A Continuous-Time Mirror Descent Approach to Sparse Phase Retrieval

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 full 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 [54], 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.