The Interconnectivity Vector: A Finite-Dimensional Vector Representation of Persistent Homology

TL;DR

This paper introduces the interconnectivity vector, a new finite-dimensional representation of persistence diagrams that captures homological feature connections, with a stabilized version proven to be stable and effective for data analysis.

Contribution

It proposes a novel vectorization method for persistence diagrams inspired by Bag-of-Words, including a stabilized version with proven stability properties.

Findings

The stabilized interconnectivity vector is stable under small data perturbations.

Both vector versions demonstrate high discriminative power on multiple datasets.

The method effectively captures the connections between homological features.

Abstract

Persistent Homology (PH) is a useful tool to study the underlying structure of a data set. Persistence Diagrams (PDs), which are 2D multisets of points, are a concise summary of the information found by studying the PH of a data set. However, PDs are difficult to incorporate into a typical machine learning workflow. To that end, two main methods for representing PDs have been developed: kernel methods and vectorization methods. In this paper we propose a new finite-dimensional vector, called the interconnectivity vector, representation of a PD adapted from Bag-of-Words (BoW). This new representation is constructed to demonstrate the connections between the homological features of a data set. This initial definition of the interconnectivity vector proves to be unstable, but we introduce a stabilized version of the vector and prove its stability with respect to small perturbations in the inputs. We evaluate both versions of the presented vectorization on several data sets and show their high discriminative power.