Regression on fixed-rank positive semidefinite matrices: a Riemannian approach

Abstract

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixed-rank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks.

References in corpus (3)

• Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
• Low-rank optimization for semidefinite convex problems
• Metric and Kernel Learning using a Linear Transformation

Cited by in corpus (9)

• Manopt, a Matlab toolbox for optimization on manifolds
• The Universal Approximation Property
• Geometric Mean Metric Learning
• Low-rank optimization for distance matrix completion
• Riemannian preconditioning
• Non-Euclidean Universal Approximation
• Rank-preserving geometric means of positive semi-definite matrices
• On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry
• On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization