Here is the schedule of talks for Fall 2025.
You can find the titles and abstracts of the talks below.
October 16th
Title: Curvature Sets and Persistent Homology
Speaker: Mario Gómez
Abstract: Given a metric space (X,d), the n-th curvature set is the set of n-by-n distance matrices generated by a sample from X with n or less points. We use this as inspiration to define the (n,k) persistence set of X is the set of k-dimensional persistence diagrams of all n-point samples from X. A major obstacle that hampers the widespread use of topological techniques in data science is the sometimes prohibitive computational cost of persistent homology. Persistence sets aim to circumvent this limitation while retaining useful geometric and topological information from the input space. We study the experimental and theoretical properties of persistence sets and compare them with the standard VR-persistent homology from the perspectives of computational efficiency and discriminative power, including in a practical shape classification task. We characterize several persistence sets of the circle, higher-dimensional spheres, surfaces with constant curvature, and a specific family of metric graphs, and show spaces that have different persistence sets but are indistinguishable by persistent homology. All in all, we believe that persistence sets can aid in data science tasks where the shape is important but the standard persistent homology algorithms are impractical.
October 30th
Title: Topology-Driven Learning for Biomedical Images – Uncertainty, Synthesis, and Prediction
Speaker: Chao Chen
Abstract: With advanced imaging techniques, we are collecting images of various complex structures such as neurons, vessels, tissues and cells. These structures encode important information about underlying biological mechanisms. To fully exploit these structures, we propose to enhance learning pipelines with topology, the branch of abstract mathematics that deals with structures such as connections, loops and branches. Under-the-hood is a formulation of the topological computation as a robust and differentiable operator. This inspires a series of novel methods for segmentation, uncertainty estimation, generation, and analysis of these topology-rich biomedical structures. We demonstrate how these methods provide a better AI support for the annotation and analysis of images of vasculature and breast tissue. In digital pathology, we demonstrate how combining rich spatial and topological characterization with deep learning techniques will enhance cell/gland segmentation, synthesis, and gene expression prediction.
December 4th
Title: Wasserstein-Cramér-Rao Theory of Unbiased Estimation
Speaker: Nicolás García Trillos
Abstract: The quantity of interest in the classical Cramér-Rao theory of unbiased estimation (i.e., the Cramér-Rao lower bound, exact efficiency in exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently sampled data set from the same distribution. In this talk, we will be interested in a quantity that represents the instability of an estimator when its value is compared to the value for an infinitesimal additive perturbation of the original data set; we refer to this as the “sensitivity” of an estimator. The resulting theory of sensitivity is based on the Wasserstein geometry in the same way that the classical theory of variance is based on the Fisher-Rao (equivalently, Hellinger) geometry. I'll present a collection of results which are analogous to the classical case: a Wasserstein-Cramér-Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models in which there exist unbiased estimators achieving the lower bound exactly, and a guarantee that Wasserstein projection estimators achieve the lower bound asymptotically. I'll discuss some simple statistical examples to illustrate the theory, sometimes revealing new optimality properties for existing estimators and other times revealing entirely new ones. I'll also discuss some of the many open questions that this work (and in fact the whole perspective this work is based on) motivates.
This talk is based on joint work with Adam Quinn Jaffe (Columbia) and Bodhisattva Sen (Columbia).