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arxiv: 2605.16785 · v1 · pith:B4GAKXG7new · submitted 2026-05-16 · 💻 cs.CV · cs.AI

Encoding Robust Topological Signatures for Hyperdimensional Computing

Arpan Kusari This is my paper

Pith reviewed 2026-05-19 21:33 UTC · model grok-4.3

classification 💻 cs.CV cs.AI
keywords hyperdimensional computingtopological signaturesrobustnessZernike momentsFourier descriptorsimage classificationMNISTedge learning
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The pith

Topology-guided hyperdimensional computing resists image corruptions by encoding holes and rotation-invariant shape signatures.

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

The paper establishes that standard pixel-based hyperdimensional computing loses accuracy under shifts like rotation, noise, or occlusion, but incorporating explicit topological primitives reverses much of that loss. Discrete holes are extracted from binarized shapes and described with RTS-invariant signatures, then projected into hypervectors and bundled in a permutation-invariant way. Learned reliability weights balance the outer-shape Zernike channel against the hole channel through late fusion of cosine similarities. Experiments on MNIST and EMNIST show the resulting representations maintain high accuracy across multiple corruption families and support lightweight online updates. The approach reaches competitive clean-data accuracy while surpassing a compact CNN trained only on clean images on several pixel-level corruptions.

Core claim

By extracting holes from binarized shapes, encoding the outer contour with spatial-pyramid Zernike moments and each hole with an intrinsic Fourier radial signature plus relative geometry, mapping every primitive to a bipolar hypervector, and aggregating variable-cardinality hole sets through permutation-invariant bundling, the method produces a single image hypervector whose cosine similarity remains stable under rotation, Gaussian noise, salt-and-pepper noise, cutout, and zoom.

What carries the argument

RTS-invariant topological descriptors (spatial-pyramid Zernike moments for the outer shape and intrinsic Fourier radial signatures for holes) mapped to hypervectors via randomized projection and role binding, then aggregated by permutation-invariant bundling.

If this is right

  • Competitive clean accuracy is achieved while robustness to multiple pixel-level corruptions markedly exceeds that of a compact CNN trained on clean data.
  • Lightweight online training further improves performance under distribution shifts without retraining from scratch.
  • Variable numbers of holes are handled naturally without fixed-size padding or truncation.
  • Late fusion with learned nonnegative reliability weights prevents any single cue from dominating the final similarity score.

Where Pith is reading between the lines

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

  • The same topological encoding could be applied to other shape-based tasks where holes carry semantic information, such as defect detection in manufacturing imagery.
  • The reliability-weighting step offers a template for fusing additional invariant descriptors in future HD pipelines without manual tuning.
  • Because the method remains prototype-based and supports online updates, it may suit continual-learning scenarios where new corruptions appear after deployment.

Load-bearing premise

Holes and other topological primitives can be extracted reliably from binarized images even after rotation, noise, occlusion, or scaling corruptions.

What would settle it

If replacing the topological signatures with random features of the same dimensionality eliminates the robustness advantage over the naive HD baseline on the same corrupted test sets, the claim that explicit topology drives the gain would be refuted.

Figures

Figures reproduced from arXiv: 2605.16785 by Arpan Kusari.

Figure 1
Figure 1. Figure 1: Confusion matrices of MNIST dataset classification after training for 20 epochs using naive [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Outer contour (blue) and hole contour (red). The shape A in our case could be framed in two parts - global geometry through the shape of outer boundary AO and geometry of the holes (if any) present inside the shape, AHi , where Hi represents the i th hole, 0 ≤ Hi ≤ Hmax. The outer contour is the usual object silhouette information which represents the boundary/appearance-based invariants commonly used in c… view at source ↗
read the original abstract

Hyperdimensional (HD) computing offers an attractive alternative to deep networks for edge learning due to its simplicity, fast prototype-based inference, and compatibility with online updates. However, standard pixel-based HD encoders are brittle: small distribution shifts such as rotation, noise, or occlusion can drastically reduce accuracy. We extract discrete topological primitives-most notably holes-from binarized shapes and pair them with rotation/translation/scale (RTS)-invariant shape signatures. Our method constructs RTS-stable descriptors for (i) the outer shape using a spatial-pyramid variant of Zernike moments and (ii) each hole using an intrinsic Fourier descriptor of its radial signature together with RTS-canonical relative geometry. Each primitive is mapped to a bipolar hypervector via randomized projection and role binding, and variable-cardinality hole sets are aggregated by permutation-invariant bundling to form a single image hypervector. To avoid over-weighting any cue, we learn nonnegative reliability weights for the Zernike and hole channels on a validation set via late fusion of cosine similarities. Experiments on MNIST and EMNIST under controlled corruptions (rotation, Gaussian noise, salt-and-pepper, cutout, zoom) show that Topology-guided HD computing substantially improves robustness compared with a naive HD baseline, maintaining high accuracy across multiple corruption families and benefiting from lightweight online training. Compared with a compact CNN trained on clean data, our method achieves competitive clean accuracy while offering markedly stronger robustness to several pixel-level corruptions, demonstrating that explicit topological structure is a practical route to robust HD representations. The code is provided at https://github.com/arpan-kusari/Topological-HDC.

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

2 major / 2 minor

Summary. The manuscript proposes a topology-guided hyperdimensional computing (HDC) encoder for robust image classification. It binarizes inputs to extract outer contours and holes, computes RTS-invariant descriptors via spatial-pyramid Zernike moments for the outer shape and intrinsic Fourier radial signatures for holes, maps each primitive to a bipolar hypervector through randomized projection and role binding, aggregates variable hole sets via permutation-invariant bundling, and learns nonnegative reliability weights on a validation set for late-fusion cosine similarity. Experiments on MNIST and EMNIST under controlled corruptions (rotation, Gaussian noise, salt-and-pepper, cutout, zoom) report substantially higher robustness than a naive pixel-based HD baseline, competitive clean accuracy versus a compact CNN trained on clean data, and benefits from lightweight online training; code is released at the provided GitHub link.

Significance. If the reported robustness gains are reproducible, the work supplies a concrete demonstration that explicit topological primitives can be encoded into HDC representations to improve resilience to pixel-level distribution shifts while retaining the simplicity and online-update advantages of HDC. The provision of code is a positive contribution to reproducibility. The central claim, however, rests on the assumption that the chosen topological descriptors remain stable under the tested corruptions.

major comments (2)
  1. [§3] §3 (Encoding pipeline): the binarization step used to extract discrete holes and outer contours is described without any threshold value, selection method (fixed, Otsu, adaptive), or morphological cleanup. Under salt-and-pepper densities above ~10% or Gaussian σ > 0.2, a fixed global threshold can erase genuine holes or introduce spurious ones, directly altering the topological primitives whose RTS-invariant signatures are claimed to confer robustness. This is load-bearing for the main experimental claim; the manuscript must specify the exact binarization procedure and report the fraction of corrupted samples on which topology is preserved.
  2. [§4] §4 (Experiments): the robustness results are presented without statistical significance tests, exact hyperparameter settings for the naive HD baseline, or ablation tables isolating the contribution of the Zernike channel versus the hole channel. Without these, it is impossible to verify whether the reported accuracy gains are attributable to the topological encoding or to other implementation choices.
minor comments (2)
  1. [Abstract and §4] The abstract and §4 refer to 'lightweight online training' but do not quantify the number of updates or the learning-rate schedule used; a brief clarification would improve reproducibility.
  2. [Figures] Figure captions should explicitly state the corruption parameters (e.g., exact σ for Gaussian noise, density for salt-and-pepper) rather than referring only to 'controlled corruptions'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major point below and have revised the paper to incorporate the suggested clarifications and additions, which we believe strengthen the presentation and verifiability of our results.

read point-by-point responses

  1. Referee: [§3] §3 (Encoding pipeline): the binarization step used to extract discrete holes and outer contours is described without any threshold value, selection method (fixed, Otsu, adaptive), or morphological cleanup. Under salt-and-pepper densities above ~10% or Gaussian σ > 0.2, a fixed global threshold can erase genuine holes or introduce spurious ones, directly altering the topological primitives whose RTS-invariant signatures are claimed to confer robustness. This is load-bearing for the main experimental claim; the manuscript must specify the exact binarization procedure and report the fraction of corrupted samples on which topology is preserved.

    Authors: We agree that the binarization procedure requires explicit specification to support the robustness claims. In the revised manuscript, Section 3 now details that we apply Otsu’s automatic thresholding to the input grayscale images, followed by morphological closing with a 3×3 structuring element and removal of connected components smaller than 5 pixels to suppress noise-induced artifacts. We have also added a new supplementary table (Table S1) that reports, for each corruption type and severity level, the percentage of test samples in which the number of detected holes remains identical to the clean-image topology. These preservation rates exceed 82% for salt-and-pepper densities ≤10% and Gaussian σ≤0.2, and drop gracefully at higher levels, directly addressing the concern that topological primitives remain sufficiently stable under the tested conditions. revision: yes

  2. Referee: [§4] §4 (Experiments): the robustness results are presented without statistical significance tests, exact hyperparameter settings for the naive HD baseline, or ablation tables isolating the contribution of the Zernike channel versus the hole channel. Without these, it is impossible to verify whether the reported accuracy gains are attributable to the topological encoding or to other implementation choices.

    Authors: We acknowledge that these omissions limit the ability to fully attribute the observed gains. The revised Section 4 now includes: (i) paired t-tests (with p-values reported in the main results table) confirming that accuracy differences versus the naive baseline are statistically significant (p<0.01) across all corruption families; (ii) an appendix table listing the complete hyperparameter set for the naive pixel-based HD baseline (D=10 000, 100 training epochs, random projection seed fixed for reproducibility); and (iii) a dedicated ablation subsection (4.3) with a table comparing Zernike-only, holes-only, and combined channels under every corruption. The ablations show that each channel contributes complementary robustness, with the late-fusion combination yielding the highest scores, thereby isolating the benefit of the topological encoding. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's core pipeline applies established, externally defined invariants (spatial-pyramid Zernike moments and intrinsic Fourier radial signatures) to binarized topological primitives, then aggregates via standard hypervector operations and late-fusion weights fitted on a held-out validation set. These steps are independent of the target robustness claims, which are supported by direct empirical comparisons on MNIST/EMNIST under controlled corruptions rather than by any self-referential reduction or fitted prediction of the final accuracy numbers. No load-bearing equation or premise collapses to a tautology or self-citation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that binarized topological primitives remain extractable and informative under corruption, plus a small number of learned fusion weights; no new physical entities are introduced.

free parameters (1)
  • nonnegative reliability weights for Zernike and hole channels
    Learned on a validation set via late fusion of cosine similarities to balance the two descriptor families.
axioms (1)
  • domain assumption Discrete topological primitives (holes) extracted from binarized shapes can be paired with RTS-invariant signatures that remain stable under the tested corruptions.
    Invoked when constructing the outer-shape Zernike descriptors and the per-hole Fourier radial signatures plus relative geometry.

pith-pipeline@v0.9.0 · 5816 in / 1529 out tokens · 47104 ms · 2026-05-19T21:33:20.804640+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
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    Relation between the paper passage and the cited Recognition theorem.

    We extract discrete topological primitives-most notably holes-from binarized shapes and pair them with rotation/translation/scale (RTS)-invariant shape signatures... using a spatial-pyramid variant of Zernike moments and an intrinsic Fourier descriptor of its radial signature

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    Frontiers in big data , volume=

    An encoding framework for binarized images using hyperdimensional computing , author=. Frontiers in big data , volume=. 2024 , publisher=

    work page 2024

  2. [2]

    2025 11th International Conference on Computing and Artificial Intelligence (ICCAI) , pages=

    Learning encoding phasors with fractional power encoding , author=. 2025 11th International Conference on Computing and Artificial Intelligence (ICCAI) , pages=. 2025 , organization=

    work page 2025

  3. [3]

    Journal of Artificial Intelligence Research , volume=

    A theoretical perspective on hyperdimensional computing , author=. Journal of Artificial Intelligence Research , volume=

    work page

  4. [4]

    Cognitive computation , volume=

    Hyperdimensional computing: An introduction to computing in distributed representation with high-dimensional random vectors , author=. Cognitive computation , volume=. 2009 , publisher=

    work page 2009

  5. [5]

    ACM Computing Surveys , volume=

    A survey on hyperdimensional computing aka vector symbolic architectures, part ii: Applications, cognitive models, and challenges , author=. ACM Computing Surveys , volume=. 2023 , publisher=

    work page 2023

  6. [6]

    Proceedings of the 2016 international symposium on low power electronics and design , pages=

    A robust and energy-efficient classifier using brain-inspired hyperdimensional computing , author=. Proceedings of the 2016 international symposium on low power electronics and design , pages=

    work page 2016

  7. [7]

    , author=

    The perceptron: a probabilistic model for information storage and organization in the brain. , author=. Psychological review , volume=. 1958 , publisher=

    work page 1958

  8. [8]

    Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    Deep neural networks are easily fooled: High confidence predictions for unrecognizable images , author=. Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    work page

  9. [9]

    Association for Computing Machinery CogArch21 , year=

    Hyperdimensional Computing and Spectral Learning , author=. Association for Computing Machinery CogArch21 , year=

    work page

  10. [10]

    Nature Reviews Physics , volume=

    Topological methods for data modelling , author=. Nature Reviews Physics , volume=. 2020 , publisher=

    work page 2020

  11. [11]

    International journal of computer vision , volume=

    On the local behavior of spaces of natural images , author=. International journal of computer vision , volume=. 2008 , publisher=

    work page 2008

  12. [12]

    Journal of Machine Learning Research , volume=

    Topology of deep neural networks , author=. Journal of Machine Learning Research , volume=

    work page

  13. [13]

    Pattern Recognition Letters , volume=

    Exploring generalized shape analysis by topological representations , author=. Pattern Recognition Letters , volume=. 2017 , publisher=

    work page 2017

  14. [14]

    and Harer, J

    Edelsbrunner, H. and Harer, J. , biburl =. Surveys on discrete and computational geometry , interhash =

    work page

  15. [15]

    Computer-Aided Design , volume=

    Shape retrieval using 3D Zernike descriptors , author=. Computer-Aided Design , volume=. 2004 , publisher=

    work page 2004

  16. [16]

    IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans , volume=

    Complex Zernike moments features for shape-based image retrieval , author=. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans , volume=. 2008 , publisher=

    work page 2008

  17. [17]

    Proceedings of the IEEE , volume=

    Gradient-based learning applied to document recognition , author=. Proceedings of the IEEE , volume=. 2002 , publisher=

    work page 2002

  18. [18]

    2017 international joint conference on neural networks (IJCNN) , pages=

    EMNIST: Extending MNIST to handwritten letters , author=. 2017 international joint conference on neural networks (IJCNN) , pages=. 2017 , organization=

    work page 2017