这篇论文提出用光滑重参数化把单纯形约束变成无约束，黎曼梯度下降比投影梯度下降更好，做张量分解和函数配准更靠谱。
该论文研究定义在单纯形乘积空间上的优化问题，涵盖低秩离散多变量概率分布学习和基于SRVF的函数数据配准。作者提出用逐元素严格凸的光滑重参数化替代乘积单纯形，将约束优化转化为流形上的无约束优化。该重参数化使流形上的二阶KKT点映射到乘积单纯形上的弱二阶KKT点。基于此提出的黎曼梯度下降（RGD）算法优于投影梯度下降（PGD），并在曲线配准中更忠实保留函数形状。
Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration
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.