Perturbation Robust Representations of Topological Persistence Diagrams

TL;DR

This paper introduces Perturbed Topological Signatures, a new robust representation of persistence diagrams on Grassmann manifolds, enabling effective fusion with deep learning for applications like shape analysis, activity recognition, and dynamical modeling.

Contribution

The paper proposes a theoretically grounded method to create perturbation robust topological representations that can be integrated with modern machine learning architectures.

Findings

Favorable recognition performance in shape, activity, and dynamical tasks.

Efficient computation compared to baseline methods.

Robustness to perturbations demonstrated in experiments.

Abstract

Topological methods for data analysis present opportunities for enforcing certain invariances of broad interest in computer vision, including view-point in activity analysis, articulation in shape analysis, and measurement invariance in non-linear dynamical modeling. The increasing success of these methods is attributed to the complementary information that topology provides, as well as availability of tools for computing topological summaries such as persistence diagrams. However, persistence diagrams are multi-sets of points and hence it is not straightforward to fuse them with features used for contemporary machine learning tools like deep-nets. In this paper we present theoretically well-grounded approaches to develop novel perturbation robust topological representations, with the long-term view of making them amenable to fusion with contemporary learning architectures. We term the proposed representation as Perturbed Topological Signatures, which live on a Grassmann manifold and hence can be efficiently used in machine learning pipelines. We explore the use of the proposed descriptor on three applications: 3D shape analysis, view-invariant activity analysis, and non-linear dynamical modeling. We show favorable results in both high-level recognition performance and time-complexity when compared to other baseline methods.