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Hyperspherical Simplex Representations for Backward-Compatible Retrieval

Simone Ricci, Niccolò Biondi, Federico Pernici, Alberto Del Bimbo

arXiv:2610.29550v1cs.CVcs.LG

TL;DR

Retrieval updates can disrupt compatibility, while many existing methods depend on the previous model. This paper uses aligned hyperspherical simplex representations from softmax outputs or logits; it reports competitive benchmark results, including substantial improvements with many updates.

  • Problem

    Backward-compatible retrieval must keep representations from different model versions directly comparable, but many existing methods require the old model as a reference, which can be difficult to access.

    Source: §1 · p. 2

  • Method

    The paper derives hyperspherical simplex representations from softmax outputs or logits and uses a deterministic linear transformation to realign existing-class prototypes when new classes change the simplex geometry.

    Source: §1 · p. 3, §5 · p. 18

  • Results

    The representations achieve competitive results on standard compatibility benchmarks, with substantial improvements in scenarios involving many model updates, including independently pretrained models from public repositories.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The analysis identifies geometric conditions for backward-compatibility inequalities to hold in expectation, and aligned regular-simplex representations satisfy those conditions under the stated assumptions.

    Source: §1 · p. 3

    The alignment transformation requires the correspondence between old and new class indices to be known.

    Source: §5 · p. 19

Abstract

from arXiv · show

Training modern AI models has become increasingly expensive, and updating deployed models can alter their internal feature representations, affecting applications built upon them. In retrieval systems, such updates often require complete gallery re-indexing, a computationally expensive process at scale. Existing backward-compatibility methods enable direct comparison of representations from updated and earlier models, avoiding re-indexing. However, they typically require compatibility-specific training, learned mapping functions, or architectural modifications, and often depend explicitly on the previous model. We propose hyperspherical simplex representations derived from softmax outputs and logits for backward-compatible retrieval across independently trained models. They leverage the simplex structure induced by softmax cross-entropy training with one-hot labels as a common geometric reference across model updates. Since this reference depends on the number of classes, adding new ones expands the simplex, misaligning existing class prototypes with their earlier counterparts. We therefore derive a deterministic linear transformation to restore their alignment. Our theoretical analysis of the formal backward-compatibility definition identifies two geometric conditions under which its inequalities hold in expectation, assuming von Mises--Fisher class distributions with increasing concentration across updates. Aligned regular-simplex representations satisfy these conditions, providing a theoretical basis for the proposed representations under the stated assumptions. Without using additional compatibility-specific losses, learned mappings, or architectural modifications, the proposed representations achieve competitive results on standard compatibility benchmarks, including expanded class sets, more expressive architectures, and independently trained models from public repositories.

1 Introduction

Model updates can disrupt retrieval behavior, while established compatibility methods often depend on old models or specialized training. The paper proposes softmax- and logit-derived hyperspherical simplex representations, analyzes their compatibility conditions, and reports competitive benchmark results without compatibility-specific losses, learned mappings, or architectural changes.

1 Introduction

  • Model updates can change retrieval rankings and disrupt applications, while existing representation-compatibility training often requires the old model as a reference.Restricted access to old architectures and weights, along with high training costs, makes that approach increasingly challenging.Source: §1 · p. 2
  • Softmax outputs and logits provide a shared simplex-based geometric reference for representations across independently trained models.When new classes change the simplex geometry, a deterministic linear transformation restores alignment between existing class prototypes and their earlier counterparts.Source: §1 · p. 3
  • Under explicit distributional and concentration assumptions, aligned regular-simplex representations satisfy geometric conditions for backward-compatibility inequalities in expectation.The paper shows that the proposed representations satisfy those conditions.Source: §1 · p. 3
  • Top-k sparsification reduces storage for hyperspherical simplex representations, using softmax rank preservation and top-k calibration as motivation.The paper reports negligible retrieval-accuracy changes without degrading compatibility.Source: §1 · p. 3
  • The proposed representations achieve competitive results on standard compatibility benchmarks, including many-update scenarios and independently pretrained models from public repositories.They do so without additional compatibility-specific losses, learned mappings, or architecture modifications.Source: §1 · p. 3

2 Related Work

Prior approaches pursue representation compatibility through reference-model training, learned mappings, or fixed classifiers, each with particular dependencies or overhead. The paper instead obtains aligned simplex representations from independently trained models and gives geometric conditions for compatibility under explicit assumptions.

