Geometric Methods for Machine Learning and Optimization
Many machine learning applications involve non-Euclidean data, such as graphs, strings or matrices. In such cases, exploiting Riemannian geometry can deliver algorithms that are computationally superior to standard (Euclidean) nonlinear programming approaches. This observation has resulted in an increasing interest in Riemannian methods in the optimization and machine learning community. In the first part of the talk, we consider the task of learning a robust classifier in hyperbolic space. Such spaces have received a surge of interest for representing large-scale, hierarchical data, due to the fact that they achieve better representation accuracy with lower dimensions. We present the first theoretical guarantees for the (robust) large-margin learning problem in hyperbolic space and discuss conditions under which hyperbolic methods are guaranteed to surpass the performance of their Euclidean counterparts. In the second part, we introduce Riemannian Frank-Wolfe (RFW) methods for constraint optimization on manifolds. Here, the goal of the theoretical analysis is two-fold: We first show that RFW converges at a nonasymptotic sublinear rate, recovering the best-known guarantees for its Euclidean counterpart. Secondly, we discuss how to implement the method efficiently on matrix manifolds. Finally, we consider applications of RFW to the computation of Riemannian centroids and Wasserstein barycenters, which are crucial subroutines in many machine learning methods.Based on joint work with Suvrit Sra (MIT) and Manzil Zaheer, Ankit Singh Rawat, Aditya Menon and Sanjiv Kumar (all Google Research).