Persistence-Augmented Neural Networks

Arnur Nigmetov; Dmitriy Morozov; Elena Xinyi Wang

arxiv: 2604.08469 · v2 · submitted 2026-04-09 · 💻 cs.LG

Persistence-Augmented Neural Networks

Elena Xinyi Wang , Arnur Nigmetov , Dmitriy Morozov This is my paper

Pith reviewed 2026-05-10 18:10 UTC · model grok-4.3

classification 💻 cs.LG

keywords topological data analysisdata augmentationneural networksMorse-Smale complexpersistencehistopathologyporous materialsgradient flow

0 comments

The pith

Injecting local topological features from the Morse-Smale complex into neural networks improves accuracy on histopathology classification and 3D material regression.

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

The paper develops a data augmentation technique that adds structured topological information to the inputs of convolutional and graph neural networks. It extracts local regions of gradient flow and their multi-scale hierarchy from the Morse-Smale complex of the data, then encodes this information in a form that networks can use directly. The procedure runs in O(n log n) time, which keeps it feasible for sizable datasets. Experiments on two real-world tasks show the augmented models beat both ordinary networks and those using global topological summaries such as persistence images. The work therefore argues that explicit local topology supplies signals that standard layers do not readily discover from raw data.

Core claim

We propose a persistence-based data augmentation framework that encodes local gradient flow regions and their hierarchical evolution using the Morse-Smale complex. This representation, compatible with both convolutional and graph neural networks, retains spatially localized topological information across multiple scales. The augmentation procedure itself has computational complexity O(n log n), making it practical for large datasets. On histopathology image classification and 3D porous material regression, the method consistently outperforms baselines and global TDA descriptors such as persistence images and landscapes. Pruning the base level of the hierarchy further reduces memory usage yet

What carries the argument

Morse-Smale complex persistence augmentation, which partitions data into hierarchical gradient-flow regions and injects their localized persistence features into network inputs.

If this is right

Networks receive explicit multi-scale topological structure without having to learn it implicitly from pixels or nodes.
The O(n log n) cost allows the augmentation to be used on large collections without prohibitive overhead.
Pruning lower hierarchy levels provides a controllable memory-performance trade-off.
The same injection approach works for both grid-based CNNs and graph-based GNNs.
Local topological descriptors outperform global summaries by preserving spatial position of features.

Where Pith is reading between the lines

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

The same local-augmentation idea could be tested on time-series or point-cloud data where gradient structure is also meaningful.
Because the injected features are derived from explicit topological partitions, they may make it easier to trace which structures drive a network's decision.
If the added features prove complementary, similar persistence-based augmentations could be built around other topological constructions besides the Morse-Smale complex.

Load-bearing premise

Local hierarchical features extracted from the Morse-Smale complex supply information that standard neural-network layers cannot easily learn from the raw data alone, and that this information remains useful after injection into CNN or GNN architectures.

What would settle it

Applying the augmentation to the histopathology or porous-material datasets and observing that test accuracy or regression error fails to improve over the unaugmented baseline or over global persistence descriptors.

Figures

Figures reproduced from arXiv: 2604.08469 by Arnur Nigmetov, Dmitriy Morozov, Elena Xinyi Wang.

**Figure 1.** Figure 1: Dual graph and simplification where ϕ(x) denotes the ordered pair of critical points reached by following the gradient path through x. Edges in E are defined between regions that are adjacent in the MS complex. Two nodes v(m1,M1) and v(m2,M2) are connected by an edge if there exists a shared boundary between their corresponding regions in the domain. Specifically, (v1, v2) iff ∃x1 ∈ v1, x2 ∈ v2 and (x1, x2… view at source ↗

