{i\}, and a strict inversion contains at most two retained neighbors, so\#​misses≤2​inv​\(πi\)\\\#\\text\{misses\}\\leq 2\\,\\mathrm\{inv\}\(\\pi\_\{i\}\)andDLPi​\(k′\)−O\-DLPi​\(k′\)=\#​misses/k′≤2​inv​\(πi\)/k′\\text\{DLP\}\_\{i\}\(k^\{\\prime\}\)\-\\text\{O\-DLP\}\_\{i\}\(k^\{\\prime\}\)=\\\#\\text\{misses\}/k^\{\\prime\}\\leq 2\\,\\mathrm\{inv\}\(\\pi\_\{i\}\)/k^\{\\prime\}\. Lemma[S1](https://arxiv.org/html/2610.41107#Thmlemma1)givesinv⁡\(πi\)≤G⋆​ℓi/ΔK\\mathrm\{inv\}\(\\pi\_\{i\}\)\\leq G^\{\\star\}\\ell\_\{i\}/\\Delta\_\{K\}; averaging overiigives Eq\. \([22](https://arxiv.org/html/2610.41107#S3.E22)\)\. ∎

##### What the plane can hold

The following bound makes the crowding remark of Section[4\.1](https://arxiv.org/html/2610.41107#S4.SS1)quantitative\. LetdegH⁡\(j\)\\deg^\{H\}\(j\)be the number of points that countjjamong theirkkhigh\-dimensional neighbors, the in\-degree ofjjin the high\-dimensionalkk\-NN digraph\.

*This proposition says that in the plane a point can be among thekknearest neighbors of at most7​k7kothers, so neighbor edges into high\-dimensional hubs beyond that count cannot be kept by any embedding\.*

###### Proposition S1\(Crowding bound for order\)

For any embedding ofnnpoints inℝ2\\mathbb\{R\}^\{2\}with distinct pairwise distances,

O\-DLP​\(k\)≤DLP​\(k\)≤1n​k​∑j=1nmin⁡\(degH⁡\(j\),7​k\)\.\\text\{O\-DLP\}\(k\)\\;\\leq\\;\\text\{DLP\}\(k\)\\;\\leq\\;\\frac\{1\}\{nk\}\\sum\_\{j=1\}^\{n\}\\min\\bigl\(\\deg^\{H\}\(j\),\\;7k\\bigr\)\.\(37\)

###### Proof 2

The first inequality is Eq\. \([8](https://arxiv.org/html/2610.41107#S3.E8)\)\. For the second, writeDLP​\(k\)=1n​k​∑i\|𝒩i∩𝒩^i\|\\text\{DLP\}\(k\)=\\frac\{1\}\{nk\}\\sum\_\{i\}\|\\mathcal\{N\}\_\{i\}\\cap\\hat\{\\mathcal\{N\}\}\_\{i\}\|with𝒩^i\\hat\{\\mathcal\{N\}\}\_\{i\}thekknearest embedded points, and count pairs by their second element:∑i\|𝒩i∩𝒩^i\|=∑j\#\{i:j∈𝒩i,j∈𝒩^i\}≤∑jmin\(degH\(j\),deglow\(j\)\)\\sum\_\{i\}\|\\mathcal\{N\}\_\{i\}\\cap\\hat\{\\mathcal\{N\}\}\_\{i\}\|=\\sum\_\{j\}\\\#\\\{i:j\\in\\mathcal\{N\}\_\{i\},\\;j\\in\\hat\{\\mathcal\{N\}\}\_\{i\}\\\}\\leq\\sum\_\{j\}\\min\(\\deg^\{H\}\(j\),\\deg^\{\\text\{low\}\}\(j\)\), wheredeglow⁡\(j\)\\deg^\{\\text\{low\}\}\(j\)is the in\-degree ofjjin the embeddedkk\-NN digraph\. Stone’s lemma\[[44](https://arxiv.org/html/2610.41107#bib.bib44)\]bounds that in\-degree: partition the plane aroundyjy\_\{j\}into seven cones of angle2​π/7<π/32\\pi/7<\\pi/3; if two pointsya,yby\_\{a\},y\_\{b\}lie in the same cone with‖ya−yj‖<‖yb−yj‖\\\|y\_\{a\}\-y\_\{j\}\\\|<\\\|y\_\{b\}\-y\_\{j\}\\\|, the angle atyjy\_\{j\}is belowπ/3\\pi/3and so‖yb−ya‖<‖yb−yj‖\\\|y\_\{b\}\-y\_\{a\}\\\|<\\\|y\_\{b\}\-y\_\{j\}\\\|, henceyay\_\{a\}is closer toyby\_\{b\}thanyjy\_\{j\}is; only thekkpoints of a cone nearest toyjy\_\{j\}can therefore haveyjy\_\{j\}among theirkknearest, anddeglow⁡\(j\)≤7​k\\deg^\{\\text\{low\}\}\(j\)\\leq 7k\. ∎

