The Interconnectivity Vector: A Finite-Dimensional Vector Representation of Persistent Homology
This work addresses the problem of integrating Persistent Homology data into machine learning workflows by providing a stable vector representation for Persistence Diagrams, which is an incremental improvement for researchers in topological data analysis.
This paper introduces the interconnectivity vector, a new finite-dimensional vector representation for Persistence Diagrams (PDs), which are summaries of Persistent Homology (PH) data. The authors propose a stabilized version of this vector and prove its stability, demonstrating high discriminative power on several datasets.
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.