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arxiv: 2606.02841 · v1 · pith:27WW7TM3new · submitted 2026-06-01 · 💻 cs.LG · math.AT

Learning Coherent Representations: A Topological Approach to Interpretability

Sigurd Gaukstad , Melvin Vaupel , Valdemar Karg{\aa}rd Olsen , Erik Hermansen , Benjamin Dunn This is my paper

Pith reviewed 2026-06-28 15:41 UTC · model grok-4.3

classification 💻 cs.LG math.AT
keywords coherenceinterpretabilityVietoris-Rips filtrationtopological data analysisneural representationsautoencodersBERT embeddingsFréchet variance
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The pith

Coherent matrices induce bounded interleaving between Vietoris-Rips filtrations of samples and features, ensuring compatible topological structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines coherence for non-negative matrices as the property that each sample activates a geometrically clustered set of features and each feature activates a clustered set of samples, with every sample and feature participating in at least one such relation. It proves that any coherent matrix produces a bounded interleaving between the Vietoris-Rips filtration built on the samples and the filtration built on the features. This topological compatibility is presented as the mechanism that makes both the individual features and the overall feature space interpretable. The authors introduce the Coh objective, derived from Fréchet variance, as a differentiable loss that can be added to training to enforce the coherence condition, and they contrast it with sparsity by noting that coherence requires geometric connectivity rather than mere rarity of activation.

Core claim

A non-negative matrix is coherent when rows attend to geometrically clustered columns and columns attend to geometrically clustered rows, with full coverage; such matrices induce a bounded interleaving between the Vietoris-Rips filtrations of the row space and column space, so that the two spaces share compatible topological structure. This constraint is enforced by minimizing the Coh objective based on Fréchet variance, which the authors apply inside auto-encoders on synthetic circle data and rotated MNIST as well as inside BERT token embeddings on language data.

What carries the argument

The coherence property of a matrix together with the Coh objective based on Fréchet variance, which together enforce geometrically clustered attendances that produce bounded interleaving of Vietoris-Rips filtrations.

If this is right

  • When data lies on a circle, coherent features must tile the circle into contiguous arcs.
  • Coherence supplies interpretability for both individual features and the geometry of the feature space itself.
  • Unlike sparsity, which only limits the number of samples per feature, coherence additionally requires that the active samples form a connected region.
  • The same coherence objective can be applied to token embeddings in language models to produce interpretable token representations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same coherence construction could be tested on data manifolds other than circles to check whether the bounded interleaving still produces human-readable features.
  • Coherence might be combined with existing regularization techniques to add explicit topological constraints without requiring full manifold knowledge.
  • If coherence improves interpretability on rotated MNIST, similar gains could appear in other rotation-equivariant or manifold-structured vision tasks.
  • The brain-inspired motivation suggests that coherence regularizers might be useful for modeling place or grid cells in artificial systems, though the paper does not test this.

Load-bearing premise

That the geometric coherence produced by the Coh objective will yield features that are meaningfully interpretable to humans on real data distributions rather than merely satisfying the abstract geometric definition.

What would settle it

Training an auto-encoder with the Coh objective on points sampled from a circle and finding that the learned features fail to form contiguous arcs.

Figures

Figures reproduced from arXiv: 2606.02841 by Benjamin Dunn, Erik Hermansen, Melvin Vaupel, Sigurd Gaukstad, Valdemar Karg{\aa}rd Olsen.

