Shoving Tubes through Shapes Gives a Sufficient and Efficient Shape Statistic

The classical persistent homology transform was introduced in the field of topological data analysis about 10 years ago, and has since been proven to be a very powerful descriptor of Euclidean shapes. The transform sends a shape X to the map associating to each direction v on the sphere $S^{n-1}$ the persistent diagram with respect to the height function h_v. The transform has been shown to be injective (it is a sufficient shape statistic: probing a shape from each direction completely describes it), and for each shape it gives a continuous map from the sphere to the space of persistence diagrams.
We introduce a generalised persistent homology transform (PHT) in which we consider arbitrary parameter spaces, and any filtration functions. In particular, we define the "distance-from-flat” PHT, where the parameter space is the Grassmannian AG(m,n) of affine subspaces of $R^n$, and the filtration functions d_P encode the distance from a given flat P. We prove that this version retains continuity and injectivity, while offering computational advantages over the classical PHT. In particular, homology in degree 0 suffices for the injectivity of the distance-from-line, so-called tubular, PHT, yielding an efficient tool that can outperform top neural networks in shape classification.

This is joint work with Adam Onus and Nina Otter.