Composite optimization for robust blind deconvolution

Charisopoulos, Vasileios; Davis, Damek; Díaz, Mateo

doi:10.48550/arxiv.1901.01624

Cited by 9 publications

(30 citation statements)

References 36 publications

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“…[24,Lemma 4.2]. In particular, a variety of practical problems in statistical signal recovery are weakly convex; see [14,15,19,34] for examples. for some closed function f : R d → R that is bounded from below on a closed set X ⊆ R d .…”

Section: Preliminariesmentioning

confidence: 99%

Stochastic optimization over proximally smooth sets

Davis,

Drusvyatskiy,

Shi

2020

Preprint

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We introduce a class of stochastic algorithms for minimizing weakly convex functions over proximally smooth sets. As their main building blocks, the algorithms use simplified models of the objective function and the constraint set, along with a retraction operation to restore feasibility. All the proposed methods come equipped with a finite time efficiency guarantee in terms of a natural stationarity measure. We discuss consequences for nonsmooth optimization over smooth manifolds and over sets cut out by weakly-convex inequalities.

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Section: Preliminariesmentioning

confidence: 99%

Stochastic optimization over proximally smooth sets

Davis,

Drusvyatskiy,

Shi

2020

Preprint

Self Cite

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“…An increasing amount of research has shown how matrix recovery problems, which in the worst case are hard, become tractable under appropriate statistical assumptions. Examples include phase retrieval [2][3][4], blind deconvolution [1,5], matrix sensing [6,7], matrix completion [8,9], and robust PCA [10,11], among others [12][13][14][15]. Convex relaxations have proven to be a great tool to tackle these problems, but they often require lifting the problem to a higher dimensional space and consequently end up being computationally expensive.…”

Section: Introductionmentioning

confidence: 99%

“…To tackle the blind deconvolution problem, [1] proposed the following nonconvex nonsmooth formulation argmin w,x f S (w, x)…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The authors of [1] designed a two-stage method based on this formulation and showed that if the measuring vectors, a i and b i , are i.i.d standard Gaussian, then their algorithm converges rapidly to a solution whenever m (d 1 + d 2 ). 1 Nonetheless, experimentally the initialization stage seems to be superfluous. Indeed, a simple randomly-initialized subgradient algorithm is successful at solving the problem most of the time provided that m/(d 1 + d 2 ) is big enough, see the experiments in the last section for support of this claim.…”

Section: Introductionmentioning

confidence: 99%

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The nonsmooth landscape of blind deconvolution

Díaz

2019

Preprint

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The blind deconvolution problem aims to recover a rank-one matrix from a set of rankone linear measurements. Recently, Charisopulos et al.[1] introduced a nonconvex nonsmooth formulation that can be used, in combination with an initialization procedure, to provably solve this problem under standard statistical assumptions. In practice, however, initialization is unnecessary. As we demonstrate numerically, a randomly initialized subgradient method consistently solves the problem. In pursuit of a better understanding of this phenomenon, we study the random landscape of this formulation. We characterize in closed form the landscape of the population objective and describe the approximate location of the stationary points of the sample objective. In particular, we show that the set of spurious critical points lies close to a codimension two subspace. In doing this, we develop tools for studying the landscape of a broader family of singular value functions, these results may be of independent interest. * people.cam.cornell.edu/md825/

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Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

Chi

Chen

2019

IEEE Trans. Signal Process.

336

284

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Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently.In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.

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Composite optimization for robust blind deconvolution

Cited by 9 publications

References 36 publications

Stochastic optimization over proximally smooth sets

Stochastic optimization over proximally smooth sets

The nonsmooth landscape of blind deconvolution

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

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