Perturbation Robust Representations of Topological Persistence Diagrams

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 theoretically well-grounded approaches to develop novel perturbation robust topological representations are presented, with the long-term view of making them amenable to fusion with contemporary learning architectures. The proposed representation lives on a Grassmann manifold and hence can be efficiently used in machine learning pipelines.

The proposed representation.The efficacy of the proposed descriptor was explored on three applications: view-invariant activity analysis, 3D shape analysis, and non-linear dynamical modeling. Favorable results in both high-level recognition performance and improved performance in reduction of time-complexity when compared to other baseline methods are obtained.