Development of Riemannian conjugate gradient methods and their applications to large-scale problems

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K17647
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kyoto University (2017-2018); Tokyo University of Science (2016)
• Principal Investigator: SATO Hiroyuki 京都大学, 情報学研究科, 特定准教授 (80734433)
• Research Collaborator: AIHARA Kensuke ; KASAI Hiroyuki ; SATO Kazuhiro
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 最適化 / アルゴリズム / 数値線形代数 / 機械学習 / 制御工学 / 応用数学 / 幾何学

Outline of Final Research Achievements

The generalized Stiefel manifold has important applications including the canonical correlation analysis. This research proposed a new retraction and its efficient implementation on this manifold, which is essential in Riemannian optimization methods such as the conjugate gradient method. As applications to control engineering, this research also proposed new algorithms for system identification and optimal model reduction problems, some of which are based on the Riemannian conjugate gradient method. Furthermore, this research generalized stochastic optimization methods, which are effective for large-scale optimization problems in machine learning with big data, to Riemannian manifolds together with their convergence analyses and numerical verification of their performances.

Free Research Field

数理最適化

Academic Significance and Societal Importance of the Research Achievements

本研究の成果によって，機械学習や制御工学などにおいて現れる最適化問題の一部をより高速に解くことができるようになった．また，リーマン多様体上の共役勾配法や確率的最適化手法の一般論を発展させたことによって，それらが適用できるような大規模な問題をより効果的に解くための土台が築かれたといえる．個別の問題に対して，本研究で提案したアルゴリズムに基づくアプローチを構築することによって，諸分野における技術のさらなる発展が期待される．