Intrinsic Topological Transforms via the Distance Kernel Embedding

Explore the concept of intrinsic topological transforms through the distance kernel embedding in this conference talk. Delve into the development of topological transforms for abstract metric measure spaces, extending beyond shapes embedded in Euclidean space. Learn about the distance kernel operator and its role in creating stable and quasi-injective topological transforms. Examine the process of pre-composing Persistent Homology Transform (PHT) or Euler Characteristic Transform (ECT) with Euclidean embeddings derived from eigenfunctions and eigenvalues of an integral operator. Discover the numerical experiments conducted to compute and compare eigenvalues, embeddings, and stability constants across various 2- and 3-manifolds. Gain insights into geometry-preserving embedding, stability results, inverse results, and coarse bounds for both continuous and finite cases of the Distance Kernel Embedding (DKE). Conclude with an exploration of results for the Persistence Kernel Transform, providing a comprehensive understanding of this innovative approach to topological analysis.

Intro
Persistent Homology Transform
Geometry-Preserving Embedding
Stability Results for the Distance Kernel Embedding (DKE)
Inverse Results for the Distance Kernel Embedding (DKE)
Coarse Bounds (Continuous Case)
Coarse Bounds (Finite Case)
Results for the Persistence Kernel Transform