# The nonsmooth landscape of phase retrieval

arXiv: https://arxiv.org/abs/1711.03247 · HTML: https://mlsanity.com/paper/1711.03247/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Nov 2017.

## Built on by (most cited first, 16 of 16)

- [Escaping mediocrity: how two-layer networks learn hard generalized linear models with SGD](https://mlsanity.com/paper/2305.18502/) (2023)
- [Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization](https://mlsanity.com/paper/2108.11832/) (2021)
- [Catapult Dynamics and Phase Transitions in Quadratic Nets](https://mlsanity.com/paper/2301.07737/) (2023)
- [Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterization](https://mlsanity.com/paper/2310.06159/) (2023)
- [Spectral Gradient Descent Mitigates Anisotropy-Driven Misalignment: A Case Study in Phase Retrieval](https://mlsanity.com/paper/2601.22652/) (2026)
- [Gradient descent provably escapes saddle points in the training of shallow ReLU networks](https://mlsanity.com/paper/2208.02083/) (2022)
- [Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithms](https://mlsanity.com/paper/2102.02837/) (2021)
- [On the Uniform Convergence of Subdifferentials in Stochastic Optimization and Learning](https://mlsanity.com/paper/2405.10289/) (2024)
- [Stable Phase Retrieval with Mirror Descent](https://mlsanity.com/paper/2405.10754/) (2024)
- [Fast Escape, Slow Convergence: Learning Dynamics of Phase Retrieval under Power-Law Data](https://mlsanity.com/paper/2511.18661/) (2025)
- [Provable Phase Retrieval with Mirror Descent](https://mlsanity.com/paper/2210.09248/) (2022)
- [Sharp description of local minima in the loss landscape of high-dimensional two-layer ReLU neural networks](https://mlsanity.com/paper/2604.09412/) (2026)
- [Convergence of Spectral Descent for Non-smooth Optimization](https://mlsanity.com/paper/2605.26977/) (2026)
- [Smoothed Robust Phase Retrieval](https://mlsanity.com/paper/2409.01570/) (2024)
- [ℓ1-norm rank-one symmetric matrix factorization has no spurious second-order stationary points](https://mlsanity.com/paper/2410.05025/) (2024)
- [Global Optimality for Constrained Exploration via Penalty Regularization](https://mlsanity.com/paper/2604.28144/) (2026)

## Abstract (arXiv)

We consider a popular nonsmooth formulation of the real phase retrieval problem. We show that under standard statistical assumptions, a simple subgradient method converges linearly when initialized within a constant relative distance of an optimal solution. Seeking to understand the distribution of the stationary points of the problem, we complete the paper by proving that as the number of Gaussian measurements increases, the stationary points converge to a codimension two set, at a controlled rate. Experiments on image recovery problems illustrate the developed algorithm and theory.