# Low Complexity Regularized Phase Retrieval

arXiv: https://arxiv.org/abs/2407.16413 · HTML: https://mlsanity.com/paper/2407.16413/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 74 references that are in the map.
- First posted on arXiv 23 Jul 2024.

## Builds on (most cited first, 21 of 21)

- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Phase Retrieval using Alternating Minimization](https://mlsanity.com/paper/1306.0160/) (2013)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems](https://mlsanity.com/paper/1706.06461/) (2017)
- [SLOPE - Adaptive variable selection via convex optimization](https://mlsanity.com/paper/1407.3824/) (2014)
- [Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow](https://mlsanity.com/paper/1506.03382/) (2015)
- [The Numerics of Phase Retrieval](https://mlsanity.com/paper/2004.05788/) (2020)
- [The Squared-Error of Generalized LASSO: A Precise Analysis](https://mlsanity.com/paper/1311.0830/) (2013)
- [Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimization](https://mlsanity.com/paper/1702.06175/) (2017)
- [Sparse Phase Retrieval: Convex Algorithms and Limitations](https://mlsanity.com/paper/1303.4128/) (2013)
- [Sparse Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms](https://mlsanity.com/paper/1311.2745/) (2013)
- [Guarantees of Total Variation Minimization for Signal Recovery](https://mlsanity.com/paper/1301.6791/) (2013)
- [Model Selection with Low Complexity Priors](https://mlsanity.com/paper/1307.2342/) (2013)
- [Low Complexity Regularization of Linear Inverse Problems](https://mlsanity.com/paper/1407.1598/) (2014)
- [On Fienup Methods for Regularized Phase Retrieval](https://mlsanity.com/paper/1702.08339/) (2017)
- [Stable Phase Retrieval with Mirror Descent](https://mlsanity.com/paper/2405.10754/) (2024)
- [Efficient Compressive Phase Retrieval with Constrained Sensing Vectors](https://mlsanity.com/paper/1507.08254/) (2015)
- [Compressed Sensing from Phaseless Gaussian Measurements via Linear Programming in the Natural Parameter Space](https://mlsanity.com/paper/1611.05985/) (2016)
- [Total variation-based reconstruction and phase retrieval for diffraction tomography](https://mlsanity.com/paper/2201.11579/) (2022)
- [Provable Phase Retrieval with Mirror Descent](https://mlsanity.com/paper/2210.09248/) (2022)
- [New Conditions for Sparse Phase Retrieval](https://mlsanity.com/paper/1310.1351/) (2013)

## Abstract (arXiv)

In this paper, we study the phase retrieval problem in the situation where the vector to be recovered has an a priori structure that can encoded into a regularization term. This regularizer is intended to promote solutions conforming to some notion of simplicity or low complexity. We investigate both noiseless recovery and stability to noise and provide a very general and unified analysis framework that goes far beyond the sparse phase retrieval mostly considered in the literature. In the noiseless case we provide sufficient conditions under which exact recovery, up to global sign change, is possible. For Gaussian measurement maps, we also provide a sample complexity bound for exact recovery. This bound depends on the Gaussian width of the descent cone at the soughtafter vector which is a geometric measure of the complexity of the latter. In the noisy case, we consider both the constrained (Mozorov) and penalized (Tikhonov) formulations. We provide sufficient conditions for stable recovery and prove linear convergence for sufficiently small noise. For Gaussian measurements, we again give a sample complexity bound for linear convergence to hold with high probability. This bound scales linearly in the intrinsic dimension of the sought-after vector but only logarithmically in the ambient dimension.