Factorization strategies for third-order tensors

By Misha Elena Kilmer, Carla D. Moravitz Martin. 2011

1,350 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Computer Networks and Communications

View Paper: https://www.semanticscholar.org/paper/9336bfdcb40532ffd0d890854fee6fd3d98254ac

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. Shuchin Aeron
5. Zhouchen Lin
6. Misha Elena Kilmer
7. Ning Hao
8. Canyi Lu

In Computational Theory and Mathematics

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

In Computer Networks and Communications

1. Ning Hao
2. Shuchin Aeron
3. Zhenming Peng
4. Landan Zhang
5. Gregory Ely
6. Canyi Lu
7. Zhouchen Lin
8. Jiashi Feng

Citations in the corpus

• Infrared Small Target Detection Based on Partial Sum of the Tensor Nuclear Norm (2019, methodology): “(t-SVD) [47] Let X ∈ R n 1 × n 2 × n 3 .”
• Tensor Factorization for Low-Rank Tensor Completion (2018, background): “F = A ∗ B and F̄ = ĀB̄ are equivalent [36] (also see Lemma 1).”
• Novel Methods for Multilinear Data Completion and De-noising Based on Tensor-SVD (2014, methodology): “For experiments we test 3 algorithms for video data completion from randomly missing entries: TNN minimization of Section 4, Low Rank Tensor Completion (LRTC) algorithm in [15], which uses the notion of tensor-n-rank [7], and the nuclear norm minimization on the vectorized video data using the algor”
• Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm (2018, methodology): “We empirically set λ = [15 , 15 , 1 .”
• 5D seismic data completion and denoising using a novel class of tensor decompositions (2015, methodology): “Kilmer and Martin (2011a) develop tSVD for third-order tensors and Martin et al. (2013) extend it to higher orders.”
• Tensor-Based Formulation and Nuclear Norm Regularization for Multienergy Computed Tomography (2013, methodology): “With this motivation, as the second approach, we propose a new tensor nuclear norm based on tensor singular value decomposition (t-SVD), which is introduced by Kilmer and Martin [39] and has been proved to be useful for applications such as facial recognition and image deblurring [40].”
• Exact Tensor Completion Using t-SVD (2015, neutral): “1 ([17], [18]): Let the t-SVD of M ∈ Rn1×n2×n3 be given by M = U ∗ S ∗VT and for k < min (n1 , n2) define Mk = ∑k i=1 U(:, i, :) ∗ S(i, i, :) ∗V(:, i, :) , Then”
• Generalized tensor function via the tensor singular value decomposition based on the T-product (2019, background): “Kilmer [29] gives the tensor singular value decomposition in 2011 (See Fig.”
• Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging (2013, methodology): “Recent work by Kilmer and Martin, [10] and Braman [2] provides a setting in which the familiar tools of linear algebra can be extended to better understand third-order tensors.”
• T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product (2019, background): “The operations bcirc, unfold and fold are defined as follows [12, 21, 22]:”
• Tensor-Tensor Products with Invertible Linear Transforms (2015, neutral): “Recently, the t-product has been introduced as a useful generalization of matrix multiplication for tensors of order three (i.e. three-way arrays) [1] and higher [2].”
• Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization (2016, methodology): “In this work, we study the Tensor Robust Principal Component (TRPCA) problem which aims to recover a low tubal rank tensor and a sparse tensor from their sum.”
• Infrared Dim and Small Target Detection via Multiple Subspace Learning and Spatial-Temporal Patch-Tensor Model (2021, background): “Besides, Definition 1 (T-Product) [51]: Let a tensor B ∈ C n 1 × l × n 3 , and the t-product A ∗ B can be defined as follows: where bcirc ( A ) denotes a block circulant matrix of size A ( 1 ) · · · A ( 3 ) ... ... . . . ... Definition 2 (t-SVD and Skinny t-SVD) [52]: Let A ∈ R n 1 × n 2 × n 3 .”