jmlr jmlr2011 jmlr2011-80 jmlr2011-80-reference knowledge-graph by maker-knowledge-mining
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Author: Gilles Meyer, Silvère Bonnabel, Rodolphe Sepulchre
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 fixedrank 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. Keywords: linear regression, positive semidefinite matrices, low-rank approximation, Riemannian geometry, gradient-based learning