2 Related Work

  • Regularization-based methods align old and new representations, while architectural feature expansion can preserve compatibility but adds complexity at successive updates.The cited discussion also notes that regularization-based backward-compatible models often do not match the discriminative performance of newly independently trained models.Source: §2 · p. 4
  • Mapping-based approaches learn functions aligning same-input representations from different models, but composing mappings across updates adds overhead and feature alignment alone does not guarantee formal compatibility inequalities.The paper contrasts these methods with its geometric analysis of the formal conditions.Source: §2 · p. 4
  • Pre-allocated fixed-classifier approaches avoid old-model reference but require a shared classifier across updates and architectural modifications.The paper's alternative obtains aligned regular-simplex geometry directly from logits or softmax outputs of independently trained models.Source: §2 · p. 4
  • The paper complements Biondi et al. (2026) by deriving closed-form geometric conditions for aligned hyperspherical representations under explicit distributional assumptions.Here, independently trained means no model uses an earlier model's parameters, representations, outputs, or compatibility losses, though task, data, initialization, and label-to-output convention may be shared.Source: §2 · pp. 3–4

3 Compatibility of Hyperspherical Simplex Representations

This section develops softmax- and logit-derived hyperspherical simplex representations, a deterministic correction for class expansion, and conditions under which aligned prototypes satisfy backward-compatibility inequalities in expectation. It also considers top-k sparsification to reduce storage while preserving class ranking.

3.2.1 Features Derived From Softmax Outputs

  • Softmax outputs approach one-hot simplex vertices, but direct L2 normalization cannot produce interclass angles above π/2; centering the simplex before normalization yields regular-simplex prototypes satisfying the angular condition.With an unchanged class set, prototype directions remain fixed across updates; the resulting representation is Probability Simplex Projection (PSP).Source: §3.2.1 · pp. 6–7, §3.2 · p. 6

3.2.2 Features Derived From Logits

  • Logit-UFM predicts that logits align with the same simplex prototype directions as softmax outputs, and normalized logits retain greater within-class spread but lower prototype alignment than softmax-derived features.On CIFAR-100 with ResNet-18, higher logit NC1, NC2 equidistance, and NC3 values indicate this greater residual variability and lower convergence toward ideal simplex geometry.Source: §3.2.2 · pp. 7–8
  • When a softmax output is uniform, centering projects it to zero, so its L2-normalization is undefined; the method defines the representation as zero in this unlikely case.Trained classifiers typically produce non-uniform output distributions.Source: §3.2.2 · p. 7

3.3 Deterministic Angular Correction Under Class Expansion

  • Adding classes changes the simplex and misaligns existing prototypes; a deterministic linear transformation removes new-coordinate components and aligns updated old-class prototypes with their earlier counterparts.The transformation depends only on simplex geometry and the class counts, so it requires neither learned parameters nor data.Source: §3.3 · pp. 8–10

3.4 Theoretical Analysis of the Backward-Compatibility Definition

  • Under independent vMF class distributions with increasing concentration, aligned prototypes separated by angles greater than π/2 ensure both compatibility inequalities hold in expectation; centered aligned regular simplices satisfy these conditions.The vMF assumption is a tractable model, not an exact characterization of learned features. Increasing concentration is motivated by observations of reduced within-class variability and is empirically supported for logits and softmax outputs across updates.Source: §3.4 · pp. 10–11
  • Aligned regular-simplex prototypes make PSP and LSP satisfy both compatibility inequalities in expectation under Theorem 3.6’s assumptions.PSP has this geometry by construction; LSP has it under the logit-UFM assumptions, and the transformation P_t,k restores alignment when classes are added. An Appendix I analysis finds positive average margins for both inequalities only when centering and angular correction are combined. The result also applies to pre-allocated d-simplex classifiers, though PSP and LSP do not require a shared fixed classifier.Source: §3.4 · p. 11