The bound bites only through the hubs of the high\-dimensional graph\[[45](https://arxiv.org/html/2610.41107#bib.bib45)\]: every neighbor edge that points into a node withdegH\>7​k\\deg^\{H\}\>7kbeyond the first7​k7kis lost in any planar embedding, whatever the loss\. Table[A\.22](https://arxiv.org/html/2610.41107#A1.T22)evaluates the bound on the eight benchmarks atk=10k=10\. The hub tail is thin everywhere: the bound is1\.0001\.000on MNIST and Covertype, where no point exceeds in\-degree7070, and at least0\.9680\.968on the rest, with CIFAR\-10 the most hub\-heavy at3\.2%3\.2\\%of edges\. The measured DLP of the best method lies between0\.180\.18and0\.650\.65\(Table[5](https://arxiv.org/html/2610.41107#S4.T5)\), so the in\-degree ceiling of the plane is far from what limits membership, and the order that survives is set by the loss and the layout it reaches\.

Table A\.22:The crowding bound of Proposition[S1](https://arxiv.org/html/2610.41107#Thmproposition1)atk=10k=10: the largest high\-dimensional in\-degree, the number of points with in\-degree above7​k=707k=70, the share of neighbor edges that no planar embedding can keep, and the right\-hand side of Eq\. \([37](https://arxiv.org/html/2610.41107#A1.E37)\)\.
##### The trading scorer

The scorer of[A\.4](https://arxiv.org/html/2610.41107#A1.SS4)computes, for each bariithat DBSCAN leaves outside every cluster, the Spearman correlationρi\\rho\_\{i\}between the ranks1,…,k′1,\\dots,k^\{\\prime\}ofii’sk′k^\{\\prime\}nearest input\-space neighbors and the ranks of those same bars by distance toyiy\_\{i\}in the embedding, withk′=10k^\{\\prime\}=10\. The order half of the marker score is1−ρi1\-\\rho\_\{i\}\.

*This proposition says that the order half of the marker score is bounded by the same per\-bar rank loss that SR\-tSNE minimizes\.*

###### Proposition S2\(The scorer reads a controlled quantity\)

Letk′≤kk^\{\\prime\}\\leq kand letℓi\\ell\_\{i\}be the hard per\-point rank loss of Section[3\.4](https://arxiv.org/html/2610.41107#S3.SS4)\. Then, for every barii,

1−ρi≤12k′​\(k′\+1\)⋅G⋆ΔK​ℓi\.1\-\\rho\_\{i\}\\;\\leq\\;\\frac\{12\}\{k^\{\\prime\}\(k^\{\\prime\}\+1\)\}\\cdot\\frac\{G^\{\\star\}\}\{\\Delta\_\{K\}\}\\;\\ell\_\{i\}\.\(38\)

###### Proof 3

Writedj=\|j−rj\|d\_\{j\}=\|j\-r\_\{j\}\|for the displacement of thejj\-th input\-space neighbor,rjr\_\{j\}being its rank among thek′k^\{\\prime\}neighbors by embedded distance, and letinvi​\(k′\)\\mathrm\{inv\}\_\{i\}\(k^\{\\prime\}\)be the number of pairs among thek′k^\{\\prime\}neighbors whose order the embedding inverts\. Spearman’s coefficient on two permutations of\{1,…,k′\}\\\{1,\\dots,k^\{\\prime\}\\\}isρi=1−6​∑jdj2/\(k′​\(k′2−1\)\)\\rho\_\{i\}=1\-6\\sum\_\{j\}d\_\{j\}^\{2\}/\\bigl\(k^\{\\prime\}\(k^\{\\prime 2\}\-1\)\\bigr\)\. The Diaconis–Graham inequality\[[46](https://arxiv.org/html/2610.41107#bib.bib46)\]gives∑jdj≤2​invi​\(k′\)\\sum\_\{j\}d\_\{j\}\\leq 2\\,\\mathrm\{inv\}\_\{i\}\(k^\{\\prime\}\), and eachdj≤k′−1d\_\{j\}\\leq k^\{\\prime\}\-1, so∑jdj2≤\(k′−1\)​∑jdj≤2​\(k′−1\)​invi​\(k′\)\\sum\_\{j\}d\_\{j\}^\{2\}\\leq\(k^\{\\prime\}\-1\)\\sum\_\{j\}d\_\{j\}\\leq 2\(k^\{\\prime\}\-1\)\\,\\mathrm\{inv\}\_\{i\}\(k^\{\\prime\}\)\. Hence1−ρi≤12​invi​\(k′\)/\(k′​\(k′\+1\)\)1\-\\rho\_\{i\}\\leq 12\\,\\mathrm\{inv\}\_\{i\}\(k^\{\\prime\}\)/\\bigl\(k^\{\\prime\}\(k^\{\\prime\}\+1\)\\bigr\)\. Every pair of the firstk′k^\{\\prime\}neighbors that the embedding inverts is a strict inversion ofπi\\pi\_\{i\}, because the two carry distinct relevancesk−j\+1k\-j\+1and the order among candidates is the order by embedded distance; soinvi​\(k′\)≤inv⁡\(πi\)\\mathrm\{inv\}\_\{i\}\(k^\{\\prime\}\)\\leq\\mathrm\{inv\}\(\\pi\_\{i\}\)\. Lemma[S1](https://arxiv.org/html/2610.41107#Thmlemma1)givesinv⁡\(πi\)≤\(G⋆−Gihard\)/ΔK=G⋆​ℓi/ΔK\\mathrm\{inv\}\(\\pi\_\{i\}\)\\leq\(G^\{\\star\}\-G^\{\\text\{hard\}\}\_\{i\}\)/\\Delta\_\{K\}=G^\{\\star\}\\ell\_\{i\}/\\Delta\_\{K\}\. ∎

