8 papers
Geometry and Stability of Supervised Learning Problems
Facundo Mémoli, Brantley Vose, Robert C. Williamson
We introduce a notion of distance between supervised learning problems, which we call the Risk distance. This distance, inspired by optimal transport, facilitates stability results…
Ephemeral persistence features and the stability of filtered chain complexes
Facundo Mémoli, Ling Zhou
We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of…
Persistence and Topological Complexity
Facundo Mémoli, Ling Zhou
Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent a…
Geometric Bounds for Persistence
Alexey Balitskiy, Baris Coskunuzer, Facundo Mémoli
In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset of a Banach space $\math…
Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
Henry Adams, Johnathan Bush, Nate Clause +13
We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two…
Grassmannian Persistence Diagrams: Special Properties in the 1-Parameter Setting
Aziz Burak Gülen, Facundo Mémoli, Zhengchao Wan
In this paper, we explore the discriminative power of Grassmannian persistence diagrams of 1-parameter filtrations, examine their relationships with other related constructions, an…