Low-rank optimization with trace norm penalty

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 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 superlinearly 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. © 2013 Society for Industrial and Applied Mathematics.

Calculation of metallic and insulating phases of V2O3 by hybrid density functionals.

The electronic structure of vanadium sesquioxide V2O3 in its different phases has been calculated using the screened exchange hybrid density functional. The hybrid functional accurately reproduces the experimental electronic properties of all three phases, the paramagnetic metal (PM) phase, the anti-ferromagnetic insulating phase, and the Cr-doped paramagnetic insulating (PI) phase. We find that a fully relaxed supercell model of the Cr-doped PI phase based on the corundum structure has a monoclinic-like local strain around the substitutional Cr atoms. This is found to drive the PI-PM transition, consistent with a Peierls-Mott transition. The PI phase has a calculated band gap of 0.15 eV, in good agreement with experiment.