Convolutional Persistence Transforms

TL;DR

This paper introduces convolutional persistence, a method combining convolutional filters with topological data analysis to better capture patterns in data over simplicial complexes, with theoretical guarantees and improved classification performance.

Contribution

It extends topological data analysis by integrating convolutional filters, proving injectivity of the resulting invariants, and demonstrating enhanced predictive power in practical tasks.

Findings

Convolutional persistence provides an injective topological invariant.

The method improves stability and flexibility over traditional persistence.

Experiments show significant gains in classification accuracy with convolutional persistence.

Abstract

In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution filter as a local motif, the persistence diagram of the resulting convolution describes the way the motif is distributed across the simplicial complex. This pipeline, which we call convolutional persistence, extends the capacity of topology to observe patterns in such data. Moreover, we prove that (generically speaking) for any two labeled complexes one can find some filter for which they produce different persistence diagrams, so that the collection of all possible convolutional persistence diagrams is an injective invariant. This is proven by showing convolutional persistence to be a special case of another topological invariant, the Persistent Homology Transform. Other advantages of convolutional persistence are improved stability, greater flexibility for data-dependent vectorizations, and reduced computational complexity for certain data types. Additionally, we have a suite of experiments showing that convolutions greatly improve the predictive power of persistence on a host of classification tasks, even if one uses random filters and vectorizes the resulting diagrams by recording only their total persistences.