  Convex optimization over the spectrahedron, i.e., the set of all real
$n\times n$ positive semidefinite matrices with unit trace, has important
applications in machine learning, signal processing and statistics, mainly as a
convex relaxation for optimization with low-rank matrices. It is also one of
the most prominent examples in the theory of first-order methods for convex
optimization in which non-Euclidean methods can be significantly preferable to
their Euclidean counterparts, and in particular the Matrix Exponentiated
Gradient (MEG) method which is based on the Bregman distance induced by the
(negative) von Neumann entropy. Unfortunately, implementing MEG requires a full
SVD computation on each iteration, which is not scalable to high-dimensional
problems.
  In this work we propose efficient implementations of MEG, both with
deterministic and stochastic gradients, which are tailored for optimization
with low-rank matrices, and only use a single low-rank SVD computation on each
iteration. We also provide efficiently-computable certificates for the correct
convergence of our methods. Mainly, we prove that under a strict
complementarity condition, the suggested methods converge from a "warm-start"
initialization with similar rates to their full-SVD-based counterparts.
Finally, we bring empirical experiments which both support our theoretical
findings and demonstrate the practical appeal of our methods.