3.5 Scalability via Top-k Sparsification

  • Because PSP and LSP have one dimension per class, gallery storage scales with both gallery size and class count, motivating top-k sparsification.The approach retains the highest-ranked components, motivated by softmax cross-entropy’s top-k calibration and softmax’s rank-preserving property (Lapin et al., 2016; Yang & Koyejo, 2020).Source: §3.5 · p. 11
  • Top-k sparsification preserves the class ranking of PSP and LSP because centering subtracts a shared constant and ℓ2-normalization rescales entries positively.Logits and softmax outputs have the same ranking (Yang & Koyejo, 2020), while the optimal softmax cross-entropy scores preserve posterior ordering (Lapin et al., 2016).Source: §3.5 · pp. 11–12
  • After ℓ2-normalization, the method retains the k largest entries and zeros the rest, enabling sparse nearest-neighbor search with negligible retrieval degradation while preserving compatibility.The paper evaluates this strategy for both PSP and LSP; it identifies PySparNN and Seismic (Bruch et al., 2024a;b) as sparse-search libraries.Source: §3.5 · p. 12

4 Experimental Results

Across extended-class, architecture-change, combined-update, and public-model settings, PSP and LSP provide backward-compatible retrieval, with PSP especially consistent across updates. The experiments evaluate image retrieval using compatibility matrices and compare cross-model performance with same-model performance.

4.1 Evaluation Setting

  • Evaluation uses image retrieval with CMC@1 and cosine distance; compatibility matrices compare cross-model entries against corresponding same-model entries.The empirical criterion is used because directly verifying the formal inequalities is impractical at scale.Source: §4.1 · p. 12

4.2.1 Extended Classes Benchmark

  • In long extended-class sequences, 50-step CIFAR-100 AC/ACA are 0.39/10.76 for PSP and 0.36/11.25 for LSP, versus 0.11/3.91 for CoReS.PSP and LSP also outperform competing methods on compatibility metrics in the 5-step ImageNet-1K and Google Landmarks v2 settings.Source: Table 1 · p. 13, Table 2 · p. 14, §4.2.1 · p. 14
  • The authors suggest sequential predecessor-based compatibility may accumulate misalignment, whereas PSP and LSP use regular-simplex geometry as a common reference across updates.This is presented as a possible explanation for degradation in several competing methods as update counts increase.Source: §4.2.1 · p. 15
  • Top-k storage is especially beneficial at large class counts: on Google Landmarks v2, k=256 reduces gallery storage from approximately 318 KiB to 2 KiB per representation.The reported reduction is approximately 159×; search-time benefit depends on the indexing and search backend.Source: §4.2.1 · p. 15

4.2.2 Extended Backbone Benchmark

  • For extended-backbone updates, LSP achieves the highest AA and ACA on CIFAR-100 and remains within 0.1–0.2 percentage points of the best methods on ImageNet-1K.Most evaluated approaches satisfy the compatibility criterion in this two-step setting.Source: Table 4 · p. 16, §4.2.2 · p. 16

4.2.3 Extended Classes and Backbone Benchmark

  • When classes and backbone both expand, all evaluated compatibility methods satisfy the criterion, while LSP matches or nearly matches the strongest methods’ AA and ACA.The encoder-feature baseline has AC=0 on both datasets; PSP also satisfies the criterion but has lower AA and ACA than LSP.Source: Table 4 · p. 16, §4.2.3 · p. 16

4.3 Compatibility Across Independently Trained Models From Public Repositories

  • Across independently pretrained model sequences, PSP and LSP improve compatibility over the encoder-feature baseline; PSP is most consistent, while LSP has higher open-set AA at lower compatibility.The ImageNet-1K and Places365 evaluations use models from public repositories without compatibility-specific training or adaptation.Source: Table 5 · p. 17, Figure 5 · p. 17, §4.3 · pp. 17–18
  • The retrieval evaluation compares closed-set ImageNet-1K test data with open-set Places365; models trained on the same ImageNet-1K classes share label-space-derived reference geometry without coordination during training.Source: §4.3 · pp. 17–18

5 Conclusions

The paper presents PSP and LSP for backward-compatible retrieval, using softmax-induced simplex geometry as a shared reference and a deterministic transformation to realign prototypes when classes are added. It reports competitive retrieval and compatibility results, while identifying objective, label-space, and theoretical-assumption boundaries.

