[论文解读] Geodesic Convex Optimization: Differentiation on Manifolds, Geodesics, and Convexity

Nisheeth K. Vishnoi|arXiv (Cornell University)|Jun 17, 2018

Point processes and geometric inequalities参考文献 13被引用 38

一句话总结

本文引入测地凸优化作为在黎曼流形上重新表述非凸问题的框架，通过使用测地线（在弯曲空间中推广直线的曲线）重新定义凸性。研究证明，在适当的黎曼度量下，Brascamp-Lieb 常数和算子容量等函数成为测地凸函数，从而可通过测地下降方法实现高效优化。

ABSTRACT

Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined by how it interacts with the standard differential structure in this space -- the Hessian of a convex function has to be positive semi-definite everywhere. However, in recent years, there is a growing demand to understand non-convexity and develop computational methods to optimize non-convex functions. Intriguingly, there is a type of non-convexity that disappears once one introduces a suitable differentiable structure and redefines convexity with respect to the straight lines, or {\em geodesics}, with respect to this structure. Such convexity is referred to as {\em geodesic convexity}. Interest in studying it arises due to recent reformulations of some non-convex problems as geodesically convex optimization problems over geodesically convex sets. Geodesics on manifolds have been extensively studied in various branches of Mathematics and Physics. However, unlike convex optimization, understanding geodesics and geodesic convexity from a computational point of view largely remains a mystery. The goal of this exposition is to introduce the first part of geodesic convex optimization -- geodesic convexity -- in a self-contained manner. We first present a variety of notions from differential and Riemannian geometry such as differentiation on manifolds, geodesics, and then introduce geodesic convexity. We conclude by showing that certain non-convex optimization problems such as computing the Brascamp-Lieb constant and the operator scaling problem have geodesically convex formulations.

研究动机与目标

• 通过在黎曼流形上利用测地线重新定义凸性，开发一种用于非凸优化的计算框架。
• 证明分析与线性代数中某些非凸问题在适当的微分结构下可转化为测地凸问题。
• 提供理解测地凸性所必需的微分几何时概念（如微分、测地线与联络）的自包含阐述。
• 证明测地凸性可通过在流形上利用曲率感知下降实现高效优化算法。
• 为将测地凸性应用于计算 Brascamp-Lieb 常数和求解算子缩放问题等提供理论基础。

提出的方法

• 利用黎曼几何定义测地线为局部长度最小化曲线，并作为流形上直线的推广。
• 应用 Levi-Civita 联络定义黎曼流形上的平行移动与协变导数。
• 通过函数在任意测地线线段中点的值不超过端点值的平均值这一条件，定义测地凸性。
• 在正定矩阵流形上使用度量 gX(U, V) = tr[X⁻¹UX⁻¹V] 定义测地结构。
• 证明在该度量下，log det(X) 和 log det(T(X)) 为测地凸函数，从而其差值也具有测地凸性。
• 将诸如 Brascamp-Lieb 常数和算子容量等非凸问题重新表述为在正定锥上的测地凸优化问题。

实验结果

研究问题

• RQ1 能否通过利用测地线重新定义凸性，将非凸优化问题在黎曼流形上重述为测地凸问题？
• RQ2 哪些微分几何时结构是定义测地凸性并确保算法可计算性的必要条件？
• RQ3 Brascamp-Lieb 常数能否通过在正定矩阵流形上的测地凸优化计算得出？
• RQ4 在正定锥的标准黎曼度量下，算子容量问题是否具有测地凸的表述形式？
• RQ5 测地凸性能否用于设计收敛且高效的优化算法，以处理此前被认为非凸的问题？

主要发现

• Brascamp-Lieb 常数可通过测地凸优化计算：BL(B, p) = exp(−1/2 inf_X∈S⁺⁺ⁿ FB,p(X))，其中 FB,p 为测地凸函数。
• 函数 FB,p(X) = ∑j pj log det(Bj X Bᵀj) − log det(X) 在正定矩阵流形上，于度量 gX(U, V) = tr[X⁻¹UX⁻¹V] 下为测地凸函数。
• 当 T(X) 为严格正定线性映射时，算子容量函数 log cap(X) = log(det(T(X))/det(X)) 在正定锥上为测地凸函数。
• log det(X) 与 log det(T(X)) 的测地凸性源于在给定黎曼度量下 log det 的测地凸性。
• 证明依赖于：当 (B, p) 为非退化的 Brascamp-Lieb 数据时，对所有 j 有 Tj(X) = Bj X Bᵀj 为严格正定，从而确保和的测地凸性。
• 该框架使得可在流形上使用测地下降方法处理此前超出凸优化范畴的问题。