**Figure 2.** Figure 2: Construction of the hierarchical graph input for GNNs. Each layer represents a [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Visualization of different levels of simplification [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

read the original abstract

Topological Data Analysis (TDA) provides tools to describe the shape of data, but integrating topological features into deep learning pipelines remains challenging, especially when preserving local geometric structure rather than summarizing it globally. We propose a persistence-based data augmentation framework that encodes local gradient flow regions and their hierarchical evolution using the Morse-Smale complex. This representation, compatible with both convolutional and graph neural networks, retains spatially localized topological information across multiple scales. Importantly, the augmentation procedure itself is efficient, with computational complexity $O(n \log n)$, making it practical for large datasets. We evaluate our method on histopathology image classification and 3D porous material regression, where it consistently outperforms baselines and global TDA descriptors such as persistence images and landscapes. We also show that pruning the base level of the hierarchy reduces memory usage while maintaining competitive performance. These results highlight the potential of local, structured topological augmentation for scalable and interpretable learning across data modalities.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete local Morse-Smale augmentation for feeding hierarchical topological structure into CNNs and GNNs, but the performance edge still needs numbers and stronger controls to be convincing.

read the letter

This paper's core move is to compute the Morse-Smale complex on the input data, extract the local gradient-flow regions and their hierarchy, and inject that as an augmentation into the network. The result stays spatially localized and multi-scale instead of collapsing to a global descriptor like a persistence image or landscape. The authors claim O(n log n) runtime and show a simple pruning step that drops the coarsest level to cut memory while keeping most of the accuracy on the two tasks they test: histopathology slide classification and 3D porous-material regression. Both tasks are natural fits for topological features, and the method is written to plug into standard convolutional or graph layers without changing the architecture much. That combination of local hierarchy, efficiency, and pruning is the part that feels new relative to earlier global TDA pipelines. The abstract positions the gains as consistent over baselines and global TDA methods, which would matter if they hold. The soft spot is exactly the one the stress-test note flags: we still do not see whether these Morse-Smale features actually carry information that a sufficiently deep or well-preprocessed network could not extract from the raw pixels or graph values alone. Without ablations against other local multi-scale descriptors (gradient pyramids, wavelets, or learned filters) and without error bars or dataset sizes, it is hard to tell how much of the reported lift is truly topological versus just better local conditioning. The central assumption therefore remains untested in the material provided. This is aimed at groups already working at the TDA-deep learning boundary, especially those handling scientific images or graphs where preserving geometric hierarchy at multiple scales is already known to help. A reader who needs a practical, scalable way to add that kind of inductive bias would get something usable from the framework even before the numbers are fully vetted. It deserves a serious referee because the procedure is clearly motivated, the complexity bound is attractive, and the target applications are concrete. I would send it out with requests for the full experimental tables, direct comparisons to non-topological local baselines, and checks on whether the hierarchy pruning actually preserves the claimed signal.

Referee Report

2 major / 2 minor

Summary. The paper proposes a persistence-based data augmentation framework that encodes local gradient flow regions and their hierarchical evolution using the Morse-Smale complex. This representation is compatible with both convolutional and graph neural networks, retains spatially localized topological information across multiple scales, and has O(n log n) computational complexity. It is evaluated on histopathology image classification and 3D porous material regression tasks, where it outperforms baselines and global TDA descriptors such as persistence images and landscapes. The work also shows that pruning the base level of the hierarchy reduces memory usage while maintaining competitive performance.

Significance. If the claimed performance gains are robust and stem specifically from the non-redundant topological signals provided by the local Morse-Smale features, this could offer a scalable route for injecting multi-scale hierarchical topological information into standard neural architectures. The efficiency bound and the pruning result are practical strengths; the compatibility with both CNNs and GNNs broadens potential impact.

major comments (2)

[Abstract] Abstract: the central claim of 'consistent outperformance' over baselines and global TDA descriptors is stated without any quantitative metrics, ablation results, error bars, or statistical tests. This leaves the magnitude and reliability of the gains unverified and makes the performance advantage impossible to assess from the provided description.
[Method] Method section (Morse-Smale augmentation procedure): the load-bearing assumption that the extracted local hierarchical gradient-flow features supply spatially localized multi-scale signals that standard CNN/GNN layers cannot readily learn from raw inputs is not tested. Direct comparisons against stronger local baselines (e.g., multi-scale gradient magnitude maps or learned local descriptors) are required to rule out redundancy; without them the augmentation may add no new inductive bias.

minor comments (2)

[Abstract] Abstract: specify what the variable n denotes in the stated O(n log n) complexity (pixels, vertices, or data points).
Ensure the experimental section reports the precise mechanism by which the persistence-augmented features are injected into the network (concatenation, embedding layer, etc.) and includes details on hyper-parameter sensitivity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below and describe the changes we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim of 'consistent outperformance' over baselines and global TDA descriptors is stated without any quantitative metrics, ablation results, error bars, or statistical tests. This leaves the magnitude and reliability of the gains unverified and makes the performance advantage impossible to assess from the provided description.