5 Conclusions

  • PSP and LSP use softmax-induced simplex geometry to align class prototypes across updates without compatibility-specific losses, learned mappings, a fixed classifier, or architectural changes.A deterministic linear transformation restores alignment of previously observed classes when new classes change the simplex geometry.Source: §5 · p. 18
  • Theoretical compatibility inequalities hold in expectation under von Mises–Fisher class distributions with increasing concentration when corresponding prototypes align and different-class prototypes have angles greater than π/2.Aligned regular-simplex representations satisfy these geometric conditions; PSP instantiates them by construction and LSP under the logit-UFM.Source: §5 · p. 18
  • PSP and LSP achieve competitive retrieval performance on standard backward-compatibility benchmarks, with particularly high compatibility across many model updates and independently trained public-repository models.The independently trained models require no additional adaptation.Source: §5 · p. 18
  • Top-k sparsification substantially reduces representation storage with negligible retrieval-accuracy changes and no compatibility degradation across evaluated settings.The strategy addresses representation dimensionality that grows linearly with the number of classes.Source: §5 · p. 18
  • The approach directly applies to models trained with softmax cross-entropy and one-hot class labels, not directly to alternative objectives such as self-supervised pretraining without classification heads or regression.The paper says this constraint can be mitigated by training a linear classifier on a frozen encoder using modest external labeled data, optimizing only the classifier.Source: §5 · p. 18
  • Backward compatibility assumes each class label retains the same output index across updates; changing class order or label definitions disrupts the geometric correspondence.A known correspondence between old and new indices permits class reordering to be undone by a fixed output-coordinate permutation.Source: §5 · pp. 18–19
  • The theoretical guarantees are expected results under von Mises–Fisher class-distribution and increasing-concentration assumptions, not an exact characterization of learned representations.The paper reports empirical compatibility improvements across evaluated settings despite this modeling boundary.Source: §5 · p. 19

Broader Impact Statement

Backward-compatible retrieval can reduce repeated gallery re-indexing costs, but preserving legacy representations may retain inherited biases and aggregate metrics may conceal uneven subgroup performance.

Broader Impact Statement

  • Compatibility can reduce re-indexing costs, yet retaining legacy representations may preserve inherited biases and aggregate metrics do not ensure uniform subgroup performance.The paper calls for disaggregated compatibility measures and identifying when re-indexing is preferable, particularly after bias-correcting, semantic, information-removal, or substantial distribution-shift updates.Source: Broader Impact Statement · p. 19

A Proof of Proposition 3.3

The proof analyzes logit directions under an idealized softmax cross-entropy optimization and reports experiments on simplex convergence. It also derives prototype comparisons across class counts by zero-padding old prototypes.

A Proof of Proposition 3.3

  • Under logit-UFM assumptions, logits remain aligned with the Eq. 2 directions as softmax outputs converge to simplex vertices.The proof treats each sample's logit vector as an independent optimization variable initialized at the origin, with gradient descent on softmax cross-entropy.Source: A Proof of Proposition 3.3 · pp. 28–29

B Convergence of PSP and LSP to a Simplex Configuration

  • Across CIFAR-100 and Tiny-ImageNet-200 with ResNet-50 and DenseNet, logits show greater feature-distribution spread and less simplex alignment than softmax outputs.The plotted training-loss and Neural Collapse metrics are consistent with the ResNet-18 CIFAR-100 results in Fig. 3.Source: B Convergence of PSP and LSP to a… · pp. 29–30, Figure 6 · p. 29

C Mathematical Derivation of Eq. 6

  • The Eq. 6 derivation compares prototypes across dimensions by zero-padding an old prototype and computing its cosine similarity with the later prototype.The passage describes the comparison procedure and misalignment angle, but does not provide a legible numerical angle result.Source: C Mathematical Derivation of Eq. 6 · p. 30

D Proof of Proposition 3.4

When classes are added, the proof shows that new-class prototypes are orthogonal to the subspace spanned by the earlier class prototypes, assuming existing class indices are preserved.

