Riemannian stochastic variance reduced gradient algorithm with retraction and vector transport

Hiroyuki Sato, Hiroyuki Kasai, Bamdev Mishra

研究成果: Article査読

8 被引用数 (Scopus)

抄録

In recent years, stochastic variance reduction algorithms have attracted considerable attention for minimizing the average of a large but finite number of loss functions. This paper proposes a novel Riemannian extension of the Euclidean stochastic variance reduced gradient (R-SVRG) algorithm to a manifold search space. The key challenges of averaging, adding, and subtracting multiple gradients are addressed with retraction and vector transport. For the proposed algorithm, we present a global convergence analysis with a decaying step size as well as a local convergence rate analysis with a fixed step size under some natural assumptions. In addition, the proposed algorithm is applied to the computation problem of the Riemannian centroid on the symmetric positive definite (SPD) manifold as well as the principal component analysis and low-rank matrix completion problems on the Grassmann manifold. The results show that the proposed algorithm outperforms the standard Riemannian stochastic gradient descent algorithm in each case.

本文言語English
ページ(範囲)1444-1472
ページ数29
ジャーナルSIAM Journal on Optimization
29
2
DOI
出版ステータスPublished - 2019
外部発表はい

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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