Statistics and Data Science Seminar
Jisu Kim
Inria Saclay, France
Statistical Inference for Geometric and Topological Data
Abstract: Geometric and topological structures can aid statistics in several ways. In high dimensional statistics, geometric structures can be used to reduce dimensionality. High dimensional data entails the curse of dimensionality, which can be avoided if there are low dimensional geometric structures. On the other hand, geometric and topological structures also provide useful information. Structures may carry scientific meaning about the data and can be used as features to enhance supervised or unsupervised learning. In this talk, I will explore how statistical inference can be done on geometric and topological structures. First, given a manifold assumption, I will explore the minimax rates of dimension estimator and reach estimator. First, given a manifold assumption, I will explore the minimax rate for estimating the dimension of the manifold. Second, also under the manifold assumption, I will explore the minimax rate for estimating the reach, which is a regularity quantity depicting how a manifold is smooth and far from self-intersecting. Third, I will investigate inference on cluster trees, which is a hierarchy tree of high-density clusters of a density function. Fourth, I will investigate inference on persistent homology of a density function, which is a representation of topological features of the density function at different levels. Third, I will present R package TDA for computing topological data analysis, which is a set of data analysis tools utilizing topology and includes persistent homology.
Wednesday September 30, 2020 at 4:00 PM in Zoom