An Introduction to Optimization on Smooth Manifolds

Optimization on Riemannian manifolds-the result of smooth geometry and optimization merging into one elegant modern framework-spans many areas of science and engineering, including machine learning, computer vision, signal processing, dynamical systems and scientific computing. This text introduces the differential geometry and Riemannian geometry concepts that will help students and researchers in applied mathematics, computer science and engineering gain a firm mathematical grounding to use these tools confidently in their research. Its charts-last approach will prove more intuitive from an optimizer's viewpoint, and all definitions and theorems are motivated to build time-tested optimization algorithms. Starting from first principles, the text goes on to cover current research on topics including worst-case complexity and geodesic convexity. Readers will appreciate the tricks of the trade for conducting research and for numerical implementations sprinkled throughout the book.

在黎曼流形上的優化-平滑幾何和優化相結合的結果-涵蓋了許多科學和工程領域，包括機器學習、計算機視覺、信號處理、動態系統和科學計算。本書介紹了微分幾何和黎曼幾何的概念，幫助應用數學、計算機科學和工程領域的學生和研究人員在研究中自信地使用這些工具。從優化者的角度來看，本書的圖表方法更直觀，所有定義和定理都是為了建立經過時間考驗的優化算法。從基本原理開始，本書還涵蓋了包括最壞情況複雜性和測地凸性在內的當前研究。讀者將欣賞到書中散佈的進行研究和數值實現的訣竅。