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Low Complexity Regularized Phase Retrieval

arXiv 2407.16413 ↗First posted 23 Jul 2024math.OC cs.CV cs.ITcited by 0

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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.

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.

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References · 21

  1. 2014Phase Retrieval via Wirtinger Flow: Theory and Algorithmscited by 125
  2. 2013Phase Retrieval using Alternating Minimizationcited by 58
  3. 2015Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systemscited by 50
  4. 2017First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problemscited by 37
  5. 2014SLOPE - Adaptive variable selection via convex optimizationcited by 33
  6. 2015Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flowcited by 23
  7. 2020The Numerics of Phase Retrievalcited by 21
  8. 2013The Squared-Error of Generalized LASSO: A Precise Analysiscited by 15
  9. 2017Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimizationcited by 13
  10. 2013Sparse Phase Retrieval: Convex Algorithms and Limitationscited by 8
  11. 2013Sparse Phase Retrieval: Uniqueness Guarantees and Recovery Algorithmscited by 8
  12. 2013Guarantees of Total Variation Minimization for Signal Recoverycited by 3
9 more
  1. 2013Model Selection with Low Complexity Priorscited by 3
  2. 2014Low Complexity Regularization of Linear Inverse Problemscited by 3
  3. 2017On Fienup Methods for Regularized Phase Retrievalcited by 3
  4. 2024Stable Phase Retrieval with Mirror Descentcited by 3
  5. 2015Efficient Compressive Phase Retrieval with Constrained Sensing Vectorscited by 2
  6. 2016Compressed Sensing from Phaseless Gaussian Measurements via Linear Programming in the Natural Parameter Spacecited by 2
  7. 2022Total variation-based reconstruction and phase retrieval for diffraction tomographycited by 2
  8. 2022Provable Phase Retrieval with Mirror Descentcited by 2
  9. 2013New Conditions for Sparse Phase Retrievalcited by 1

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