Geometric Methods for the Statistical Analysis of Non-Euclidean Data and Networks
Dena Asta, Carnegie Mellon University
In this talk, I will describe applications of geometry to large-scale data analysis. An overriding theme is that an understanding of the relevant geometric structure in the data is useful for efficient and large-scale statistical analyses. In the first part, I will discuss geometric methods for non-parametric methods on non-Euclidean spaces. With tools from differential geometry, I develop a general kernel density estimator, for a large class of symmetric spaces, and then derive a minimax rate for this estimator comparable to the Euclidean case. In the second part, I will discuss a geometric approach to network inference, joint work with Cosma Shalizi, that uses the above estimator on hyperbolic spaces. We propose a more general, principled statistical approach to network comparison, based on the non-parametric inference and comparison of densities on hyperbolic manifolds from sample networks.