Phase Retrieval by Alternating Minimization With Random Initialization

Zhang, Teng

doi:10.1109/tit.2020.2971211

Cited by 11 publications

(7 citation statements)

References 28 publications

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“…This assumption has been removed by Waldspurger in [53], which showed local converge without sample splitting. Convergence from a random initialization has been established in [54], however using a (suboptimal) sample size at the order of n 3/2 .…”

Section: Related Work and Discussionmentioning

confidence: 99%

Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing

Lee¹,

Stöger²

2022

Preprint

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We consider the problem of reconstructing rank-one matrices from random linear measurements, a task that appears in a variety of problems in signal processing, statistics, and machine learning. In this paper, we focus on the Alternating Least Squares (ALS) method. While this algorithm has been studied in a number of previous works, most of them only show convergence from an initialization close to the true solution and thus require a carefully designed initialization scheme. However, random initialization has often been preferred by practitioners as it is model-agnostic. In this paper, we show that ALS with random initialization converges to the true solution with εaccuracy in O(log n + log(1/ε)) iterations using only a near-optimal amount of samples, where we assume the measurement matrices to be i.i.d. Gaussian and where by n we denote the ambient dimension. Key to our proof is the observation that the trajectory of the ALS iterates only depends very mildly on certain entries of the random measurement matrices. Numerical experiments corroborate our theoretical predictions.

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Section: Related Work and Discussionmentioning

confidence: 99%

Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing

Lee¹,

Stöger²

2022

Preprint

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“…For instance, in [Chen, Chi, Fan, and Ma, 2019], the map T is not exactly contractive, which makes it difficult to upper bound ||T (z (t−1) ) − T (z (k,t−1) )||; the authors achieve this through a second leave-one-out argument, nested in the first one. [Zhang, 2020] can also be seen as a (quite extreme) instance of this principle, applied to alternating projections for phase retrieval in a setting where the number of measurement vectors is much larger than the dimension of the unknown signal. In this article, T k is not a variant of T : it is a random map, with the same distribution as T but independent from it.…”

Section: General Principlementioning

confidence: 99%

Lecture notes on non-convex algorithms for low-rank matrix recovery

Waldspurger¹

2021

Preprint

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Low-rank matrix recovery problems are inverse problems which naturally arise in various fields like signal processing, imaging and machine learning. They are non-convex and NP-hard in full generality. It is therefore a delicate problem to design efficient recovery algorithms and to provide rigorous theoretical insights on the behavior of these algorithms. The goal of these notes is to review recent progress in this direction for the class of so-called "non-convex algorithms", with a particular focus on the proof techniques.Although they aim at presenting very recent research works, these notes have been written with the intent to be, as much as possible, accessible to non-specialists.These notes were written for an eight-hour lecture at Collège de France. The original version, in French, is available online 1 and the videos of the lecture can be found on the Collège de France website 2 .The beginning takes inspiration from the review articles [Davenport and Romberg, 2016] and .

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“…Though we have not given the attraction radius around the solution, it seems that GPS/RGPS shows global convergence starting from a random initialization when the ratio m/n is large enough for Gaussian phase retrieval in all our numerical experiments. At the process of preparing this manuscript, a close work on the global convergence on alternating minimization for Gaussian phase retrieval is uploaded to arXiv [25]. The requirement is m/ log 3 m ≥ M n 3/2 log 1/2 n as n, m → ∞.…”

Section: Proof First We Havementioning

confidence: 99%

Solving phase retrieval via graph projection splitting

Zhao

2020

Inverse Problems

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Phase retrieval with prior information can be cast as a nonsmooth and nonconvex optimization problem. To decouple the signal and measurement variables, we introduce an auxiliary variable and reformulate it as an optimization with an equality constraint. We then solve the reformulated problem by graph projection splitting (GPS), where the two proximity subproblems and the graph projection step can be solved efficiently. With slight modification, we also propose a robust graph projection splitting (RGPS) method to stabilize the iteration for noisy measurements. Contrary to intuition, RGPS outperforms GPS with fewer iterations to locate a satisfying solution even for noiseless case. Based on the connection between GPS and Douglas-Rachford iteration, under mild conditions on the sampling vectors, we analyze the fixed point sets and provide the local convergence of GPS and RGPS applied to noiseless phase retrieval without prior information. For noisy case, we provide the error bound of the reconstruction. Compared to other existing methods, thanks for the splitting approach, GPS and RGPS can efficiently solve phase retrieval with prior information regularization for general sampling vectors which are not necessarily isometric. For Gaussian phase retrieval, compared to existing gradient flow approaches, numerical results show that GPS and RGPS are much less sensitive to the initialization. Thus they markedly improve the phase transition in noiseless case and reconstruction in the presence of noise respectively. GPS shows sharpest phase transition among existing methods including RGPS, while it needs more iterations than RGPS when the number of measurement is large enough. RGPS outperforms GPS in terms of stability for noisy measurements. When applying RGPS to more general non-Gaussian measurements with prior information, such as support, sparsity and TV minimization, RGPS either outperforms state-of-the-art solvers or can be combined with state-of-the-art solvers to improve their reconstruction quality. *

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Phase Retrieval by Alternating Minimization With Random Initialization

Cited by 11 publications

References 28 publications

Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing

Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing

Lecture notes on non-convex algorithms for low-rank matrix recovery

Solving phase retrieval via graph projection splitting

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