The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is characterized by an efficient factorization that makes t
he trace norm differentiable in the search space and the computation of duality gap numerically tractable. The search space is nonlinear but is equipped with a particular Riemannian structure that leads to efficient computations. We present a second-order trust-region algorithm with a guaranteed quadratic rate of convergence. Overall, the proposed optimization scheme converges super-linearly to the global solution while maintaining complexity that is linear in the number of rows and columns of the matrix. To compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor-corrector approach that outperforms the naive warm-restart approach on the fixed-rank quotient manifold. The performance of the proposed algorithm is illustrated on problems of low-rank matrix completion and multivariate linear regression.
We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It i
s thus very effective for solving programs that have a low rank solution. The factorization $X=Y Y^T$ evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem.
