Weighted Matrix Completion From Non-Random, Non-Uniform Sampling Patterns

Foucart, Simon; Needell, Deanna; Pathak, Reese; Plan, Yaniv; Wootters, Mary

doi:10.1109/tit.2020.3039308

Cited by 23 publications

(13 citation statements)

References 35 publications

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“…Succinctly, their works give a way of measuring which deterministic sampling patterns,

, are “good” with respect to a weight matrix H . In contrast to these two works, [ 19 ] is interested in the problem of whether one can find a weight matrix H and create an efficient algorithm to find an estimate

for an underlying low-rank matrix M from a sampling pattern

and noisy samples

such that

is small.…”

Section: Related Work Background and Problem Statementmentioning

confidence: 99%

“…In particular, one of our theoretical results is that we generalize the upper bounds for weighted recovery of low-rank matrices from deterministic sampling patterns in [ 19 ] to the upper bound of tensor weighted recovery. The details of the connection between our result and the matrix setting result in [ 19 ] is discussed in Section 3 .…”

Section: Related Work Background and Problem Statementmentioning

confidence: 99%

“…In matrix completion, there are mature algorithms [ 11 ], theoretical foundations [ 12 , 13 , 14 ] and various applications [ 15 , 16 , 17 , 18 ] that pave the way for solving the tensor completion problem in high-order tensors. Recently, Foucart et al [ 19 ] proposed a simple algorithm for matrix completion for general deterministic sampling patterns, and raised the following questions: given a deterministic sampling pattern

and corresponding (possibly noisy) observations of the matrix entries, what type of recovery error can we expect? In what metric?…”

Section: Introductionmentioning

confidence: 99%

“…How can we efficiently implement recovery? These were investigated in [ 19 ] by introducing an appropriate weighted error metric for matrix recovery of the form

, where M is the true underlying low-rank matrix,

refers to the recovered matrix, and H is a best rank-1 matrix approximation for the sampling pattern

. In this regard, similar questions arise for the problem of tensor completion with deterministic sampling patterns.…”

Section: Introductionmentioning

confidence: 99%

“…The simulation results show that when the sampling pattern is non-uniform, the use of weights in the weighted HOSVD algorithm is essential, and the results of the weighted HOSVD algorithm can provide a very good initialization for the total variation minimization algorithm which can dramatically reduce the iterative steps without lose the accuracy. In doing so, we extend the weighted matrix completion results of [ 19 ] to the tensor setting.…”

Section: Introductionmentioning

confidence: 99%

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HOSVD-Based Algorithm for Weighted Tensor Completion

2021

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Matrix completion, the problem of completing missing entries in a data matrix with low-dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog that attempts to impute missing tensor entries from similar low-rank type assumptions. In this paper, we study the tensor completion problem when the sampling pattern is deterministic and possibly non-uniform. We first propose an efficient weighted Higher Order Singular Value Decomposition (HOSVD) algorithm for the recovery of the underlying low-rank tensor from noisy observations and then derive the error bounds under a properly weighted metric. Additionally, the efficiency and accuracy of our algorithm are both tested using synthetic and real datasets in numerical simulations.

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“…Succinctly, their works give a way of measuring which deterministic sampling patterns,

for an underlying low-rank matrix M from a sampling pattern

and noisy samples

such that

is small.…”

Section: Related Work Background and Problem Statementmentioning

confidence: 99%

Section: Related Work Background and Problem Statementmentioning

confidence: 99%

and corresponding (possibly noisy) observations of the matrix entries, what type of recovery error can we expect? In what metric?…”

Section: Introductionmentioning

confidence: 99%

“…How can we efficiently implement recovery? These were investigated in [ 19 ] by introducing an appropriate weighted error metric for matrix recovery of the form

, where M is the true underlying low-rank matrix,

refers to the recovered matrix, and H is a best rank-1 matrix approximation for the sampling pattern

. In this regard, similar questions arise for the problem of tensor completion with deterministic sampling patterns.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

HOSVD-Based Algorithm for Weighted Tensor Completion

2021

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Bi-directional matrix completion for highly incomplete multi-label learning via co-embedding predictive side information

Xia,

Tang,

Wang

2023

Appl Intell

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Toward Efficient Direct Dynamics Studies of Chemical Reactions: A Novel Matrix Completion Algorithm

Quiton

Chae

Bac

et al. 2022

J. Chem. Theory Comput.

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This paper describes the development and testing of a polynomial variety-based matrix completion (PVMC) algorithm. Our goal is to reduce computational effort associated with reaction rate coefficient calculations using variational transition state theory with multidimensional tunneling (VTST-MT). The algorithm recovers eigenvalues of quantum mechanical Hessians constituting the minimum energy path (MEP) of a reaction using only a small sample of the information, by leveraging underlying properties of these eigenvalues. In addition to the low-rank property that constitutes the basis for most matrix completion (MC) algorithms, this work introduces a polynomial constraint in the objective function. This enables us to sample matrix columns unlike most conventional MC methods that can only sample elements, which makes PVMC readily compatible with quantum chemistry calculations as sampling a single column requires one Hessian calculation. For various types of reactionsS N 2, hydrogen atom transfer, metal–ligand cooperative catalysis, and enzyme chemistrywe demonstrate that PVMC on average requires only six to seven Hessian calculations to accurately predict both quantum and variational effects.

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Weighted Matrix Completion From Non-Random, Non-Uniform Sampling Patterns

Cited by 23 publications

References 35 publications

HOSVD-Based Algorithm for Weighted Tensor Completion

HOSVD-Based Algorithm for Weighted Tensor Completion

Bi-directional matrix completion for highly incomplete multi-label learning via co-embedding predictive side information

Toward Efficient Direct Dynamics Studies of Chemical Reactions: A Novel Matrix Completion Algorithm

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