Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics
TLDR
In this paper, the authors define rank (X) as the minimum number of triads whose sum is X, and dim1(X) to be the dimensionality of the space of matrices generated by the 1-slabs of X.About:
This article is published in Linear Algebra and its Applications.The article was published on 1977-01-01 and is currently open access. It has received 1644 citations till now. The article focuses on the topics: Rank (linear algebra) & Rank of an abelian group.read more
Citations
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Journal ArticleDOI
Tensor Decompositions and Applications
Tamara G. Kolda,Brett W. Bader +1 more
TL;DR: This survey provides an overview of higher-order tensor decompositions, their applications, and available software.
Journal ArticleDOI
PARAFAC. Tutorial and applications
TL;DR: The multi-way decomposition method PARAFAC is a generalization of PCA to higher order arrays, but some of the characteristics of the method are quite different from the ordinary two-way case.
Journal ArticleDOI
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
TL;DR: The aim of this paper is to introduce a few key notions and applications connected to sparsity, targeting newcomers interested in either the mathematical aspects of this area or its applications.
Book
Stable recovery of sparse overcomplete representations in the presence of noise
TL;DR: This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system and shows that similar stability is also available using the basis and the matching pursuit algorithms.
Book
Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation
TL;DR: This book provides a broad survey of models and efficient algorithms for Nonnegative Matrix Factorization (NMF), including NMFs various extensions and modifications, especially Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD).
References
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Book
The Design and Analysis of Computer Algorithms
Alfred V. Aho,John E. Hopcroft +1 more
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Journal ArticleDOI
Analysis of individual differences in multidimensional scaling via an n-way generalization of 'eckart-young' decomposition
J. Douglas Carroll,Jih-Jie Chang +1 more
TL;DR: In this paper, an individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common "psychological space" and a corresponding method of analyzing similarities data is proposed, involving a generalization of Eckart-Young analysis to decomposition of three-way (or higher-way) tables.
Journal ArticleDOI
Gaussian elimination is not optimal
TL;DR: In this paper, Cook et al. gave an algorithm which computes the coefficients of the product of two square matrices A and B of order n with less than 4. 7 n l°g 7 arithmetical operations (all logarithms in this paper are for base 2).
Journal ArticleDOI
Vermeidung von Divisionen.
K. Ramachandra,Volker Strassen +1 more
TL;DR: In this article, it was shown that the use of divisions does not decrease the number of (*,/)-operations for multiplication of general matrices, and that multiplication of orthogonal matrices does not increase the computational complexity.
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Analysis of individual differences in multidimensional scaling via an n-way generalization of 'eckart-young' decomposition
J. Douglas Carroll,Jih-Jie Chang +1 more