TY - JOUR

T1 - Riemannian adaptive stochastic gradient algorithms on matrix manifolds

AU - Kasai, Hiroyuki

AU - Jawanpuria, Pratik

AU - Mishra, Bamdev

N1 - Publisher Copyright:
Copyright © 2019, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2019/2/4

Y1 - 2019/2/4

N2 - Adaptive stochastic gradient algorithms in the Euclidean space have attracted much attention lately. Such explorations on Riemannian manifolds, on the other hand, are relatively new, limited, and challenging. This is because of the intrinsic non-linear structure of the underlying manifold and the absence of a canonical coordinate system. In machine learning applications, however, most manifolds of interest are represented as matrices with notions of row and column subspaces. In addition, the implicit manifold-related constraints may also lie on such subspaces. For example, the Grassmann manifold is the set of column subspaces. To this end, such a rich structure should not be lost by transforming matrices to just a stack of vectors while developing optimization algorithms on manifolds. We propose novel stochastic gradient algorithms for problems on Riemannian matrix manifolds by adapting the row and column subspaces of gradients. Our algorithms are provably convergent and they achieve the convergence rate of order O(log(T )/√T), where T is the number of iterations. Our experiments illustrate the efficacy of the proposed algorithms on several applications.

AB - Adaptive stochastic gradient algorithms in the Euclidean space have attracted much attention lately. Such explorations on Riemannian manifolds, on the other hand, are relatively new, limited, and challenging. This is because of the intrinsic non-linear structure of the underlying manifold and the absence of a canonical coordinate system. In machine learning applications, however, most manifolds of interest are represented as matrices with notions of row and column subspaces. In addition, the implicit manifold-related constraints may also lie on such subspaces. For example, the Grassmann manifold is the set of column subspaces. To this end, such a rich structure should not be lost by transforming matrices to just a stack of vectors while developing optimization algorithms on manifolds. We propose novel stochastic gradient algorithms for problems on Riemannian matrix manifolds by adapting the row and column subspaces of gradients. Our algorithms are provably convergent and they achieve the convergence rate of order O(log(T )/√T), where T is the number of iterations. Our experiments illustrate the efficacy of the proposed algorithms on several applications.

UR - http://www.scopus.com/inward/record.url?scp=85093602784&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85093602784&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85093602784

JO - Nuclear Physics A

JF - Nuclear Physics A

SN - 0375-9474

ER -