Leave-one-out Approach for Matrix Completion: Primal and Dual Analysis

Ding, Lijun; Chen, Yudong

doi:10.48550/arxiv.1803.07554

Cited by 11 publications

(15 citation statements)

References 35 publications

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“…It has been proved that with high probability X is the unique solution to (Matrix-Completion) [DC18]. Hence the matrix…”

Section: Linear Independence and Uniqueness Of Primalmentioning

confidence: 99%

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On the simplicity and conditioning of low rank semidefinite programs

Ding

Udell²

2020

Preprint

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Low rank matrix recovery problems appear widely in statistics, combinatorics, and imaging. One celebrated method for solving these problems is to formulate and solve a semidefinite program (SDP). It is often known that the exact solution to the SDP with perfect data recovers the solution to the original low rank matrix recovery problem. It is more challenging to show that an approximate solution to the SDP formulated with noisy problem data acceptably solves the original problem; arguments are usually ad hoc for each problem setting, and can be complex.In this note, we identify a set of conditions that we call regularity that limit the error due to noisy problem data or incomplete convergence. In this sense, regular SDPs are robust: regular SDPs can be (approximately) solved efficiently at scale; and the resulting approximate solutions, even with noisy data, can be trusted. Moreover, we show that regularity holds generically, and also for many structured low rank matrix recovery problems, including the stochastic block model, Z2 synchronization, and matrix completion. Formally, we call an SDP regular if it has a surjective constraint map, admits a unique primal and dual solution pair, and satisfies strong duality and strict complementarity.However, regularity is not a panacea: we show the Burer-Monteiro formulation of the SDP may have spurious second-order critical points, even for a regular SDP with a rank 1 solution.

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“…It has been proved that with high probability X is the unique solution to (Matrix-Completion) [DC18]. Hence the matrix…”

Section: Linear Independence and Uniqueness Of Primalmentioning

confidence: 99%

“…The matrix Y 0 is actually a dual certificate for X for (Matrix-Completion). We follow the construction procedure in [DC18]. First set k 0 : = C 0 log(µr ) for some large enough numerical constant C 0 .…”

Section: B1 Proof Of Lemmamentioning

confidence: 99%

On the simplicity and conditioning of low rank semidefinite programs

Ding

Udell²

2020

Preprint

Self Cite

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“…, where i ∈ [K] is chosen to satisfy (17). It follows directly from the definition of W that W F = 1.…”

Section: Upper Bound For Blind Deconvolutionmentioning

confidence: 99%

“…For the case of very small ranks this result can be refined further[17]. Namely one can remove one of the two log-factors at the cost of an r 3 -dependence on the rank.…”

mentioning

confidence: 99%

On the convex geometry of blind deconvolution and matrix completion

Krahmer¹,

Stöger²

2019

Preprint

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Low-rank matrix recovery from structured measurements has been a topic of intense study in the last decade and many important problems like matrix completion and blind deconvolution have been formulated in this framework. An important benchmark method to solve these problems is to minimize the nuclear norm, a convex proxy for the rank. A common approach to establish recovery guarantees for this convex program relies on the construction of a so-called approximate dual certificate. However, this approach provides only limited insight in various respects. Most prominently, the noise bounds exhibit seemingly suboptimal dimension factors. In this paper we take a novel, more geometric viewpoint to analyze both the matrix completion and the blind deconvolution scenario. We find that for both these applications the dimension factors in the noise bounds are not an artifact of the proof, but the problems are intrinsically badly conditioned. We show, however, that bad conditioning only arises for very small noise levels: Under mild assumptions that include many realistic noise levels we derive near-optimal error estimates for blind deconvolution under adversarial noise.

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“…There are iterative methods, for specific problems, that provably converge to minimizers [9,12,[17][18][19]. Analysis of these methods are of two types: those based on studying the iterate sequence [20][21][22], and those based on characterizing the landscape of smooth loss functions [17,23,24].…”

Section: Introductionmentioning

confidence: 99%

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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Leave-one-out Approach for Matrix Completion: Primal and Dual Analysis

Cited by 11 publications

References 35 publications

On the simplicity and conditioning of low rank semidefinite programs

On the simplicity and conditioning of low rank semidefinite programs

On the convex geometry of blind deconvolution and matrix completion

The nonsmooth landscape of blind deconvolution

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