Polynomial Representation for Persistence Diagram

Zhichao Wang, Qian Li, Gang Li, Guandong Xu; Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 6123-6132

Persistence diagram (PD) has been considered as a compact descriptor for topological data analysis (TDA). Unfortunately, PD cannot be directly used in machine learning methods since it is a multiset of points. Recent efforts have been devoted to transforming PDs into vectors to accommodate machine learning methods. However, they share one common shortcoming: the mapping of PDs to a feature representation depends on a pre-defined polynomial. To address this limitation, this paper proposes an algebraic representation for PDs, i.e., polynomial representation. In this work, we discover a set of general polynomials that vanish on vectorized PDs and extract the task-adapted feature representation from these polynomials. We also prove two attractive properties of the proposed polynomial representation, i.e., stability and linear separability. Experiments also show that our method compares favorably with state-of-the-art TDA methods.