Authors: We agree that the abstract would benefit from quantitative support. In the revised manuscript we will update the abstract to report concrete performance deltas (e.g., accuracy gains on the histopathology task and error reductions on the regression task), reference the error bars and statistical tests already present in the experimental tables, and briefly note the ablation results on hierarchy pruning. revision: yes
Referee: [Method] Method section (Morse-Smale augmentation procedure): the load-bearing assumption that the extracted local hierarchical gradient-flow features supply spatially localized multi-scale signals that standard CNN/GNN layers cannot readily learn from raw inputs is not tested. Direct comparisons against stronger local baselines (e.g., multi-scale gradient magnitude maps or learned local descriptors) are required to rule out redundancy; without them the augmentation may add no new inductive bias.

Authors: We acknowledge that additional local baselines would more rigorously isolate the contribution of the hierarchical Morse-Smale features. While our existing experiments already show consistent gains over both standard CNN/GNN architectures and global TDA descriptors, we will add direct comparisons against multi-scale gradient magnitude maps and other learned local descriptors in the revised experiments. These new results will be used to quantify whether the topological hierarchy supplies non-redundant signals beyond what local gradient or descriptor-based augmentations provide. revision: yes

Circularity Check

0 steps flagged

No circularity: procedural augmentation is independently constructible

full rationale

The paper defines a new data-augmentation pipeline that computes the Morse-Smale complex on input data, extracts hierarchical persistence features at multiple scales, and injects the resulting local descriptors into CNN or GNN layers. This construction is algorithmic (O(n log n) complexity stated explicitly) and does not rely on any fitted parameter, self-referential equation, or prior result by the same authors to justify its correctness. Performance claims are supported by external empirical comparisons against baselines and global TDA descriptors on held-out datasets; nothing in the derivation chain reduces to a tautology or to a quantity defined in terms of the target prediction. The central assumption (that the extracted features supply non-redundant inductive bias) is testable independently of the paper and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard assumptions from Morse theory and computational topology; no new free parameters, ad-hoc constants, or invented entities are introduced in the abstract.

axioms (1)

domain assumption A Morse-Smale complex can be reliably computed from gradient information on the input data and yields a useful hierarchical decomposition of local flow regions.
Invoked when the paper states that the complex encodes local gradient flow regions and their evolution.

pith-pipeline@v0.9.0 · 5457 in / 1310 out tokens · 135970 ms · 2026-05-10T18:10:05.214714+00:00 · methodology

discussion (0)

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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a persistence-based data augmentation framework that encodes local gradient flow regions and their hierarchical evolution using the Morse-Smale complex... simplification levels... inter-level edges that track the merging of regions across simplification steps.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

hierarchical simplification via persistence... sequence of increasing thresholds ε1 < ε2 < ⋯ < εk, and iteratively cancel all persistence pairs

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Clique topology reveals intrinsic geometric structure in neural correlations.Proceedings of the National Academy of Sciences, 112(44):13455–13460, 2015

Chad Giusti, Eva Pastalkova, Carina Curto, and Vladimir Itskov. Clique topology reveals intrinsic geometric structure in neural correlations.Proceedings of the National Academy of Sciences, 112(44):13455–13460, 2015

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Persistent homology analysis of protein structure, flexibility, and folding.Int

Kelin Xia and Guo-Wei Wei. Persistent homology analysis of protein structure, flexibility, and folding.Int. J. Numer. Method. Biomed. Eng., 30(8):814–844, August 2014

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Perslay: A neural network layer for persistence diagrams and new graph topo- logical signatures

Mathieu Carri` ere, Frederic Chazal, Yuichi Ike, Theo Lacombe, Martin Royer, and Yuhei Umeda. Perslay: A neural network layer for persistence diagrams and new graph topo- logical signatures. In Silvia Chiappa and Roberto Calandra, editors,Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, volume 108 ofProce...

