  We study discrete-time mirror descent applied to the unregularized empirical
risk in matrix sensing. In both the general case of rectangular matrices and
the particular case of positive semidefinite matrices, a simple potential-based
analysis in terms of the Bregman divergence allows us to establish convergence
of mirror descent -- with different choices of the mirror maps -- to a matrix
that, among all global minimizers of the empirical risk, minimizes a quantity
explicitly related to the nuclear norm, the Frobenius norm, and the von Neumann
entropy. In both cases, this characterization implies that mirror descent, a
first-order algorithm minimizing the unregularized empirical risk, recovers
low-rank matrices under the same set of assumptions that are sufficient to
guarantee recovery for nuclear-norm minimization. When the sensing matrices are
symmetric and commute, we show that gradient descent with full-rank factorized
parametrization is a first-order approximation to mirror descent, in which case
we obtain an explicit characterization of the implicit bias of gradient flow as
a by-product.
