Low-rank optimization with trace norm penalty

Bamdev Mishra (1) , Gilles Meyer (1) , Francis Bach (2, 3) , Rodolphe Sepulchre (1)

1 Institut Montefiore - Department of Electrical Engineering and Computer Science
2 SIERRA - Statistical Machine Learning and Parsimony
3 LIENS - Laboratoire d'informatique de l'école normale supérieure

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 the 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.

Mots clés: trace norm; Riemannian optimization; trust region; regularization path; predictor-corrector on quotient manifold; matrix completion; multivariate linear regression

Domaines: Apprentissage [cs.LG]; Optimisation et contrôle [math.OC]

Bamdev Mishra, Gilles Meyer, Francis Bach, Rodolphe Sepulchre. Low-rank optimization with trace norm penalty. SIAM Journal on Optimization, 2013, 23 (4), pp.2124-2149. ⟨10.1137/110859646⟩. ⟨hal-00924110⟩