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Certifying Robustness via Topological Representations

Jens Agerberg, Andrea Guidolin, Andrea Martinelli, Pepijn Roos Hoefgeest, David Eklund, Martina Scolamiero

TL;DR

This work introduces the Stable Rank Network (SRN), a two-stage architecture that learns Lipschitz, discriminative representations from persistence diagrams by coupling a learnable stable-rank vectorization with a 1-Lipschitz neural network. By preserving Lipschitz continuity with respect to $W_p$ (and Bottleneck) distances, SRN yields certified $ε$-robustness at test time, addressing adversarial vulnerabilities of diagram-based pipelines. The authors demonstrate competitive accuracy and robust performance on the ORBIT5K dataset, and derive theoretical stability guarantees for stable ranks and their impact on robustness. The approach blends topological data analysis with Lipschitz theory to enable principled robustness, offering a pathway to robust, topology-aware machine learning pipelines.

Abstract

We propose a neural network architecture that can learn discriminative geometric representations of data from persistence diagrams, common descriptors of Topological Data Analysis. The learned representations enjoy Lipschitz stability with a controllable Lipschitz constant. In adversarial learning, this stability can be used to certify $ε$-robustness for samples in a dataset, which we demonstrate on the ORBIT5K dataset representing the orbits of a discrete dynamical system.

Certifying Robustness via Topological Representations

TL;DR

This work introduces the Stable Rank Network (SRN), a two-stage architecture that learns Lipschitz, discriminative representations from persistence diagrams by coupling a learnable stable-rank vectorization with a 1-Lipschitz neural network. By preserving Lipschitz continuity with respect to (and Bottleneck) distances, SRN yields certified -robustness at test time, addressing adversarial vulnerabilities of diagram-based pipelines. The authors demonstrate competitive accuracy and robust performance on the ORBIT5K dataset, and derive theoretical stability guarantees for stable ranks and their impact on robustness. The approach blends topological data analysis with Lipschitz theory to enable principled robustness, offering a pathway to robust, topology-aware machine learning pipelines.

Abstract

We propose a neural network architecture that can learn discriminative geometric representations of data from persistence diagrams, common descriptors of Topological Data Analysis. The learned representations enjoy Lipschitz stability with a controllable Lipschitz constant. In adversarial learning, this stability can be used to certify -robustness for samples in a dataset, which we demonstrate on the ORBIT5K dataset representing the orbits of a discrete dynamical system.
Paper Structure (22 sections, 2 theorems, 25 equations, 3 figures, 3 tables)

This paper contains 22 sections, 2 theorems, 25 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Persistent homology and learnable vectorizations
  4. Lipschitz neural networks and robustness at test time
  5. Methods
  6. The Stable Rank Network and its robustness properties
  7. Adversarial examples in spaces of persistence diagrams
  8. Results
  9. Conclusion
  10. Persistent homology and distances
  11. Persistence diagrams
  12. Wasserstein distance between persistence diagrams
  13. Stable ranks
  14. Definition of stable ranks
  15. Computation of stable ranks
  16. ...and 7 more sections

Key Result

proposition 1

Let $p\in [1,\infty]$ and let $F:\mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0}$ be a reparameterization which is $K$-Lipschitz for some $K>0$, that is, $\lvert F(b) - F(a) \rvert \le K \lvert b-a\rvert$ for all $a,b\in \mathbb{R}_{\ge 0}$. Then for all $D,D'\in \mathcal{PD}$.

Figures (3)

  • Figure 1: An example sample for each class in the ORBIT5K dataset.
  • Figure 2: A Persistent Homology Machine Learning pipeline using Stable Rank Network.
  • Figure 3: Distribution of certified $\epsilon$-robustness for correctly classified samples of the test set, for the different classes.

Theorems & Definitions (4)

  • proposition 1
  • proof
  • corollary 1
  • proof