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arxiv: 2512.07988 · v3 · submitted 2025-12-08 · 💻 cs.LG · cs.GR· cs.HC

HOLE: Homological Observation of Latent Embeddings for Neural Network Interpretability

Sudhanva Manjunath Athreya , Paul Rosen This is my paper

Pith reviewed 2026-05-16 23:51 UTC · model grok-4.3

classification 💻 cs.LG cs.GRcs.HC
keywords persistent homologyneural network interpretabilitylatent embeddingstopological data analysisclass separationmodel robustnessfeature disentanglement
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The pith

Persistent homology on neural network activations reveals topological patterns tied to class separation and robustness.

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

Deep learning models succeed at tasks but keep their internal representations hard to inspect. This paper presents HOLE, which computes persistent homology on the activations inside network layers to extract shape-based features such as connected components and holes. These features are shown through visualizations including cluster flow diagrams, blob graphs, and heatmap dendrograms that track how data structure changes from layer to layer. The work finds that the resulting topological signatures align with improved class separation, disentangled features, and greater resistance to input changes or compression. This supplies a geometric perspective that can complement existing ways of probing model behavior.

Core claim

HOLE extracts topological features from intermediate activations using persistent homology and visualizes them with cluster flow diagrams, blob graphs, and heatmap dendrograms. Evaluation on discriminative models shows these features associate with class separation, feature disentanglement, and robustness to perturbations and compression.

What carries the argument

Persistent homology applied directly to the intermediate activations of a neural network to track topological evolution across layers.

Load-bearing premise

That the topological invariants computed on activations reflect meaningful semantic properties such as class separation instead of unrelated geometric artifacts of the embedding spaces.

What would settle it

A test on matched models known to differ sharply in class separation that finds no corresponding difference in their persistent homology barcodes or persistence diagrams at the same layers.

Figures

Figures reproduced from arXiv: 2512.07988 by Paul Rosen, Sudhanva Manjunath Athreya.

Figure 1
Figure 1. Figure 1: HOLE provides global interpretability via multiple visualization techniques: (left) Sankey flows (layer-wise represen [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example (left) persistence diagram and (rght) barcode. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: HOLE overview shows how during inference, neural net [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (o) The input dataset was used to generate (a-d,i-k) dis [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Examples of the visualizations used to support tasks [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Input noise robustness evaluation on CIFAR-10. Various [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of (a-d) Sankey diagrams for ViT encoder [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of (left) Blob graphs and (right) Sankey dia [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: ResNet-34 persistent dendrogram + heatmap before and [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Blob visualizations of ViT encoder layer 11 activations [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
read the original abstract

Deep learning models have achieved remarkable success across various domains, yet their learned representations and decision-making processes remain largely opaque and hard to interpret. This work introduces HOLE (Homological Observation of Latent Embeddings), a method for analyzing and interpreting discriminative neural networks through persistent homology. HOLE extracts topological features from intermediate activations and presents them using a suite of visualization techniques, including cluster flow diagrams, blob graphs, and heatmap dendrograms. These tools facilitate the examination of representation structure and quality across layers. We evaluate HOLE using a range of discriminative models, focusing on representation quality, interpretability across layers, and robustness to input perturbations and model compression. The results indicate that topological analysis reveals patterns associated with class separation, feature disentanglement, and model robustness, providing a complementary perspective for understanding and improving deep learning systems.

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

3 major / 2 minor

Summary. The paper introduces HOLE, a method that applies persistent homology to intermediate activations of discriminative neural networks to extract topological features, which are then visualized via cluster flow diagrams, blob graphs, and heatmap dendrograms. It claims this provides insights into representation quality, layer-wise interpretability, class separation, feature disentanglement, and robustness to perturbations and compression, evaluated across a range of models.

