Tensor Completion Algorithms in Big Data Analytics

Song, Qingquan; Ge, Hancheng; Caverlee, James; Hu, Xia

doi:10.1145/3278607

Cited by 166 publications

(128 citation statements)

References 205 publications

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“…The last example is missing value prediction or data completion for which a variety of tensor methods has been proposed, for a review see [87]. Presently, the developed toolbox only allows missing values for CP decomposition which limits this demonstration to the Bayesian CP model.…”

Section: Tensor Completionmentioning

confidence: 99%

The probabilistic tensor decomposition toolbox

Hinrich¹,

Madsen²,

Mørup³

2020

Mach. Learn.: Sci. Technol.

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This article introduces the probabilistic tensor decomposition toolbox -a MATLAB toolbox for tensor decomposition using Variational Bayesian inference and Gibbs sampling. An introduction and overview of probabilistic tensor decomposition and its connection with classical tensor decomposition methods based on maximum likelihood is provided. We subsequently describe the probabilistic tensor decomposition toolbox which encompasses the Canonical Polyadic, Tucker, and Tensor Train decomposition models. Currently, unconstrained, non-negative, orthogonal, and sparse factors are supported. Bayesian inference forms a principled way of incorporating prior knowledge, prediction of held-out data, and estimating posterior probabilities. Furthermore, it facilitates automatic model order determination, automatic regularization on factors (e.g. sparsity), and inherently penalizes model complexity which is beneficial when inferring hierarchical models, such as heteroscedastic noise modelling. The toolbox allows researchers to easily apply Bayesian tensor decomposition methods without the need to derive or implement these methods themselves. Furthermore, it serves as a reference implementation for comparing existing and new tensor decomposition methods. The software is available from https://github.com/JesperLH/prob-tensor-toolbox/. © 2020 The Author(s). Published by IOP Publishing Ltd Mach. Learn.: Sci. Technol. 1 (2020) 025011 J L Hinrich et al Bayesian tensor decompositionFor comprehensive reviews of maximum likelihood based tensor decomposition methods the reader is referred to existing tensor decomposition reviews [6-9, 22, 23]. We presently provide a short overview of Bayesian tensor decomposition. For brevity, this section will drop the prefix Bayesian and unless otherwise stated all models are based on fully Bayesian inference.Currently, probabilistic models are primarily based on the Tucker and CP decomposition. The earliest works using Tucker decomposition are found in [24,25] where both the core array and factors follow a normal distribution. The latter also considered enforcing sparsity on the core array to automatically learn the most prominent multi-linear interactions. However, neither of these methods were fully Bayesian as they relied on maximum-a-posteriori estimation which provides a point estimate of the posterior distribution. The first fully Bayesian Tucker model with normal factors using Gibbs sampling was proposed in [26] and the indeterminacy and structure of the core array was explored in [27]. The Tucker decomposition was extended to handle missing values and sparse noise based on variational Bayesian (VB) inference [28] whereas the Tucker model with an infinite core size was explored in [29][30][31] with factors specified as Gaussian processes and inference based on VB. Extensions of the Tucker model to count data was explored in [32,33] using the Poisson likelihood and either Gamma or Dirichlet factors. For categorical data, a Tucker model with multinomial likelihood and factors was considered in [34]....

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Section: Tensor Completionmentioning

confidence: 99%

The probabilistic tensor decomposition toolbox

Hinrich¹,

Madsen²,

Mørup³

2020

Mach. Learn.: Sci. Technol.

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“…Interpretation using Lagrange duality and improved reconstruction performance were studied in [15]. Extension of compressed sensing ideas to low rank matrix and tensor estimation and completion was achieved rapidly after the vector case [11], [13], [9], [18], [42], [37], [17], [39] and the case of unknown noise variance was studied in [26]. The Lagrangian approach of [15] and also extended to the matrix setting in [1].…”

Section: Second Stage Setmentioning

confidence: 99%

A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids

Chrétien¹,

Clarkson²

2020

Journal of Industrial &Amp; Management Optimization

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State estimation in power grids is a crucial step for monitoring and control tasks. It was shown that the state estimation problem can be solved using a convex relaxation based on semi-definite programming. In the present paper, we propose a fast algorithm for solving this relaxation. Our approach uses the Bürer Monteiro factorisation in a special way that solves the problem on the sphere and estimates the scale in a Gauss-Seidel fashion. Simulation results confirm the promising behaviour of the method.

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“…Moreover, multi-linear decomposition models prevent cross-modal interactions, which is particularly useful for image representations. Such a low-rank image completion methodology is mostly based on the concept of tensor completion issues that have been extensively studied [32][33][34][35][36]. There are many tensor decomposition models that are used for image completion tasks, including the fundamental ones, such as CANDECOMP/PARAFAC (CP) [37,38] and Tucker decomposition [39][40][41], as well as tensor networks, such as tensor ring [42], tensor train [43,44], hierarchical Tucker decomposition [45], and other tensor decomposition models [46].…”

Section: Introductionmentioning

confidence: 99%

Image Completion with Hybrid Interpolation in Tensor Representation

Zdunek¹,

Sadowski²

2020

Applied Sciences

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The issue of image completion has been developed considerably over the last two decades, and many computational strategies have been proposed to fill-in missing regions in an incomplete image. When the incomplete image contains many small-sized irregular missing areas, a good alternative seems to be the matrix or tensor decomposition algorithms that yield low-rank approximations. However, this approach uses heuristic rank adaptation techniques, especially for images with many details. To tackle the obstacles of low-rank completion methods, we propose to model the incomplete images with overlapping blocks of Tucker decomposition representations where the factor matrices are determined by a hybrid version of the Gaussian radial basis function and polynomial interpolation. The experiments, carried out for various image completion and resolution up-scaling problems, demonstrate that our approach considerably outperforms the baseline and state-of-the-art low-rank completion methods.

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Tensor Completion Algorithms in Big Data Analytics

Cited by 166 publications

References 205 publications

The probabilistic tensor decomposition toolbox

The probabilistic tensor decomposition toolbox

A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids

Image Completion with Hybrid Interpolation in Tensor Representation

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