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The paper addresses the problem of learning a regression model parameterizedby a fixed-rank positive semidefinite matrix. The focus is on the nonlinearnature of the search space and on scalability to high-dimensional problems. Themathematical developments rely on the theory of gradient descent algorithmsadapted to the Riemannian geometry that underlies the set of fixed-rankpositive semidefinite matrices. In contrast with previous contributions in theliterature, no restrictions are imposed on the range space of the learnedmatrix. The resulting algorithms maintain a linear complexity in the problemsize and enjoy important invariance properties. We apply the proposedalgorithms to the problem of learning a distance function parameterized by apositive semidefinite matrix. Good performance is observed on classicalbenchmarks.

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