Facial Recognition Using Tensor-Tensor Decompositions

Hao, Ning; Kilmer, Misha E.; Braman, Karen; Hoover, Randy C.

doi:10.1137/110842570

Cited by 165 publications

(142 citation statements)

References 17 publications

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“…The following definitions and properties are introduced in [12,21,22]. We introduce definitions of transpose, identity and orthogonal of tensors as follows.…”

Section: The Tensor T-productmentioning

confidence: 99%

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

Miao

Wei

2020

Commun. Appl. Math. Comput.

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“…The following definitions and properties are introduced in [12,21,22]. We introduce definitions of transpose, identity and orthogonal of tensors as follows.…”

Section: The Tensor T-productmentioning

confidence: 99%

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

Miao

Wei

2020

Commun. Appl. Math. Comput.

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“…Working in the Fourier domain conveniently reduces the number of arithmetic operations [22], and since the operation is separable in the third dimension it allows for parallelism. Although the representation of the training patches in the third-order tensor resembles the matrix formulation, it is not a re-formulation of the matrix problem packaged as tensors.…”

Section: Tensor Factorization Via T-productmentioning

confidence: 99%

A tensor-based dictionary learning approach to tomographic image reconstruction

2016

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We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that a) we use a third-order tensor representation for our images and b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization problem with sparsity constraints. The reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary. Numerical results show that our tensor formulation leads to very sparse representations of both the training images and the reconstructions due to the ability of representing repeated features compactly in the dictionary.

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“…1), which gives a new tensor representation and compression idea based on the tensor T-product method especially for third order tensors. The tensor singular value decomposition of tensor A ∈ C m×n×p is given by [17,29,30] A = U * S * V H ,…”

Section: Introductionmentioning

confidence: 99%

Generalized tensor function via the tensor singular value decomposition based on the T-product

Miao

Wei

2020

Linear Algebra and its Applications

101

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In this paper, we present the definition of generalized tensor function according to the tensor singular value decomposition (T-SVD) based on the tensor T-product. Also, we introduce the compact singular value decomposition (T-CSVD) of tensors, from which the projection operators and Moore Penrose inverse of tensors are obtained. We establish the Cauchy integral formula for tensors by using the partial isometry tensors and applied it into the solution of tensor equations. Then we establish the generalized tensor power and the Taylor expansion of tensors. Explicit generalized tensor functions are listed. We define the tensor bilinear and sesquilinear forms and proposed theorems on structures preserved by generalized tensor functions. For complex tensors, we established an isomorphism between complex tensors and real tensors. In the last part of our paper, we find that the block circulant operator established an isomorphism between tensors and matrices. This isomorphism is used to prove the F-stochastic structure is invariant under generalized tensor functions. The concept of invariant tensor cones is raised.

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Facial Recognition Using Tensor-Tensor Decompositions

Cited by 165 publications

References 17 publications

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

A tensor-based dictionary learning approach to tomographic image reconstruction

Generalized tensor function via the tensor singular value decomposition based on the T-product

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