流形优化系列短课程3：二阶优化算法及商流形

Riemannian Optimization Series 3: Second Order Optimization Methods and Quotient Manifolds

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三期。本期课程主要包括两部分内容。第一部分是介绍线性空间的嵌入子流形的二阶相关概念及优化算法。第二部分是简要介绍抽象流形的定义并由其引申 出商流形的概念。如时间允许，再简单回顾流形上优化算法的进展并介绍流形上优化软件包的使用。

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 3 of the Riemannian optimization short courses. This short course includes two main parts. In the first part, we introduce the second order Riemannian geometry on embedded submanifold of linear spaces. In the second part, we briefly introduce the abstract definition of manifolds and give an example that is not an embedded submanifold, i.e., quotient manifold. If time permits, we briefly review the current state of Riemannian optimization fields and give an instruction of Riemannian optimization package.

课程信息 Course Information