Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration

Abstract

We consider optimization problems defined on product spaces of simplices. Examples of this class of problems include learning low-rank discrete multivariate probability distributions via simplex constrained tensor decomposition and performing functional data registration under the Square Root Velocity Function (SRVF) representation. In this work, we demonstrate the feasibility of replacing the product simplex with a smooth, elementwise strictly convex reparameterization, resulting in an unconstrained optimization problem on a manifold. We show that performing such a reparameterization results in the second order Karush-Kuhn-Tucker (KKT) points on the smooth manifold being mapped to the weak second order KKT points on the product simplex. This leads to a Riemannian Gradient Descent (RGD) algorithm for solving the reparameterized problem, which outperforms Projected Gradient Descent (PGD), and provides a more faithful representation of the original function shapes while performing curve registration.