Figure 1
Figure 1. Figure 1: Given an auto-encoder with a non-negative activation function, we can treat the encoded latent space as a matrix M, whose rows are samples and columns are features. Most often, the topology of the samples and features are vastly different. We regularize these spaces to be topologically similar by creating an explicit interleaving between the filtered simplicial complexes induced by the latent samples and l… view at source ↗
Figure 2
Figure 2. Figure 2: Coherent vs non-coherent matrices derived from circular state space. Left: Coherence ε = 0.18. Right: Non-coherent ε = 1.46. In rows one and three, we show PCA projection of rows and columns colored by the activation of a column and row. In row two and four we show persistence diagrams of rows and columns where we highlight the most persistent H1. Note that only the coherent matrix exhibits matching circul… view at source ↗
Figure 3
Figure 3. Figure 3: Two Circles Toy experiment. UMAP projections of latent samples and latent features , with persistence diagrams for each. We highlight the two most persistent H1 features, representing the two circles. Samples are colored by activation of a single feature; features are colored by activation on a single sample. Only the COH model gives the expected disjoint circular topology in both spaces [PITH_FULL_IMAGE:… view at source ↗
Figure 4
Figure 4. Figure 4: Single Digit Experiment. For a representative seed we show the UMAP projections of latent samples and latent features, with persistence diagrams for each. We highlight the single most persistent H1 features, representing the expected circle. Samples are colored by activation of a single feature and features are colored by activation on a single sample. Only the COH model gives the expected circular topolog… view at source ↗
Figure 5
Figure 5. Figure 5: Toy experiment: sphere. We replicate the two circle toy experiment with a sphere. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Toy experiment: torus. We replicate the two circle toy experiment with a torus [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Double Digit Experiment. For two different seeds we show UMAP projections of latent samples and latent features, with persistence diagrams for each. We highlight the two most persistent H1 features, representing the two expected circles. Samples are colored by activation of a single feature and features are colored by activation on a single sample. The seeds are picked as to show the diversity in the learn… view at source ↗
Figure 8
Figure 8. Figure 8: Double Digit Experiment (COH + L1). For two different seeds we show UMAP projections of latent samples and latent features, with persistence diagrams for each. We highlight the two most persistent H1 features, representing the two expected circles. Samples are colored by activation of a single feature and features are colored by activation on a single sample. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Single digit experiment. We plot UMAP projections of latent samples and latent features colored by a latent feature and sample representatively. For a random sample of features we plot the weighted sum of the original images, the random features corresponds to the coloring of the latent samples. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Single digit experiment. We analyse at a representative latent space from the single digit experiment using circular coordinates from persistence (co)homology, and compare it against the true generating angles. Second row we color the latent samples by circular coordinates. In the last row we plot circular coordinates against the true angle. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Double digit experiment (Separated). We plot UMAP projections of latent samples and latent features colored by a latent feature and sample representatively. For a random sample of features we plot the weighted sum of the original images, the random features corresponds to the coloring of the latent samples. We note COH nicely distinguishes the two classes into two circles. Features corresponding to the la… view at source ↗
Figure 12
Figure 12. Figure 12: Double digit experiment (Merged). We plot UMAP projections of latent samples and latent features colored by a latent feature and sample representatively. For a random sample of features we plot the weighted sum of the original images, the random features corresponds to the coloring of the latent samples. We note, in this case, COH has merged the two classes into the same circle, but have excellent angular… view at source ↗
Figure 13
Figure 13. Figure 13: Double digit experiment (L 1 and COH) . We plot UMAP projections of latent samples and latent features colored by a latent feature and sample representatively. For a random sample of features we plot the weighted sum of the original images, the random features corresponds to the coloring of the latent samples. Using COH and L1 together we get a much cleaner results. 21 [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 14
Figure 14. Figure 14: Double digit experiment (L 1 and COH). We find a separated a latent space in the double digit experiment using circular coordinates from persistence (co)homology, and compare it against the true generating angles. Note that we have both good angle tuning and component tuning of the features. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Double digit experiment (Separated). We find a separated latent space in the double digit experiment using circular coordinates from persistence (co)homology, and compare it against the true generating angles. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Double digit experiment (Merged). We analyse here a merged latent space from the double digit experiment using circular coordinates from persistence (co)homology, and compare it against the true generating angles. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Mean ± std over 5 seeds for the overlap of the top 20 tokens per feature against the 20-nearest-neighbor token neighborhood in the Vanilla embedding. Scoring interpretability using LLMs. For the first seed of each of the two non-negative token embeddings, we examine the top 10 tokens for each of the 256 features. We feed this list of features into Claude Opus 4.7 and ask it to assign a binary score to eac… view at source ↗
Figure 18
Figure 18. Figure 18: For the first seed, we plot the average best feature overlap with a token neighborhood of the Vanilla model at each epoch. We also track the average coherence score per epoch. 10 0 10 20 15 10 5 0 5 10 15 Coh 26 24 22 20 18 6 8 10 12 Softplus 32 30 28 26 24 14 16 18 20 22 Vanilla 6 8 10 12 14 2 0 2 4 6 8 10 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 2 1 0 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 0.05 0.10 0.15 … view at source ↗
Figure 19
Figure 19. Figure 19: UMAP projections of the token and feature embeddings for the first seed. The plots are colored by the values of a single token vector and a single feature vector. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Again for the first seed: for each feature we plot, on the x-axis, its distance to its 20 nearest feature vectors, and on the y-axis, the overlap of its top 20 tokens with those of the neighboring feature. In words, each point answers: how far am I from that feature (x-axis), and how similar am I to it in terms of top-20 tokens (y-axis)? This showcases that the coherent embedding has more meaningful geome… view at source ↗
read the original abstract

Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction cells respond to contiguous regions of state space. A non-negative matrix is coherent if each row (sample) attends to geometrically clustered columns (features) and vice versa, and in addition every sample is well described by some feature and every feature is needed by some sample. We prove that coherent matrices induce a bounded interleaving between the Vietoris-Rips filtrations of samples and features, guaranteeing that both spaces share compatible topological structure. This geometric constraint facilitates interpretability. For example, if data lies on a circle, coherent features must tile that circle into contiguous arcs. We introduce Coh, a differentiable objective function based on Fr\'echet variance that enforces coherence during training. Unlike sparsity, which bounds how many samples a feature activates on, coherence bounds which samples, requiring geometric connectivity rather than only rarity. This yields not just interpretable features but an interpretable feature space. We validate Coh in an auto-encoder using synthetic and rotated MNIST datasets and in a token embedding of BERT using language data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript defines coherence as a geometric property of non-negative matrices requiring bidirectional clustering of rows (samples) to columns (features) and vice versa, plus coverage conditions ensuring every sample is described by some feature and every feature is used by some sample. It proves that any matrix satisfying this definition induces a bounded interleaving between the Vietoris-Rips filtrations on the sample space and feature space. The paper introduces the differentiable Coh objective, based on Fréchet variance, to enforce coherence during training, contrasts it with sparsity, and validates the approach in an autoencoder on synthetic and rotated MNIST data as well as in BERT token embeddings on language data.

Significance. If the central claim holds, the work supplies a parameter-free topological guarantee that links a geometric matrix property directly to compatible persistent homology between dual spaces, providing a rigorous alternative to sparsity for interpretability. The proof is presented as a direct consequence of the coherence definition with no evident additional assumptions or fitted quantities, which strengthens the contribution. Experiments illustrate the objective's effect on feature geometry, though quantitative measures of improved human interpretability are not reported.

minor comments (3)
  1. [§4] §4 (experiments): the rotated MNIST and BERT results would benefit from explicit comparison metrics (e.g., human annotation agreement or downstream task performance) to substantiate the interpretability claim beyond qualitative examples.
  2. [§3] The definition of the Coh objective in terms of Fréchet variance is introduced without an explicit equation number; adding one would improve traceability to the coherence conditions.
  3. [Figure 2] Figure captions for the synthetic circle example could more clearly annotate the interleaving bound value derived in the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for the positive evaluation of its significance. The recommendation of minor revision is noted. However, the report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines coherence as a geometric property on non-negative matrices (bidirectional clustering plus coverage). It then proves that any matrix meeting this definition induces a bounded interleaving between the Vietoris-Rips filtrations of its row and column spaces. This is presented as a direct mathematical consequence of the definition itself, with no reduction to fitted parameters, self-citations, or ansatzes. The Coh objective is introduced separately as a differentiable loss to encourage the property during training; its empirical success is an independent question. No load-bearing self-citation chains, self-definitional steps, or renamed known results appear in the abstract or described claims. The derivation chain is therefore self-contained against external topological benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the newly defined coherence property and the unverified proof that it induces bounded interleaving in Vietoris-Rips filtrations; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Coherent matrices induce a bounded interleaving between Vietoris-Rips filtrations of samples and features
    Stated as the key theorem in the abstract.
invented entities (1)
  • coherence property no independent evidence
    purpose: Geometric constraint ensuring clustered activations for interpretability
    Newly introduced definition combining non-negative matrix properties with geometric connectivity

pith-pipeline@v0.9.1-grok · 5766 in / 1095 out tokens · 23855 ms · 2026-06-28T15:41:19.524850+00:00 · methodology

