Alternating Least Squares Tensor Completion in The TT-Format

Grasedyck, Lars; Kluge, Melanie; Krämer, Sebastian

doi:10.48550/arxiv.1509.00311

Cited by 6 publications

(8 citation statements)

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“…There are several tensor decompositions, and all these papers derive some optimization procedure for one of them, namely, CP decomposition, Tucker decomposition or TT/MPS decomposition. The simplest technique is the alternating least squares [10]. It just finds the solution iteratively at each iteration minimizing the objective function w.r.t.…”

Section: Resultsmentioning

confidence: 99%

Tensor Completion via Gaussian Process Based Initialization

Kapushev¹,

Oseledets²,

Burnaev³

2019

Preprint

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In this paper, we consider the tensor completion problem representing the solution in the tensor train (TT) format. It is assumed that tensor is high-dimensional, and tensor values are generated by an unknown smooth function. The assumption allows us to develop an efficient initialization scheme based on Gaussian Process Regression and TT-cross approximation technique. The proposed approach can be used in conjunction with any optimization algorithm that is usually utilized in tensor completion problems. We empirically justify that in this case the reconstruction error improves compared to the tensor completion with random initialization. As an additional benefit, our technique automatically selects rank thanks to using the TT-cross approximation technique.

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Section: Resultsmentioning

confidence: 99%

Tensor Completion via Gaussian Process Based Initialization

Kapushev¹,

Oseledets²,

Burnaev³

2019

Preprint

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“…To solve the tensor completion problem with TT decomposition, Wang et al [30] and Grasedyck et al [6] developed algorithms that iteratively solve minimization problems with respect to G k for each k = 1, . .…”

Section: Related Workmentioning

confidence: 99%

“…[30] assumed that the TT rank is given. Grasedyck et al [6] proposed a grid search method. However, the TT rank is determined by a single parameter (i.e., R 1 = • • • = R K−1 ) and the search method lacks its generality.…”

Section: Related Workmentioning

confidence: 99%

“…In contrast, such a relationship has not been studied for TT decomposition yet. Second, standard TT decomposition algorithms, such as alternating least squares (ALS) [6,30] , require a huge computational cost. The main bottleneck arises from the singular value decomposition (SVD) operation to an "unfolding" matrix, which is reshaped from the input tensor.…”

Section: Introductionmentioning

confidence: 99%

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On Tensor Train Rank Minimization: Statistical Efficiency and Scalable Algorithm

Imaizumi,

Maehara,

Hayashi

2017

Preprint

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Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations. First, we introduce a convex relaxation of the TT decomposition problem and derive its error bound for the tensor completion task. Next, we develop an alternating optimization method with a randomization technique, in which the time complexity is as efficient as the space complexity is. In experiments, we numerically confirm the derived bounds and empirically demonstrate the performance of our method with a real higher-order tensor.

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“…The low-rank tensor decomposition paradigm allows for extracting the most meaningful and informative latent structures of a tensor, which usually contain heterogeneous and multi-aspect data. The Canonical Polyadic (CP) decomposition [20,28,25], the Tucker or the multilinear decomposition [44,13,14], and the tensor-train (TT) decomposition [36,15,38] are among the most fundamental tensor decomposition forms. Other variants include hierarchical tensor representations [12,39,40] and PARAFAC2 models [39].…”

Section: Introductionmentioning

confidence: 99%

New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition

Dong,

Gao,

Guan

et al. 2021

Preprint

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We propose new Riemannian preconditioned algorithms for low-rank tensor completion via the polyadic decomposition of a tensor. These algorithms exploit a non-Euclidean metric on the product space of the factor matrices of the low-rank tensor in the polyadic decomposition form. This new metric is designed using an approximation of the diagonal blocks of the Hessian of the tensor completion cost function, thus has a preconditioning effect on these algorithms. We prove that the proposed Riemannian gradient descent algorithm globally converges to a stationary point of the tensor completion problem, with convergence rate estimates using the Lojasiewicz property. Numerical results on synthetic and real-world data suggest that the proposed algorithms are more efficient in memory and time compared to state-of-the-art algorithms. Moreover, the proposed algorithms display a greater tolerance for overestimated rank parameters in terms of the tensor recovery performance, thus enable a flexible choice of the rank parameter.

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Alternating Least Squares Tensor Completion in The TT-Format

Cited by 6 publications

References 0 publications

Tensor Completion via Gaussian Process Based Initialization

Tensor Completion via Gaussian Process Based Initialization

On Tensor Train Rank Minimization: Statistical Efficiency and Scalable Algorithm

New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition

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