用光滑重参数化解决单纯形乘积空间优化难题，提升概率张量分解与函数配准精度。
Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration
- 将单纯形乘积空间转为光滑流形，实现无约束优化
- 重参数化后二阶KKT点映射更准确，性能优于投影梯度法
- 适合做概率张量分解与函数曲线配准的研究者参考
我们研究定义在单纯形乘积空间上的优化问题，典型应用包括通过单纯形约束张量分解学习低秩离散多变量概率分布，以及在平方根速度函数（SRVF）表示下进行函数数据配准。本文证明，可通过元素级严格凸的光滑重参数化，将原乘积单纯形替换为一个光滑流形，从而将原问题转化为流形上的无约束优化。我们证明，该重参数化使流形上的二阶卡鲁什-库恩-塔克（KKT）点映射为原单纯形上的弱二阶KKT点。基于此，提出一种黎曼梯度下降（RGD）算法，其性能优于投影梯度下降（PGD），并在函数配准时更忠实地保留原始函数形状。
原文摘要 · Abstract (English)
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.
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