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Reference graph

Works this paper leans on

42 extracted references · 6 canonical work pages

  1. [1]

    Gardner and Erik Hermansen and Marius Pachitariu and Yoram Burak and Nils A

    Richard J. Gardner and Erik Hermansen and Marius Pachitariu and Yoram Burak and Nils A. Baas and Benjamin A. Dunn and May-Britt Moser and Edvard I. Moser , title =. Nature , year =

  2. [2]

    Discrete

    Vin de Silva and Dmitriy Morozov and Mikael Vejdemo-Johansson , title =. Discrete. 2011 , volume =. doi:10.1007/s00454-011-9344-x , url =

    work page doi:10.1007/s00454-011-9344-x 2011

  3. [3]

    C. H. Dowker , journal =. Homology Groups of Relations , urldate =

  4. [4]

    Proximity of Persistence Modules and Their Diagrams , booktitle =

    Chazal, Fr. Proximity of Persistence Modules and Their Diagrams , booktitle =. 2009 , location =. doi:10.1145/1542362.1542407 , note =

    work page doi:10.1145/1542362.1542407 2009

  5. [5]

    Distill , year =

    Olah, Chris and Cammarata, Nick and Schubert, Ludwig and Goh, Gabriel and Petrov, Michael and Carter, Shan , title =. Distill , year =

  6. [6]

    Bronstein and Joan Bruna and Taco Cohen and Petar Velickovic , title =

    Michael M. Bronstein and Joan Bruna and Taco Cohen and Petar Velickovic , title =. CoRR , volume =. 2021 , url =. 2104.13478 , timestamp =

    Pith/arXiv arXiv 2021

  7. [7]

    , title =

    Carlsson, Gunnar E. , title =. Bulletin of the American Mathematical Society , volume =. 2009 , doi =

    2009

  8. [8]

    Topology-Preserving Deep Image Segmentation , url =

    Hu, Xiaoling and Li, Fuxin and Samaras, Dimitris and Chen, Chao , booktitle =. Topology-Preserving Deep Image Segmentation , url =

  9. [9]

    Manifold-tiling Localized Receptive Fields are Optimal in Similarity-preserving Neural Networks , url =

    Sengupta, Anirvan and Pehlevan, Cengiz and Tepper, Mariano and Genkin, Alexander and Chklovskii, Dmitri , booktitle =. Manifold-tiling Localized Receptive Fields are Optimal in Similarity-preserving Neural Networks , url =

  10. [10]

    Proceedings of the 36th International Conference on Machine Learning , pages =

    Connectivity-Optimized Representation Learning via Persistent Homology , author =. Proceedings of the 36th International Conference on Machine Learning , pages =. 2019 , editor =

    2019

  11. [11]

    Perea and Luis Scoccola and Christopher J

    Jose A. Perea and Luis Scoccola and Christopher J. Tralie , title =. 2023 , publisher =

    2023

  12. [12]

    Lee and Haim Sompolinsky , title =

    Uri Cohen and SueYeon Chung and Daniel D. Lee and Haim Sompolinsky , title =. Nature Communications , year =

  13. [13]

    Neural Computation , year =

    Erik Rybakken and Nils Baas and Benjamin Dunn , title =. Neural Computation , year =

  14. [14]

    C. H. Dowker , title =. Annals of Mathematics , year =

  15. [15]

    Topological methods , booktitle =

    Anders Bj. Topological methods , booktitle =. 1996 , editor =

    1996

  16. [16]

    1994 , publisher =

    Algebraic Topology , author =. 1994 , publisher =. doi:10.1007/978-1-4684-9322-1 , note =

    work page doi:10.1007/978-1-4684-9322-1 1994

  17. [17]

    2018 , publisher =

    Tralie, Christopher and Saul, Nathaniel and Bar-On, Rann , title =. 2018 , publisher =. doi:10.21105/joss.00925 , url =

    work page doi:10.21105/joss.00925 2018

  18. [18]

