Intrinsic Topological Transforms via the Distance Kernel Embedding
Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined were the Persistent Homology Transform (PHT) and Euler Characteristic Transform (ECT), both of which apply to shapes embedded in Euclidean space. The contribution of this paper is to define topological transforms for abstract metric measure spaces. Our proposed pipeline is to pre-compose the PHT or ECT with a Euclidean embedding derived from the eigenfunctions and eigenvalues of an integral operator. To that end, we define and study an integral operator called the distance kernel operator, and demonstrate that it gives rise to stable and quasi-injective topological transforms. We conclude with some numerical experiments, wherein we compute and compare the eigenfunctions and eigenvalues of our operator across a range of standard 2- and 3-manifolds.
Topological Transforms
Persistent Homology
Inverse Problems
Spectral Geometry
Algebraic Topology
Topological Data Analysis
Mathematics of computing~Algebraic topology
56:1-56:15
Regular Paper
https://arxiv.org/abs/1912.02225
Clément
Maria
Clément Maria
INRIA Sophia Antipolis-Méditerranée, Valbonne, France
Steve
Oudot
Steve Oudot
INRIA Saclay, Palaiseau, France
Elchanan
Solomon
Elchanan Solomon
Department of Mathematics, Duke University, Durham, NC USA
10.4230/LIPIcs.SoCG.2020.56
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Clément Maria, Steve Oudot, and Elchanan Solomon
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