D Proof of Proposition 3.4

  • When class count increases from Ck to Ct, each newly introduced class prototype is orthogonal to the subspace spanned by the earlier class prototypes.The proof extends the result from one added class to multiple added classes by zero-padding the old prototypes.Source: D Proof of Proposition 3.4 · pp. 30, 32
  • For the one-class-addition proof, existing classes retain their coordinate indices; the argument treats the d=1 case as trivial and calculates for d≥2.The new class adds a simplex vertex and shifts the simplex center before its prototype is compared with zero-padded old prototypes.Source: D Proof of Proposition 3.4 · p. 30
  • The derivation establishes orthogonality by showing that a generic zero-padded old prototype and the new-class prototype have zero dot product.The proof states that this holds for every earlier class prototype, making each new-class prototype orthogonal to their span.Source: D Proof of Proposition 3.4 · p. 32

E Proof of Proposition 3.5

The transformation P_t,k maps updated prototypes into the earlier feature space, aligning prototypes for classes already present. At retrieval time, correction is applied to incoming queries while the gallery remains fixed.

E Proof of Proposition 3.5

  • P_t,k aligns each transformed prototype for a class already present at step k with its counterpart at that step.The proof extends from adding one class to updates that introduce multiple classes.Source: E Proof of Proposition 3.5 · pp. 32–33

F Pseudocode

  • PSP or LSP must be chosen before deployment and used consistently for gallery indexing and future queries.They are alternative feature definitions, not post-hoc transformations of arbitrary deployed embeddings.Source: F Pseudocode · p. 33
  • P_t,k is needed only when the updated model has more classes; it maps the query into the gallery’s earlier feature space without re-indexing the gallery.When class counts are equal, no cross-update transformation is required.Source: F Pseudocode · p. 33
  • The transformation is an inference-time correction, not a training component; training adds no previous-model dependence, compatibility loss, alignment module, or architectural change.In backward-compatible retrieval, only incoming queries are corrected; stored gallery features are left unchanged.Source: F Pseudocode · p. 33

G Theoretical Analysis of the Backward-Compatibility Definition

The theoretical analysis gives conditions for backward-compatibility inequalities to hold in expectation under a vMF model with increasing concentration. Aligned regular-simplex prototypes satisfy the required geometry, while the guarantee remains conditional on its assumptions.

G Theoretical Analysis of the Backward-Compatibility Definition

  • Under independent class-conditional vMF features whose concentration increases across updates, compatibility holds in expectation when corresponding prototypes align and distinct-class prototypes have angles greater than π/2.The expected-distance inequalities are illustrated across feature-space dimensions under these assumptions.Source: G Theoretical Analysis of the… · p. 35, Figure 7 · p. 36
  • The guarantee depends on increased concentration around class prototypes, not on classification accuracy; as concentration approaches zero, expected same- and different-class distances become less separated.The theorem requires updated representations to be more concentrated than the old ones.Source: G Theoretical Analysis of the… · p. 36
  • Aligned regular-simplex prototypes centered at the origin satisfy the geometric conditions and therefore the compatibility inequalities in expectation under Theorem 3.6’s assumptions.The simplex has pairwise angles greater than π/2.Source: G Theoretical Analysis of the… · p. 37
  • Simplex ETF geometry alone does not ensure corresponding prototypes align across independently obtained representation spaces, because a common orthogonal rotation preserves within-space geometry while changing orientation.Alignment is an additional condition needed for the expectation-level compatibility result.Source: G Theoretical Analysis of the… · p. 38

H Empirical Evidence of Increased Concentration Across Model Updates

Across the tested updates, logits and softmax outputs generally become more concentrated around class prototypes. The evidence supports the increasing-concentration assumption, but the trend is not universal for every class under domain shift.

Extended Classes Scenario

Independently Trained Models in Public Repositories Scenario

  • With ImageNet-1K-pretrained ResNet-18 and ResNet-50 models evaluated on CIFAR-10, updated-model logits are more concentrated for 7 of 10 classes and softmax outputs for all 10.This is observed in an open-set evaluation where the test distribution differs from training.Source: H Empirical Evidence of Increased… · p. 39, Figure 8 · p. 39

H Empirical Evidence of Increased Concentration Across Model Updates

  • A few classes show less concentrated logits under domain shift, so increased concentration holds predominantly rather than for every class.The authors interpret the observed overall trend as empirical evidence consistent with increased vMF concentration.Source: H Empirical Evidence of Increased… · p. 39

I Empirical Analysis of the Centering Projection and the Angular Correction

The centering projection and angular correction play complementary roles in the softmax-based representation. Only their combination, PSP, yields positive average margins for both compatibility inequalities across all evaluated model pairs.

