Persistence B-Spline Grids: Stable Vector Representation of Persistence Diagrams Based on Data Fitting
This provides a stable vector representation for persistence diagrams, addressing a bottleneck in integrating topological data analysis with machine learning, though it appears incremental as it builds on existing data fitting techniques.
The paper tackles the problem of finding a stable vector representation for persistence diagrams in topological data analysis, proposing persistence B-spline grids (PBSG) based on data fitting, which is theoretically proven stable with respect to the 1-Wasserstein distance and shown effective on synthetic, random, dynamical system, and 3D CAD model datasets.
Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram (PD), which is a summary diagram for representing topological features. From the perspective of data fitting, a stable vector representation, namely, persistence B-spline grid (PBSG), is proposed based on the efficient technique of progressive-iterative approximation for least-squares B-spline function fitting. We theoretically prove that the PBSG method is stable with respect to the metric of 1-Wasserstein distance defined on the PD space. The proposed method was tested on a synthetic data set, data sets of randomly generated PDs, data of a dynamical system, and 3D CAD models, showing its effectiveness and efficiency