# A Continuous-Time Mirror Descent Approach to Sparse Phase Retrieval

arXiv: https://arxiv.org/abs/2010.10168 · HTML: https://mlsanity.com/paper/2010.10168/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2020.

## Built on by (most cited first, 9 of 9)

- [Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity](https://mlsanity.com/paper/2106.09524/) (2021)
- [(S)GD over Diagonal Linear Networks: Implicit Regularisation, Large Stepsizes and Edge of Stability](https://mlsanity.com/paper/2302.08982/) (2023)
- [Label noise (stochastic) gradient descent implicitly solves the Lasso for quadratic parametrisation](https://mlsanity.com/paper/2206.09841/) (2022)
- [More is Less: Inducing Sparsity via Overparameterization](https://mlsanity.com/paper/2112.11027/) (2021)
- [Implicit Regularization in Matrix Sensing via Mirror Descent](https://mlsanity.com/paper/2105.13831/) (2021)
- [Nearly Minimax-Optimal Rates for Noisy Sparse Phase Retrieval via Early-Stopped Mirror Descent](https://mlsanity.com/paper/2105.03678/) (2021)
- [Acceleration and Implicit Regularization in Gaussian Phase Retrieval](https://mlsanity.com/paper/2311.12888/) (2023)
- [Policy Regret for Embedding Model Routing: Contextual Bandits with Low-Rank Experts](https://mlsanity.com/paper/2606.14929/) (2026)
- [Entropic Mirror Descent for Linear Systems: Polyak's Stepsize and Implicit Bias](https://mlsanity.com/paper/2505.02614/) (2025)

## Abstract (arXiv)

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^star∈ℝn with minimum (in modulus) non-zero entry on the order of | x^star |2/√k from k2 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.