Third-Order Tensors as Linear Operators on a Space of Matrices

Braman, Karen

doi:10.1016/j.laa.2010.05.025

Cited by 193 publications

(207 citation statements)

References 1 publication

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“…The recently emerged t-product [7], [1], [6], [22], which is based on 1D circular convolution, can be described by our tensor model. Let's take the t-product defined in [22] for example -Given tensors A and B of size m × n, if we leave A alone and constrain B satisfying…”

Section: B Connection To T-productmentioning

confidence: 99%

“…To evaluate the efficacy of the t-product [7], [1], [6], our tSRC is developed as two variants, tSRC-1D and tSRC-2D. Classifier tSRC-1D is tSRC via tensor sparse representation offered by the t-product, but with the constraint that the only non-zero scalar entries of tensor β k are in β k (0, :) for all k = 1, 2, · · · , N .…”

Section: B Evaluations On the Raw Mnist Datamentioning

confidence: 99%

“…Inspired by the recently reported technique called t-product [7], [1], [6], we extend SRC to its tensor-based variant while still retaining our constrained subspace model. We demonstrate in experiments that tSRC outperforms SRC and its non-tensor variant (Yang's method) in classification accuracy.…”

Section: Introductionmentioning

confidence: 99%

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Supervised classification via constrained subspace and tensor sparse representation

Liao

Maybank

Zhang

et al. 2017

2017 International Joint Conference on Neural Networks (IJCNN)

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Abstract-SRC, a supervised classifier via sparse representation, has rapidly gained popularity in recent years and can be adapted to a wide range of applications based on the sparse solution of a linear system. First, we offer an intuitive geometric model called constrained subspace to explain the mechanism of SRC. The constrained subspace model connects the dots of NN, NFL, NS, NM. Then, inspired from the constrained subspace model, we extend SRC to its tensor-based variant, which takes as input samples of high-order tensors which are elements of an algebraic ring. A tensor sparse representation is used for query tensors. We verify in our experiments on several publicly available databases that the tensor-based SRC called tSRC outperforms traditional SRC in classification accuracy. Although demonstrated for image recognition, tSRC is easily adapted to other applications involving underdetermined linear systems.

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Section: B Connection To T-productmentioning

confidence: 99%

Section: B Evaluations On the Raw Mnist Datamentioning

confidence: 99%

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Supervised classification via constrained subspace and tensor sparse representation

Liao

Maybank

Zhang

et al. 2017

2017 International Joint Conference on Neural Networks (IJCNN)

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“…In this section, we review the t-product proposed by Kilmer and Martin (2011b) and further analyzed by Braman (2010) and Kilmer et al (2013). We will focus on 3D tensors for ease of exposition and interpretation.…”

Section: The T-productmentioning

confidence: 99%

5D seismic data completion and denoising using a novel class of tensor decompositions

et al. 2015

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We have developed a novel strategy for simultaneous interpolation and denoising of prestack seismic data. Most seismic surveys fail to cover all possible source-receiver combinations, leading to missing data especially in the midpoint-offset domain. This undersampling can complicate certain data processing steps such as amplitude-variationwith-offset analysis and migration. Data interpolation can mitigate the impact of missing traces. We considered the prestack data as a 5D multidimensional array or otherwise referred to as a 5D tensor. Using synthetic data sets, we first found that prestack data can be well approximated by a lowrank tensor under a recently proposed framework for tensor singular value decomposition (tSVD). Under this low-rank assumption, we proposed a complexity-penalized algorithm for the recovery of missing traces and data denoising. In this algorithm, the complexity regularization was controlled by tuning a single regularization parameter using a statistical test. We tested the performance of the proposed algorithm on synthetic and real data to show that missing data can be reliably recovered under heavy downsampling. In addition, we demonstrated that compressibility, i.e., approximation of the data by a low-rank tensor, of seismic data under tSVD depended on the velocity model complexity and shot and receiver spacing. We further found that compressibility correlated with the recovery of missing data because high compressibility implied good recovery and vice versa.

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“…Further, it has been shown by Braman [30] that under this multiplication operation, R ×m×n is a free module over a commutative ring with unity where the "scalars" are R 1×1×n tuples. In addition, it has been shown in [30] and [31] that all linear transformations on the space R ×m×n can be represented by multiplication by a third-order tensor.…”

Section: A Mathematical Preliminariesmentioning

confidence: 99%

Pose estimation from a single image using tensor decomposition and an algebra of circulants

Hoover

Braman

Hao

2011

2011 IEEE/RSJ International Conference on Intelligent Robots and Systems

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Dimensionality reduction and object classification (recognition and pose estimation) serve as important tools in robotics, robotic vision, and industrial automation. The current paper presents a new approach to dimensionality reduction and object classification of three-dimensional rigid objects. The approach is based upon recent developments in tensor decompositions and a newly defined algebra of circulants. In particular, it is shown that under the right tensor multiplication operator, a third order tensor can be written as a product of third order tensors in which the left and right tensors are tensor-orthogonal and the inner-tensor is a diagonal tensor of singular-tuples. This new development allows for a proper tensor singular value decomposition (SVD) to be defined and has natural extension to tensor principal component analysis (PCA). Comparisons are made with traditional PCA and it is shown that the current approach is capable of recovering significantly more information from an image sequence using a much smaller subspace dimension. Further, it is shown that for most objects, accurate pose estimation can be performed from a single subspace dimension.

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Third-Order Tensors as Linear Operators on a Space of Matrices

Cited by 193 publications

References 1 publication

Supervised classification via constrained subspace and tensor sparse representation

Supervised classification via constrained subspace and tensor sparse representation

5D seismic data completion and denoising using a novel class of tensor decompositions

Pose estimation from a single image using tensor decomposition and an algebra of circulants

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