I Empirical Analysis of the Centering Projection and the Angular Correction

  • Without centering, softmax features remain in the nonnegative orthant, so different-class margins stay negative even when query restriction makes same-class margins positive.The centering projection removes this limitation; the comparison uses a 5-to-10-class CIFAR-10 update with ResNet-18.Source: I Empirical Analysis of the Centering… · pp. 39–40
  • Only PSP yields positive average margins for both compatibility inequalities across all evaluated model pairs.The analysis compares average margins and counts pairs for which both margins are positive.Source: I Empirical Analysis of the Centering… · p. 40, Table 6 · p. 40
  • Centering places class prototypes in a regular simplex with inter-class angles greater than π/2, while P_t,k restores cross-update alignment lost under class expansion.The two operations jointly support the geometric conditions identified by Theorem 3.6.Source: I Empirical Analysis of the Centering… · p. 40

J Implementation Details For Standard Compatibility Benchmarks

The appendix specifies training and data-partitioning settings for CIFAR-100, ImageNet-1K, and Google Landmarks v2 experiments.

J Implementation Details For Standard Compatibility Benchmarks

  • CIFAR-100 experiments use 32×32 images and train ResNet-18 or ResNet-50 for 120 epochs, with class-disjoint partitions for Extended Classes.Training uses batch size 128, SGD, and a temperature factor of 12 in cross-entropy loss.Source: J Implementation Details For Standard… · p. 41
  • ImageNet-1K experiments use 224×224 images and train ResNet-18 or ResNet-50 for 90 epochs, partitioning classes evenly across Extended Classes update steps.Training uses batch size 1536 and SGD.Source: J Implementation Details For Standard… · p. 41
  • Google Landmarks v2 experiments train an ImageNet-1K-initialized ResNet-18 for 30 epochs, with the first update using 24,393 classes.Remaining classes are divided into T −1 class-disjoint subsets.Source: J Implementation Details For Standard… · p. 41

K Compatibility Matrices for 5-Step Update Scenario in the Extended Classes Benchmark

In the five-step CIFAR-100 Extended Classes scenario, PSP and LSP show the strongest compatibility while retaining competitive retrieval performance.

K Compatibility Matrices for 5-Step Update Scenario in the Extended Classes Benchmark

  • PSP and LSP achieve the highest compatibility while maintaining competitive retrieval performance across the five-step update sequence.Figure 9 presents CMC@1 compatibility matrices for the scenario.Source: K Compatibility Matrices for 5-Step… · p. 41

L Additional Results on Independently Trained Models from Public Repositories

Across independently pretrained public models, PSP and LSP improve backward compatibility, with the relative trade-off between compatibility and retrieval depending on the setting. Linear probes also extend the approach to self-supervised and vision-language encoders.

L Additional Results on Independently Trained Models from Public Repositories

L.1 Extension to Self-Supervised and Vision-Language Encoders via Linear Probing

  • Frozen-backbone linear probes on ImageNet-1K produce PSP and LSP features from classifier outputs, with normalized encoder features as the baseline.The protocol covers DINO and CLIP families.Source: §L.1 · p. 43, Figure 13 · p. 45, Figure 14 · p. 45
  • For DINO, PSP achieves AC = 1.0 and ACA = 72.00, while LSP achieves AC = 0.67 and ACA = 47.02.The baseline has reasonable same-model retrieval but negligible cross-model retrieval; Figure 13 shows the compatibility matrices.Source: §L.1 · p. 44, Figure 13 · p. 45
  • For CLIP, PSP achieves AC = 1.0 and ACA = 59.20, while LSP improves over the baseline but remains below PSP.Figure 14 shows the CLIP-family compatibility matrices.Source: §L.1 · p. 44, Figure 14 · p. 45
  • The linear-probe protocol offers a practical way to apply PSP and LSP to pretrained encoders without native classification heads.Across DINO and CLIP, both representations achieve substantially higher compatibility metrics than the baseline.Source: §L.1 · p. 44, Table 7 · p. 44
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