Douglas-Rachford feasibility methods for matrix completion problems

Artacho, Francisco J. Aragón; Borwein, Jonathan M.; Tam, Matthew Kyle

doi:10.21914/anziamj.v55i0.7470

Cited by 3 publications

(4 citation statements)

References 28 publications

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“…Also, the possibility of taking pairwise or q-wise (q < m) circumcenters is in the scope of our future studies, as well as deriving the sharp rate for CRM. Positive experiments together with its geometrical appeal, may reveal CRM as an option for solving highly important feasibility problems (even nonconvex) involving (affine) subspaces, with very promising behavior in several applications, e.g., the matrix completion problem [13], the basis pursuit problem [14] and the nonconvex sparse affine feasibility problem [15].…”

Section: Discussionmentioning

confidence: 99%

On the linear convergence of the circumcentered-reflection method

Behling

Bello-Cruz

Santos

2018

Operations Research Letters

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In order to accelerate the Douglas-Rachford method we recently developed the circumcentered-reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.

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Section: Discussionmentioning

confidence: 99%

On the linear convergence of the circumcentered-reflection method

Behling

Bello-Cruz

Santos

2018

Operations Research Letters

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Section: Introductionmentioning

confidence: 99%

“…In more general cases, the relationship with the proximal point algorithm was studied in Reference 21. As the fruitful application of DR algorithm in nonconvex problems, 22,23 the related theoretical analysis of splitting algorithms also attracted more and more experts and scholars. 24,25 To faster solve nonconvex and nonsmooth minimization problem problems, recently, an inertial technology was proposed and becomes a research hotspot.…”

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confidence: 99%

An inertial Douglas–Rachford splitting algorithm for nonconvex and nonsmooth problems

Feng

Zhang

et al. 2021

Concurrency and Computation

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In the fields of wireless communication and data processing, there are varieties of mathematical optimization problems, especially nonconvex and nonsmooth problems.For these problems, one of the biggest difficulties is how to improve the speed of solution. To this end, here we mainly focused on a minimization optimization model that is nonconvex and nonsmooth. Firstly, an inertial Douglas-Rachford splitting (IDRS) algorithm was established, which incorporate the inertial technology into the framework of the Douglas-Rachford splitting algorithm. Then, we illustrated the iteration sequence generated by the proposed IDRS algorithm converges to a stationary point of the nonconvex nonsmooth optimization problem with the aid of the Kurdyka-Łojasiewicz property. Finally, a series of numerical experiments were carried out to prove the effectiveness of our proposed algorithm from the perspective of signal recovery. The results are implicit that the proposed IDRS algorithm outperforms another algorithm.

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“…Recently, FBS and DRS are found to numerically converge for certain nonconvex problems, for example, FBS for image restoration [22], dictionary learning, and matrix decomposition [25], and DRS for nonconvex feasibility problem [14], matrix completion [1], and phase retrieval [7]. Theoretically, their iterates have been shown to converge to stationary points in some nonconvex settings [2,14,26,11].…”

Section: Introductionmentioning

confidence: 99%

An Envelope for Davis–Yin Splitting and Strict Saddle-Point Avoidance

Liu

Yin

2019

J Optim Theory Appl

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It is known that operator splitting methods based on Forward Backward Splitting (FBS), Douglas-Rachford Splitting (DRS), and Davis-Yin Splitting (DYS) decompose a difficult optimization problems into simpler subproblem under proper convexity and smoothness assumptions. In this paper, we identify an envelope (an objective function) whose gradient descent iteration under a variable metric coincides with DYS iteration. This result generalizes the Moreau envelope for proximal-point iteration and the envelopes for FBS and DRS iterations identified by Patrinos, Stella, and Themelis.Based on the new envelope and the Stable-Center Manifold Theorem, we further show that, when FBS or DRS iterations start from random points, they avoid all strict saddle points with probability one. This result extends the similar results by Lee et al. from gradient descent to splitting methods. 4When h = 0, (5) simplifies to Douglas-Rachford Splitting iteration,When g = 0, P γg reduces to Id and thus (5) simplifies towhich is Forward-Backward Splitting iteration slightly generalized by including the linear operator L.When f = 0, P γf reduces to Id and (5) simplifies to Backward-Forward Splitting,When f = g = 0, (5) simplifies to gradient descent iteration Derivation of envelopeNow we show that, (5) can be written as gradient descent iteration of an envelope function under the following assumption.Assumption 1.

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Douglas-Rachford feasibility methods for matrix completion problems

Abstract: In this paper we give general recommendations for successful application of the Douglas-Rachford reflection method to convex and non-convex real matrix-completion problems. These guidelines are demonstrated by various illustrative examples.

Cited by 3 publications

References 28 publications

On the linear convergence of the circumcentered-reflection method

On the linear convergence of the circumcentered-reflection method

An inertial Douglas–Rachford splitting algorithm for nonconvex and nonsmooth problems

An Envelope for Davis–Yin Splitting and Strict Saddle-Point Avoidance

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