On Polynomial Time Methods for Exact Low-Rank Tensor Completion

Dong, Xueyi; Yuan, Ming

doi:10.1007/s10208-018-09408-6

Cited by 54 publications

(56 citation statements)

References 33 publications

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“…After the present paper was accepted for publication, and independently from our work, two groups posted related results on the tensor completion problem [19,27,28]. We then state our main results on tensor completion: Section 3 presents the unfolding-based algorithm, and Section 4 presents the more specialized algorithm for overcomplete 3-tensors.…”

Section: Organization Of the Papermentioning

confidence: 99%

“…As noted above, for our unfolding algorithm we study the column spaces of partially revealed matrices with large aspect ratio; our results on this are presented in Section 6. After the present paper was accepted for publication, and independently from our work, two groups posted related results on the tensor completion problem [19,27,28].…”

Section: Organization Of the Papermentioning

confidence: 99%

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Spectral Algorithms for Tensor Completion

Montanari

Sun

2018

Comm Pure Appl Math

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In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees have been proved, for third-order tensors, using the sixth level of the sum-ofsquares (SOS) semidefinite programming hierarchy. However, the SOS approach does not scale well to large problem instances. By contrast, spectral methodsbased on unfolding or matricizing the tensor-are attractive for their low complexity, but have been believed to require a much larger sample size.This paper presents two main contributions. First, we propose a new method, based on unfolding, which outperforms naive ones for symmetric k th -order tensors of rank r. For this result we make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third-order tensors, our algorithm matches the SOS method in terms of sample size (requiring about rd 3=2 revealed entries), subject to a worse rank condition (r d 3=4 rather than r d 3=2 ). We complement this result with a different spectral algorithm for third-order tensors in the overcomplete (r d ) regime. Under a random model, this second approach succeeds in estimating tensors of rank d Ä r d 3=2 from about rd 3=2 revealed entries.

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Section: Organization Of the Papermentioning

confidence: 99%

Section: Organization Of the Papermentioning

confidence: 99%

Spectral Algorithms for Tensor Completion

Montanari

Sun

2018

Comm Pure Appl Math

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“…The algorithm presented here is similar in spirit to those developed earlier by Keshavan et al (2010a, 2010b), Xia and Yuan (2019). A key difference is that we introduce an explicit rule of gradient descent update where each iteration on Grassmannians is calibrated with orthogonal rotations.…”

Section: Initial Estimatementioning

confidence: 88%

Statistical Inferences of Linear Forms for Noisy Matrix Completion

Dong

Yuan

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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We introduce a flexible framework for making inferences about general linear forms of a large matrix based on noisy observations of a subset of its entries. In particular, under mild regularity conditions, we develop a universal procedure to construct asymptotically normal estimators of its linear forms through double‐sample debiasing and low‐rank projection whenever an entry‐wise consistent estimator of the matrix is available. These estimators allow us to subsequently construct confidence intervals for and test hypotheses about the linear forms. Our proposal was motivated by a careful perturbation analysis of the empirical singular spaces under the noisy matrix completion model which might be of independent interest. The practical merits of our proposed inference procedure are demonstrated on both simulated and real‐world data examples.

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“…As discussed in several related work (e.g., [14,59,84,106,107]), once we obtain reliable estimates of the subspace spanned by the tensor factors, we can further exploit the tensor structure to estimate the unknown tensor. Indeed, in many tensor completion algorithms, subspace estimation serves as a crucial initial step for tensor completion.…”

Section: Corollary 42 (Symmetric Tensor Completion)mentioning

confidence: 99%

“…However, the above-mentioned approach might lead to suboptimal performance when the row dimension and the column dimension of the matrix differ dramatically. This issue has already been recognized in multiple contexts, including but not limited to unfolding-based spectral methods for tensor estimation [57,84,106,108,114] and spectral methods for biclustering [47]. Motivated by this suboptimality issue, an alternative is to look at the "sample Gram matrix" which, as one expects, shares the same leading left singular space as the original observed data matrix.…”

Section: Further Related Workmentioning

confidence: 99%

Subspace estimation from unbalanced and incomplete data matrices: ℓ2,∞ statistical guarantees

Cai

Chi

et al. 2021

Ann. Statist.

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This paper is concerned with estimating the column space of an unknown low-rank matrix A ∈ R d 1 ×d 2 , given noisy and partial observations of its entries. There is no shortage of scenarios where the observations-while being too noisy to support faithful recovery of the entire matrix-still convey sufficient information to enable reliable estimation of the column space of interest. This is particularly evident and crucial for the highly unbalanced case where the column dimension d 2 far exceeds the row dimension d 1 , which is the focal point of the current paper.We investigate an efficient spectral method, which operates upon the sample Gram matrix with diagonal deletion. While this algorithmic idea has been studied before, we establish new statistical guarantees for this method in terms of both 2 and 2,∞ estimation accuracy, which improve upon prior results if d 2 is substantially larger than d 1 . To illustrate the effectiveness of our findings, we derive matching minimax lower bounds with respect to the noise levels, and develop consequences of our general theory for three applications of practical importance: (1) tensor completion from noisy data, (2) covariance estimation/principal component analysis with missing data and (3) community recovery in bipartite graphs. Our theory leads to improved performance guarantees for all three cases.

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On Polynomial Time Methods for Exact Low-Rank Tensor Completion

Cited by 54 publications

References 33 publications

Spectral Algorithms for Tensor Completion

Spectral Algorithms for Tensor Completion

Statistical Inferences of Linear Forms for Noisy Matrix Completion

Subspace estimation from unbalanced and incomplete data matrices: ℓ2,∞ statistical guarantees

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