The Persistent Homology Transform (PHT) is an injective shape descriptor that uniquely encodes constructible sets via their persistence diagrams across a continuous family of filtrations. While theoretically powerful, its direct computation requires scanning a shape with infinitely many filtrations, rendering it intractable in practice.

In this talk we will introduce the distance-from-flat PHT on affine Grassmannians and discuss a uniform finite-approximation theorem for this class of transforms, valid for all constructible sets in a compact domain. A central difficulty is that the affine Grassmannian is non-compact and the canonical assignment of a distance function to each flat fails to be Lipschitz with respect to classical metrics on the affine Grassmannian. We address this by introducing a new Hausdorff-based metric adapted to our setting, and derive asymptotic upper bounds on the number of filtrations sufficient to approximate the transform up to any prescribed error. Finally, we will also discuss algorithmic strategies for selecting flats in practice.

This internship took place at the Laboratoire de Mathématiques d’Orsay.

References:

[1] Katharine Turner, Sayan Mukherjee, and Doug M Boyer. Persistent homology transform for modeling shapes and surfaces. Information and Inference: A Journal of the IMA, 3(4):310–344, 2014.

[2] Justin Curry, Sayan Mukherjee, and Katharine Turner. How many directions determine a shape and other sufficiency results for two topological transforms. Transactions of the American Mathematical Society, Series B, 9(32):1006–1043, 2022.

[3] Adam Onus, Nina Otter, Renata Turkes. Shoving tubes through shapes gives a sufficient and efficient shape statistic, https://arxiv.org/abs/2412.18452, 2024.