    ICML , year=

    Topological autoencoders , author=. ICML , year=

  19. [19]

    2023 , journal=

    Towards Monosemanticity: Decomposing Language Models With Dictionary Learning , author=. 2023 , journal=

    2023

  20. [20]

    Higgins, Irina and others , booktitle=. -

  21. [21]

    ICML , year=

    Emergence of separable manifolds in deep language representations , author=. ICML , year=

  22. [22]

    Proceedings of the 12th International Conference on Learning Representations (ICLR) , year =

    Hoagy Cunningham and Aidan Ewart and Logan Riggs and Robert Huben and Lee Sharkey , title =. Proceedings of the 12th International Conference on Learning Representations (ICLR) , year =

  23. [23]

    Olshausen and David J

    Bruno A. Olshausen and David J. Field , title =. Nature , year =

  24. [24]

    Moser , title =

    Torkel Hafting and Marianne Fyhn and Sturla Molden and May-Britt Moser and Edvard I. Moser , title =. Nature , year =

  25. [25]

    Taube and Robert U

    Jeffrey S. Taube and Robert U. Muller and James B. Ranck Jr. , title =. Journal of Neuroscience , year =

  26. [26]

    2018 , publisher =

    McInnes, Leland and Healy, John and Saul, Nathaniel and Großberger, Lukas , title =. 2018 , publisher =. doi:10.21105/joss.00861 , url =

    work page doi:10.21105/joss.00861 2018

  27. [27]

    Ripser: efficient computation of

    Bauer, Ulrich , journal=. Ripser: efficient computation of. 2021 , publisher=

    2021

  28. [28]

    arXiv preprint arXiv:2209.10652 , year=

    Toy models of superposition , author=. arXiv preprint arXiv:2209.10652 , year=

    Pith/arXiv arXiv

  29. [29]

    BERT : Pre-training of Deep Bidirectional Transformers for Language Understanding

    Devlin, Jacob and Chang, Ming-Wei and Lee, Kenton and Toutanova, Kristina. BERT : Pre-training of Deep Bidirectional Transformers for Language Understanding. Proceedings of the 2019 Conference of the North A merican Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers). 2019. doi:10.18653/v...

    work page doi:10.18653/v1/n19-1423 2019

  30. [30]

    International Conference on Learning Representations , year=

    Pointer Sentinel Mixture Models , author=. International Conference on Learning Representations , year=

  31. [31]

    Topological methods , booktitle =

    Bj. Topological methods , booktitle =

  32. [32]

    Mediterranean Journal of Mathematics , volume =

    Brun, Morten and Salbu, Lars-Arne , title =. Mediterranean Journal of Mathematics , volume =

  33. [33]

    2407.15454 , archiveprefix =

    Brun, Morten and Grinberg, Darij , title =. 2407.15454 , archiveprefix =

    arXiv

  34. [34]

    2408.13136 , archiveprefix =

    Yoon, Iris , title =. 2408.13136 , archiveprefix =

    Pith/arXiv arXiv

  35. [35]

    A functorial

    Chowdhury, Samir and M. A functorial. Journal of Applied and Computational Topology , volume =

  36. [36]

    Rips complexes as nerves and a functorial

    Virk,. Rips complexes as nerves and a functorial. Mediterranean Journal of Mathematics , volume =

  37. [37]

    Journal of Applied and Computational Topology , volume =

    Robinson, Michael , title =. Journal of Applied and Computational Topology , volume =

  38. [38]

    Persistence stability for geometric complexes , journal =

    Chazal, Fr. Persistence stability for geometric complexes , journal =

  39. [39]

    Publications Math

    Segal, Graeme , title =. Publications Math

  40. [40]

    , title =

    Dugger, Daniel and Isaksen, Daniel C. , title =. Mathematische Zeitschrift , volume =

  41. [41]

    2310.11529 , archiveprefix =

    Vaupel, Melvin and Dunn, Benjamin , title =. 2310.11529 , archiveprefix =

    arXiv

  42. [42]

    2026 , howpublished =

    Anthropic , title =. 2026 , howpublished =

    2026