Tensor–tensor products with invertible linear transforms

Kernfeld, Eric; Kilmer, Misha E.; Aeron, Shuchin

doi:10.1016/j.laa.2015.07.021

Cited by 230 publications

(209 citation statements)

References 17 publications

Supporting

Mentioning

208

Contrasting

Unclassified

Order By: Relevance

“…We construct a block diagonal matrix based on the frontal slices of

{\hat{𝒜}}_{boldΦ}

as follows:

\overset{true‾}{𝒜_{boldΦ}} = blockdiag ({\hat{𝒜}}_{boldΦ}) : = (\begin{array}{cccc} {\overset{𝒜}{true}}_{Φ}^{false(1 false)} \\ {\overset{𝒜}{true}}_{Φ}^{false(2 false)} \\ ⋱ \\ {\overset{𝒜}{true}}_{Φ}^{false(n_{3} false)} \end{array}),

The block diagonal matrix can be converted into a tensor by the following fold operator:

fold (blockdiag ({\hat{𝒜}}_{boldΦ})) = {\hat{𝒜}}_{boldΦ} .

Kernfeld et al defined the ⋆ L ‐product between two tensors by the slices products in the transformed domain, where L is an arbitrary invertible transform. In this article, we are mainly interested in the t‐product which is based on unitary transformations.…”

Section: Transformed Tensor Svdmentioning

confidence: 99%

Robust tensor completion using transformed tensor singular value decomposition

Song

Zhang

2020

Numerical Linear Algebra App

144

View full text Add to dashboard Cite

In this article, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by using other unitary transform matrices than that by using discrete Fourier transform matrix. This would be more effective for robust tensor completion. Experimental results for hyperspectral, video and face datasets have shown that the recovery performance for the robust tensor completion problem by using transformed tensor SVD is better in peak signal-to-noise ratio than that by using Fourier transform and other robust tensor completion methods. K E Y W O R D Slow-rank, robust tensor completion, sparsity, transformed tensor singular value decomposition, unitary transform matrix

show abstract

“…We construct a block diagonal matrix based on the frontal slices of

{\hat{𝒜}}_{boldΦ}

as follows:

\overset{true‾}{𝒜_{boldΦ}} = blockdiag ({\hat{𝒜}}_{boldΦ}) : = (\begin{array}{cccc} {\overset{𝒜}{true}}_{Φ}^{false(1 false)} \\ {\overset{𝒜}{true}}_{Φ}^{false(2 false)} \\ ⋱ \\ {\overset{𝒜}{true}}_{Φ}^{false(n_{3} false)} \end{array}),

The block diagonal matrix can be converted into a tensor by the following fold operator:

fold (blockdiag ({\hat{𝒜}}_{boldΦ})) = {\hat{𝒜}}_{boldΦ} .

Section: Transformed Tensor Svdmentioning

confidence: 99%

Robust tensor completion using transformed tensor singular value decomposition

Song

Zhang

2020

Numerical Linear Algebra App

144

View full text Add to dashboard Cite

show abstract

“…Our second objective is to use our local tSVD feature vectors to determine if a pair of test images contain the same digit. To solve this problem, we consider each comparison (8) to be a feature for a particular image P j instead of minimizing over the number of classes. More specifically, we construct a 1 × 10 vector of features for each of our 10,000 test images.…”

Section: B Numerical Results: Identificationmentioning

confidence: 99%

Image classification using local tensor singular value decompositions

Newman

Kilmer

Horesh

2017

2017 IEEE 7th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

Self Cite

View full text Add to dashboard Cite

From linear classifiers to neural networks, image classification has been a widely explored topic in mathematics, and many algorithms have proven to be effective classifiers. However, the most accurate classifiers typically have significantly high storage costs, or require complicated procedures that may be computationally expensive. We present a novel (nonlinear) classification approach using truncation of local tensor singular value decompositions (tSVD) that robustly offers accurate results, while maintaining manageable storage costs. Our approach takes advantage of the optimality of the representation under the tensor algebra described to determine to which class an image belongs. We extend our approach to a method that can determine specific pairwise match scores, which could be useful in, for example, object recognition problems where pose/position are different. We demonstrate the promise of our new techniques on the MNIST data set.

show abstract

“…Note: This construction can be generalized considerably as recently shown in [11], but in this paper we restrict ourselves to using circular convolution to define t-SVD.…”

Section: B Tensors As Linear Operatorsmentioning

confidence: 99%

Tensor completion via optimization on the product of matrix manifolds

Girson

Aeron

2015

2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

Self Cite

View full text Add to dashboard Cite

We present a method for tensor completion using optimization on low-rank matrix manifolds. Our notion of tensorrank is based on the recently proposed framework of tensorSingular Value Decomposition (t-SVD) in [1], [2]. In contrast to convex optimization methods used in [1] that operate in a highdimensional space, in the manifold setting, one works directly in the reduced dimensionality space and is thus able to significantly reduce the computational costs [3], [4]. In this paper we focus on 3-D data and under the tensor algebraic framework of [1], [2] we show that a 3-D tensor of fixed tubal-rank can be seen as an element of the product manifold of fixed low-rank matrices in the Fourier domain. The tensor completion problem then reduces to finding the best approximation to the sampled data on this product manifold.Further, for 3-D data we consider and compare recovery performance under two approaches. In the first approach one samples entire mode-3 fibers of the tensor, which we refer to as tubal-sampling. The second approach employs element-wise sampling and we simply refer to this method as sampling. For these two types of sampling approaches, we present simulation results for surveillance video data and show that recovery under random sampling has better performance compared to the random tubal-sampling.

show abstract

Tensor–tensor products with invertible linear transforms

Cited by 230 publications

References 17 publications

Robust tensor completion using transformed tensor singular value decomposition

Robust tensor completion using transformed tensor singular value decomposition

Image classification using local tensor singular value decompositions

Tensor completion via optimization on the product of matrix manifolds

Contact Info

Product

Resources

About