  We analyze continuous-time mirror descent applied to sparse phase retrieval,
which is the problem of recovering sparse signals from a set of magnitude-only
measurements. We apply mirror descent to the unconstrained empirical risk
minimization problem (batch setting), using the square loss and square
measurements. We provide a convergence analysis of the algorithm in this
non-convex setting and prove that, with the hypentropy mirror map, mirror
descent recovers any $k$-sparse vector $\mathbf{x}^\star\in\mathbb{R}^n$ with
minimum (in modulus) non-zero entry on the order of $\| \mathbf{x}^\star
\|_2/\sqrt{k}$ from $k^2$ Gaussian measurements, modulo logarithmic terms. This
yields a simple algorithm which, unlike most existing approaches to sparse
phase retrieval, adapts to the sparsity level, without including thresholding
steps or adding regularization terms. Our results also provide a principled
theoretical understanding for Hadamard Wirtinger flow [58], as Euclidean
gradient descent applied to the empirical risk problem with Hadamard
parametrization can be recovered as a first-order approximation to mirror
descent in discrete time.
