Certifying Robustness via Topological Representations

2025-01-18

AI Summary

This work addresses the sensitivity of machine learning features to small input perturbations by proposing a neural network framework that jointly incorporates topological awareness and Lipschitz stability. Methodologically, it pioneers the deep integration of persistent diagram embeddings with Lipschitz-constrained neural networks, enabling learnable, discriminative geometric representations while rigorously bounding the model’s Lipschitz constant—thereby providing provable ε-robustness certification for each input sample. The key contribution lies in unifying the structural expressiveness of topological data analysis with the theoretical stability guarantees of Lipschitz neural networks, overcoming the long-standing limitations of non-differentiability and optimization difficulty inherent in conventional topological features. Experiments on the ORBIT5K dynamical-system trajectory dataset demonstrate sample-level robustness verification, confirming strong alignment between theoretical stability bounds and empirical robust performance.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Natural Language Processing: Safety and Robustness
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $epsilon$-robustness for samples in a dataset, which we demonstrate on the ORBIT5K dataset representing the orbits of a discrete dynamical system.

Problem

• Shape Feature Recognition
• Stability of Features
• Robustness in Data Sets

Innovation

• Stable Shape Features
• Robust Machine Learning
• Orbital Motion Data