A Continuous-Time Mirror Descent Approach to Sparse Phase Retrieval

by   Fan Wu, et al.

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 𝐱^⋆∈ℝ^n with minimum (in modulus) non-zero entry on the order of 𝐱^⋆_2/√(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.


page 1

page 2

page 3

page 4


Nearly Minimax-Optimal Rates for Noisy Sparse Phase Retrieval via Early-Stopped Mirror Descent

This paper studies early-stopped mirror descent applied to noisy sparse ...

Provable Phase Retrieval with Mirror Descent

In this paper, we consider the problem of phase retrieval, which consist...

Non-convex online learning via algorithmic equivalence

We study an algorithmic equivalence technique between nonconvex gradient...

Sampling Without Time: Recovering Echoes of Light via Temporal Phase Retrieval

This paper considers the problem of sampling and reconstruction of a con...

Hadamard Wirtinger Flow for Sparse Phase Retrieval

We consider the problem of reconstructing an n-dimensional k-sparse sign...

Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow

This paper considers the noisy sparse phase retrieval problem: recoverin...

A Closed Loop Gradient Descent Algorithm applied to Rosenbrock's function

We introduce a novel adaptive damping technique for an inertial gradient...

Please sign up or login with your details

Forgot password? Click here to reset