Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration
2026-08-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study optimization problems involving multiple simplexes, which often appear in tasks like learning certain probability models and aligning functional data. They propose converting these simpler but constrained problems into smooth, unconstrained ones on curved spaces (manifolds) by reparameterizing the variables. This approach links key optimality conditions between the original and new problems, allowing them to use a Riemannian Gradient Descent method. Their method works better than traditional Projected Gradient Descent and preserves the natural shape of functions more accurately during data alignment.
product simplextensor decompositionfunctional data registrationSquare Root Velocity Function (SRVF)Karush-Kuhn-Tucker (KKT) conditionsreparameterizationmanifold optimizationRiemannian Gradient Descent (RGD)Projected Gradient Descent (PGD)
Authors
Shashwat Kumar, Arafat Rahman, Anuj Srivastava, P. -A. Absil
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.