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Krishnapriyan, Maciej Haranczyk, and Dmitriy Morozov

Aditi S. Krishnapriyan, Maciej Haranczyk, and Dmitriy Morozov. Topological descriptors help predict guest adsorption in nanoporous materials.The Journal of Physical Chemistry C, 124(17):9360–9368, Apr 2020

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A survey of topological machine learning methods.Frontiers in Artificial Intelligence, Volume 4 - 2021, 2021

Felix Hensel, Michael Moor, and Bastian Rieck. A survey of topological machine learning methods.Frontiers in Artificial Intelligence, Volume 4 - 2021, 2021

work page 2021

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Per- sistence images: A stable vector representation of persistent homology.Journal of Machine Learning Research, 18(8):1–35, 2017

Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Per- sistence images: A stable vector representation of persistent homology.Journal of Machine Learning Research, 18(8):1–35, 2017

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Statistical topological data analysis using persistence landscapes.Journal of Machine Learning Research, 16(3):77–102, 2015

Peter Bubenik. Statistical topological data analysis using persistence landscapes.Journal of Machine Learning Research, 16(3):77–102, 2015

work page 2015

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Oudot, and Maks Ovsjanikov

Mathieu Carri` ere, Steve Y. Oudot, and Maks Ovsjanikov. Stable topological signatures for points on 3d shapes.Computer Graphics Forum, 34(5):1–12, August 2015

work page 2015

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Hierarchical morse complexes for piecewise linear 2-manifolds.Proceedings of the Seventeenth Annual Symposium on Computational Geometry, page 70–79, 2001

Herbert Edelsbrunner, John Harer, and Afra Zomorodian. Hierarchical morse complexes for piecewise linear 2-manifolds.Proceedings of the Seventeenth Annual Symposium on Computational Geometry, page 70–79, 2001

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Chen Cai, Nikolaos Vlassis, Lucas Magee, Ran Ma, Zeyu Xiong, Bahador Bahmani, Teng- Fong Wong, Yusu Wang, and WaiChing Sun. Equivariant geometric learning for digital rock physics: Estimating formation factor and effective permeability tensors from morse graph.International Journal for Multiscale Computational Engineering, 21(5):1–24, 2023

work page 2023

[15] [15]

Topology-aware segmentation using discrete morse theory

Xiaoling Hu, Yusu Wang, Fuxin Li, Dimitris Samaras, and Chao Chen. Topology-aware segmentation using discrete morse theory. In9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net, 2021

work page 2021

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Morse complexes for shape segmentation and homological analysis: discrete models and algorithms.Comput

Leila De Floriani, Ulderico Fugacci, Federico Iuricich, and Paola Magillo. Morse complexes for shape segmentation and homological analysis: discrete models and algorithms.Comput. Graph. Forum, 34(2):761–785, May 2015

work page 2015

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Gyulassy and Vijay Natarajan

A. Gyulassy and Vijay Natarajan. Topology-based simplification for feature extraction from 3d scalar fields. InVIS 05. IEEE Visualization, 2005., pages 535–542, 2005

work page 2005

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Hansen, Hans Hagen, and Valerio Pascucci

Attila Gyulassy, Natallia Kotava, Mark Kim, Charles D. Hansen, Hans Hagen, and Valerio Pascucci. Direct feature visualization using morse-smale complexes.IEEE Transactions on Visualization and Computer Graphics, 18(9):1549–1562, September 2012. 13

work page 2012

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Topological autoen- coders

Michael Moor, Max Horn, Bastian Rieck, and Karsten Borgwardt. Topological autoen- coders. In Hal Daum´ e III and Aarti Singh, editors,Proceedings of the 37th International Conference on Machine Learning, volume 119 ofProceedings of Machine Learning Re- search, pages 7045–7054. PMLR, 13–18 Jul 2020

work page 2020

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Perea, Elizabeth Munch, and Firas A

Jose A. Perea, Elizabeth Munch, and Firas A. Khasawneh. Approximating continuous functions on persistence diagrams using template functions.Foundations of Computational Mathematics, jun 2022

work page 2022

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