Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging

By Misha Elena Kilmer, Karen S. Braman, Ning Hao, Randy C. Hoover. SIAM Journal on Matrix Analysis and Applications, 2013

1,223 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/4f143cbc63e4c202db77566dac0f1f08c0774a45

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mathematics

1. Yimin Wei
2. Liqun Qi
3. Yun Miao
4. Zhouchen Lin
5. Canyi Lu
6. Shuchin Aeron
7. Misha Elena Kilmer
8. Wei David Liu

In Computational Mechanics

1. Zhouchen Lin
2. Canyi Lu
3. Jiashi Feng
4. Liqun Qi
5. Yun Miao
6. Yimin Wei
7. Shuicheng Yan
8. Shuchin Aeron

In Computational Theory and Mathematics

1. Liqun Qi
2. Yimin Wei
3. Ning Hao
4. Shuchin Aeron
5. Zhouchen Lin
6. Yun Miao
7. Gregory Ely
8. Canyi Lu

Citations in the corpus

• 5D seismic data completion and denoising using a novel class of tensor decompositions (2015, methodology): “Seismic data completion using tensor singular value decomposition Based on tSVD, one can define the notion of the tensor rank (Kilmer et al., 2013).”
• On Unifying Multi-view Self-Representations for Clustering by Tensor Multi-rank Minimization (2016, background): “…interpretability in the original domain, i.e., ||bcirc(X )||∗ measures the rank of bcirc(X ) by comparing every row and every column of frontal slices over the third dimension, which exploits structural information of a tensor deeper than the monotonous matrix nuclear norm of certain unfolding.”
• Tensor-Tensor Products with Invertible Linear Transforms (2015, neutral): “In [3, 4, 8], the authors exploit the fact that the t-product between n tensors is in fact equivalent to the discrete convolution of two vectors.”
• Essential Tensor Learning for Multi-View Spectral Clustering (2018, methodology): “The CANDECOMP/PARAFAC (CP) [12], [13], Tucker [14] and tensor Singular Value Decomposition (t-SVD) [15] are three main tensor decomposition techniques.”
• Tensor Factorization for Low-Rank Tensor Completion (2018, methodology): “Definition 6 (Tensor Multi-Rank and Tubal Rank [33]): For any A ∈ Rn1×n2×n3 , its multi-rank rankm(A) is a vector defined as r = (rank( Ā(1)); · · · ; Ā(n3))).”
• Novel Methods for Multilinear Data Completion and De-noising Based on Tensor-SVD (2014, methodology): “Measures of tensor complexity using t-SVD We now define two measures of tensor complexity based on the proposed t-SVD: the tensor multi-rank, proposed in [11], and the novel tensor tubal rank.”
• Tensor-Based Formulation and Nuclear Norm Regularization for Multienergy Computed Tomography (2013, neutral): “A K -way tensor is a multi-linear structure in R N 1 × N 2 ×⋯× N K .”
• T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product (2019, background): “On the other hand, the tensor T-product introduced by Kilmer [21] has been proved to be of great use in many areas, such as image processing [21, 29, 35, 37], computer vision [12], signal processing [7, 24, 25], low rank tensor recovery and robust tensor PCA [23, 24], data completion and denoising [”
• Generalized tensor function via the tensor singular value decomposition based on the T-product (2019, methodology): “Since tensor A has the T-SVD decomposition [28, 29] taking “bcirc” on both sides of the equation bcirc( where bcirc( U ) and bcirc( V H ) are unitary matrices of order n 1 n 3 × n 1 n 3 and n 2 n 3 × n 2 n 3 respectively, and is a block circulant matrix in C n 1 n 3 × n 2 n 3 .”
• Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm (2018, background): “(Tensor tubal rank) [14], [34] For A ∈ n 1 × n 2 × n 3 , the tensor tubal rank, denoted as rank t ( A ) , is defined as the number of nonzero singular tubes of S , where S is from the t-SVD of A = U ∗ S ∗ V ∗ .”
• Exact Tensor Completion Using t-SVD (2015, background): “Some of the notations and definitions of tensors are discussed in [25], [1], [2] and some notations are novel in this section.”
• Parallel matrix factorization for low-rank tensor completion (2013, neutral): “Different from the above discussed models that use tensor n -rank, the model in [32] employs the so-called tubal-rank based on the recently proposed tensor singular value decomposition (t-SVD) [8].”