Rank and optimal computation of generic tensors
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TLDR
The typical rank (= maximal border rank) of tensors of a given size and the set of optimal bilinear computations of typical TensorRank are investigated and it is shown that for the size ( n, n, 3) with n odd, the complement of theSet of Tensors of maximal borderRank is a hypersurface.About:
This article is published in Linear Algebra and its Applications.The article was published on 1983-07-01 and is currently open access. It has received 259 citations till now. The article focuses on the topics: Rank (linear algebra) & Invariants of tensors.read more
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Tensor Decomposition for Signal Processing and Machine Learning
Nicholas D. Sidiropoulos,Lieven De Lathauwer,Xiao Fu,Kejun Huang,Evangelos E. Papalexakis,Christos Faloutsos +5 more
TL;DR: The material covered includes tensor rank and rank decomposition; basic tensor factorization models and their relationships and properties; broad coverage of algorithms ranging from alternating optimization to stochastic gradient; statistical performance analysis; and applications ranging from source separation to collaborative filtering, mixture and topic modeling, classification, and multilinear subspace learning.
Book
Classical Algebraic Geometry: A Modern View
TL;DR: In this paper, the authors present a survey of the geometry of lines and cubic surfaces, including determinantal equations, theta characteristics, and the Cremona transformations.
BookDOI
Algebraic Statistics for Computational Biology
Lior Pachter,Bernd Sturmfels +1 more
TL;DR: The Four Themes are presented: Parametric inference, tree construction using Singular Value Decomposition, analysis of point mutations in vertebrate genomes, extended statistical models from trees to splits graphs, and applications of interval methods to phylogenetics.
Algebraic Complexity Theory.
TL;DR: Algebraic complexity theory as mentioned in this paper is a project of lower bounds and optimality, which unifies two quite different traditions: mathematical logic and the theory of recursive functions, and numerical algebra.
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Symmetric Tensors and Symmetric Tensor Rank
TL;DR: In this paper, the authors studied various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors and showed that symmetric rank is equal in a number of cases and that they always exist in an algebraically closed field.
References
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Book
Introduction to Lie Algebras and Representation Theory
TL;DR: In this paper, Semisimple Lie Algebras and root systems are used for representation theory, isomorphism and conjugacy theorem, and existence theorem for representation.
Book
Basic Algebraic Geometry
TL;DR: The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds as discussed by the authors, and is suitable for beginning graduate students.
Journal ArticleDOI
Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics
TL;DR: 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.
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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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O(n 2.7799 ) Complexity for n*n Approximate Matrix Multiplication.
TL;DR: A new class of algorithms approximating the result with arbitrary precision is introduced’ APA-algorithms (Arbitrary-Precision-Approximating).