流形优化系列短课程2：嵌入子流形及一阶优化算法

Riemannian Optimization Series 2: Embedded Submanifolds and First Order Optimization Methods

课程简介

黎曼流形上的优化，也被称为黎曼优化或流形优化，考虑最小化定义在黎曼流形上的实值函数。如今信息时代的数据规模越来越大，许多问题中的高维数据会落在低维流形上，分析处理 这些数据往往会将其转换成为黎曼优化问题。这些领域包括但不限于信号处理，机器学习，推荐系统，网络成分分析与分类，计算机视觉与图形学等。这使得黎曼优化在这几年受到越 来越多的关注。

此课程为流形优化系列短课程第二期。本期课程通过线性空间的嵌入子流形介绍流形上优化所需的概念及性质。主要内容包括欧式空间的黎曼嵌入子流形的判断方法，黎曼梯度定义及计算， 收缩(Retraction)的定义及设计，流形上线性搜索算法的步长选取方法，一阶算法及其理论结果。本课程我们不要求有微分流形相关背景。

Abstract

Optimization on Riemannian mainfolds, also called Riemannian optimization, considers finding an optimum of a real-valued function defined on a Riemannian manifold. Riemannian optimization has been a topic of much interest over the past few years due to many important applications, e.g., blind source separation, computations on symmetric positive matrices, low-rank learning, graph similarity, community detection, and elastic shape analysis.

This is the Series 2 of the Riemannian optimization short courses. In this short course, we introduce Riemannian optimization through embedded submanifold of linear spaces without requiring preliminaries on manifold geometry. The main contents include concepts of Riemannian embedded submanifold, Riemannian gradient and its computations, Retraction, and first order optimization methods on Riemannian manifolds.

课程信息 Course Information