Titolo: Topological Data Analysis: Invariance Properties, Statistical Problems, and Topological Machine Learning
Abstract: I will present some of my contributions to Topological Data Analysis through the problem of building rigorous and usable TDA pipelines. Starting from a function, mesh, or point cloud, one builds a filtration and extracts a topological summary, such
as a persistence diagram, merge tree, or Reeb graph. I will discuss how these summaries encode coordinate-free and invariant information, and why turning this idea into a robust mathematical pipeline requires carefully designed metrics, stability results,
and estimation procedures, using tools from topology, geometry, statistics, and optimal transport. This perspective also exposes a second difficulty: even once stable summaries have been constructed, many of them naturally live in non-linear metric spaces,
making standard statistical and machine-learning methods difficult to apply directly. I will therefore conclude with recent work on Persistence Spheres, an explicit Hilbert-space embedding of persistence diagrams with provable bi-continuity, aimed at making
topological summaries compatible with modern machine-learning pipelines.
Un caro saluto a tutti,
Laura Sangalli
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Laura Maria Sangalli
MOX - Dipartimento di Matematica
Politecnico di Milano
Piazza Leonardo da Vinci 32
20133 Milano - Italy
(+39) 02 2399 4554
laura.sangalli@polimi.it https://sangalli.faculty.polimi.it