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

arXiv: https://arxiv.org/abs/2608.02576 · HTML: https://mlsanity.com/paper/2608.02576/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 30 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Implicit Regularization for Optimal Sparse Recovery](https://mlsanity.com/paper/1909.05122/) (2019)
- [Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms](https://mlsanity.com/paper/2007.11446/) (2020)
- [The effect of smooth parametrizations on nonconvex optimization landscapes](https://mlsanity.com/paper/2207.03512/) (2022)
- [Tensors, Learning, and 'Kolmogorov Extension' for Finite-alphabet Random Vectors](https://mlsanity.com/paper/1712.00205/) (2017)
- [From the simplex to the sphere: Faster constrained optimization using the Hadamard parametrization](https://mlsanity.com/paper/2112.05273/) (2021)
- [Information-theoretic Feature Selection via Tensor Decomposition and Submodularity](https://mlsanity.com/paper/2010.16181/) (2020)
- [Minimizing low-rank models of high-order tensors: Hardness, span, tight relaxation, and applications](https://mlsanity.com/paper/2210.11413/) (2022)
- [A Shape-Based Functional Index for Objective Assessment of Pediatric Motor Function](https://mlsanity.com/paper/2501.04721/) (2025)

## Abstract (arXiv)

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.