The constant is uniform: it is fixed bykk,mmandk′k^\{\\prime\}, and it does not grow withnn\. At the setting used throughout this paper,k=m=k′=10k=m=k^\{\\prime\}=10, it is8\.8×1028\.8\\times 10^\{2\}\. The proposition identifies which objective drives the quantity down: under SR\-tSNE what remains of1−ρi1\-\\rho\_\{i\}after training is the order the plane could not restore, whereas under t\-SNE it also carries the order the objective never asked for\. The min–max normalization of[A\.4](https://arxiv.org/html/2610.41107#A1.SS4)then ranks bars by that residual\.

*This corollary says that a finite weight on the hard rank term makes the whole loss prefer the correct order of every pair\.*

###### Corollary S1\(A finite weight suffices\)

Let the embedding have squared diameter at mostR2R^\{2\}\. The total lossKL\+λ​ℒrankhard\\mathrm\{KL\}\+\\lambda\\mathcal\{L\}^\{\\text\{hard\}\}\_\{\\text\{rank\}\}, read on the distance arrays of Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3), strictly prefers the correct order for every inverted pair of every point whenever

λ\>λ⋆=n​G⋆​ln⁡\(1\+R2\)ΔK​maxi,j<l​\(pi​l−pi​j\)\+\.\\lambda\\;\>\\;\\lambda^\{\\star\}\\;=\\;\\frac\{n\\,G^\{\\star\}\\,\\ln\(1\+R^\{2\}\)\}\{\\Delta\_\{K\}\}\\max\_\{i,\\;j<l\}\\,\(p\_\{il\}\-p\_\{ij\}\)^\{\+\}\.\(39\)Sincepa​b≤1/np\_\{ab\}\\leq 1/nfor every pair,λ⋆≤G⋆​ln⁡\(1\+R2\)/ΔK\\lambda^\{\\star\}\\leq G^\{\\star\}\\ln\(1\+R^\{2\}\)/\\Delta\_\{K\}, a bound that does not grow withnn\.

##### Proof of Corollary[S1](https://arxiv.org/html/2610.41107#Thmcorollary1)

For an inverted pair, Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)\(a\) bounds the KL preference from below by−\(pi​l−pi​j\)\+​ln⁡\(1\+R2\)\-\(p\_\{il\}\-p\_\{ij\}\)^\{\+\}\\ln\(1\+R^\{2\}\), since0<ln⁡\(1\+di,j2\)−ln⁡\(1\+di,l2\)≤ln⁡\(1\+R2\)0<\\ln\(1\+d^\{2\}\_\{i,j\}\)\-\\ln\(1\+d^\{2\}\_\{i,l\}\)\\leq\\ln\(1\+R^\{2\}\)\. Theorem[3](https://arxiv.org/html/2610.41107#Thmtheorem3)\(c\) bounds the rank preference from below byλ​ΔK/\(n​G⋆\)\\lambda\\Delta\_\{K\}/\(nG^\{\\star\}\)\. The sum is positive whenλ\\lambdaexceedsλ⋆\\lambda^\{\\star\}\. Finallypa​b=\(pb\|a\+pa\|b\)/2​n≤1/np\_\{ab\}=\(p\_\{b\|a\}\+p\_\{a\|b\}\)/2n\\leq 1/nbecause each conditional distribution sums to one\. The statement is about the hard rank loss; the soft loss of Eqs\. \([2](https://arxiv.org/html/2610.41107#S3.E2)\)–\([4](https://arxiv.org/html/2610.41107#S3.E4)\) converges to it asτ→0\\tau\\to 0\(Theorem[2](https://arxiv.org/html/2610.41107#Thmtheorem2)\), so for any pair with a fixed positive distance gap the soft preference converges to the hard one\. ∎