Significance. If validated with appropriate controls, HOLE could provide a useful topological lens for neural network interpretability that complements existing activation-based or attribution methods, potentially helping identify structural changes across layers or under model modifications.

major comments (3)
  1. Abstract and evaluation sections: the claims of revealing patterns associated with class separation, feature disentanglement, and model robustness are supported only by qualitative descriptions; no quantitative metrics (e.g., persistence diagram distances, classification accuracies on topological features), baselines (e.g., random networks or untrained models), or statistical analysis are reported to substantiate the interpretability conclusions.
  2. Method and experiments: the central assumption that persistent homology barcodes on activation point clouds encode learned class structure (rather than incidental geometry of the input manifold or any Lipschitz embedding) is not tested via controls such as randomly initialized networks, label-shuffled training, or linear probes on the same inputs; without these, the interpretability interpretation remains ungrounded.
  3. Evaluation claims: the robustness analysis to input perturbations and model compression lacks specific comparisons (e.g., before/after compression persistence diagrams or correlation with accuracy drops) that would make the robustness findings load-bearing rather than observational.
minor comments (2)
  1. The visualization techniques (cluster flow diagrams, blob graphs) would benefit from explicit pseudocode or parameter settings (e.g., filtration thresholds, distance metrics used in Vietoris-Rips) to allow reproducibility.
  2. Notation for persistent homology features (e.g., barcodes, persistence diagrams) should be defined more formally with reference to standard definitions to avoid ambiguity for readers unfamiliar with topological data analysis.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We address each major comment below and will revise the manuscript to incorporate quantitative controls and comparisons where they strengthen the claims without altering the core contribution.

read point-by-point responses

  1. Referee: Abstract and evaluation sections: the claims of revealing patterns associated with class separation, feature disentanglement, and model robustness are supported only by qualitative descriptions; no quantitative metrics (e.g., persistence diagram distances, classification accuracies on topological features), baselines (e.g., random networks or untrained models), or statistical analysis are reported to substantiate the interpretability conclusions.

    Authors: We agree that the current version relies primarily on qualitative visualizations. In the revised manuscript we will add quantitative metrics, including Wasserstein distances between persistence diagrams across layers and models, as well as baseline comparisons against randomly initialized networks. Statistical tests will be included to support the reported patterns. revision: yes

  2. Referee: Method and experiments: the central assumption that persistent homology barcodes on activation point clouds encode learned class structure (rather than incidental geometry of the input manifold or any Lipschitz embedding) is not tested via controls such as randomly initialized networks, label-shuffled training, or linear probes on the same inputs; without these, the interpretability interpretation remains ungrounded.

    Authors: This is a fair criticism. While the original experiments focus on trained models, we will add the suggested controls—randomly initialized networks and label-shuffled training—in the revised version. These experiments will directly test whether the observed topological signatures arise from learned class structure rather than input geometry alone. revision: yes

  3. Referee: Evaluation claims: the robustness analysis to input perturbations and model compression lacks specific comparisons (e.g., before/after compression persistence diagrams or correlation with accuracy drops) that would make the robustness findings load-bearing rather than observational.

    Authors: We accept that more explicit quantitative links are needed. The revised manuscript will include direct before-and-after persistence diagram comparisons under compression and perturbations, together with reported correlations between topological changes and accuracy drops. revision: yes

Circularity Check

0 steps flagged

No circularity: purely observational application of standard persistent homology

full rationale

The paper introduces HOLE as a visualization and analysis pipeline that applies off-the-shelf persistent homology (Vietoris-Rips or equivalent) to intermediate activation point clouds and then renders the resulting barcodes via cluster-flow diagrams, blob graphs, and dendrograms. No equations are presented that derive a new quantity from fitted parameters, no predictions are made that are statistically forced by the same data used to demonstrate them, and no uniqueness theorems or ansatzes are smuggled in via self-citation. All reported patterns (class separation, disentanglement, robustness) are empirical observations from the computed topological features; they are not shown to be equivalent to the input activations by construction. The method is therefore self-contained as an observational tool and receives a score of 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that topological invariants computed via persistent homology on activation vectors carry semantic meaning for model behavior; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Persistent homology applied to point clouds in activation space yields features that reflect representation quality and robustness.
    Invoked when the abstract claims the extracted features reveal class separation and disentanglement.

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