发表机构
Johns Hopkins University; University of Virginia; UCLouvain（约翰斯·霍普金斯大学; 弗吉尼亚大学; 鲁汶大学（法语））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对单纯形乘积空间上的优化问题，提出用光滑重参数化替换单纯形乘积，得到性能优于投影梯度下降的黎曼梯度下降算法，可用于概率张量分解与函数数据配准。
AI 中文摘要
我们研究定义在单纯形乘积空间上的优化问题，这类问题的例子包括通过单纯形约束张量分解学习低秩离散多元概率分布，以及在平方根速度函数（SRVF）表示下进行函数数据配准。本工作证明了用逐元素严格凸的光滑重参数化替换单纯形乘积的可行性，从而得到流形上的无约束优化问题。我们表明，这种重参数化操作会将光滑流形上的二阶Karush-Kuhn-Tucker（KKT）点映射到单纯形乘积上的弱二阶KKT点。这催生了一种用于求解重参数化问题的黎曼梯度下降（RGD）算法，该算法的性能优于投影梯度下降（PGD），且在执行曲线配准时能更忠实地表示原始函数形状。
英文摘要
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.
Commentssubmitted to Journal of Optimization Theory and Applications (JOTA)
