Persistence Codebooks for Topological Data Analysis

TL;DR

This paper introduces persistence codebooks, a novel fixed-size vector representation for persistence diagrams in topological data analysis, enabling more effective integration with machine learning workflows.

Contribution

It adapts bag-of-words, VLAD, and Fisher vectors to create stable, discriminative, fixed-size representations of persistence diagrams, improving performance and efficiency.

Findings

Achieves state-of-the-art results on multiple datasets.

Provides a stable and discriminative representation with theoretical guarantees.

Operates significantly faster than existing methods.

Abstract

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine with typical machine learning workflows. In this paper we introduce persistence codebooks, a novel expressive and discriminative fixed-size vectorized representation of PDs. To this end, we adapt bag-of-words (BoW), vectors of locally aggregated descriptors (VLAD) and Fischer vectors (FV) for the quantization of PDs. Persistence codebooks represent PDs in a convenient way for machine learning and statistical analysis and have a number of favorable practical and theoretical properties including 1-Wasserstein stability. We evaluate the presented representations on several heterogeneous datasets and show their (high) discriminative power. Our approach achieves state-of-the-art performance and beyond in much less time than alternative approaches.