The Experts below are selected from a list of 1413 Experts worldwide ranked by ideXlab platform
Lieven De Lathauwer - One of the best experts on this subject based on the ideXlab platform.
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canonical polyadic decomposition of third order tensors reduction to generalized eigenvalue decomposition
SIAM Journal on Matrix Analysis and Applications, 2014Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical polyadic decomposition (CPD) of a third-order tensor is decomposition in a minimal number of Rank-1 tensors. We call an algorithm algebraic if it is guaranteed to find the decomposition when it is exact and if it relies only on standard linear algebra (essentially sets of linear equations and matrix factorizations). The known algebraic algorithms for the computation of the CPD are limited to cases where at least one of the factor matrices has Full Column Rank. In this paper we present an algebraic algorithm for the computation of the CPD in cases where none of the factor matrices has Full Column Rank. In particular, we show that if the famous Kruskal condition holds, then the CPD can be found algebraically.
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canonical polyadic decomposition of third order tensors reduction to generalized eigenvalue decomposition
arXiv: Spectral Theory, 2013Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical Polyadic Decomposition (CPD) of a third-order tensor is decomposition in a minimal number of Rank-$1$ tensors. We call an algorithm algebraic if it is guaranteed to find the decomposition when it is exact and if it only relies on standard linear algebra (essentially sets of linear equations and matrix factorizations). The known algebraic algorithms for the computation of the CPD are limited to cases where at least one of the factor matrices has Full Column Rank. In the paper we present an algebraic algorithm for the computation of the CPD in cases where none of the factor matrices has Full Column Rank. In particular, we show that if the famous Kruskal condition holds, then the CPD can be found algebraically.
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on the uniqueness of the canonical polyadic decomposition of third order tensors part ii uniqueness of the overall decomposition
SIAM Journal on Matrix Analysis and Applications, 2013Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical polyadic (also known as Candecomp/Parafac) decomposition (CPD) of a higher-order tensor is decomposition into a minimal number of Rank-$1$ tensors. In Part I, we gave an overview of existing results concerning uniqueness and presented new, relaxed, conditions that guarantee uniqueness of one factor matrix. In Part II we use these results for establishing overall CPD uniqueness in cases where none of the factor matrices has Full Column Rank. We obtain uniqueness conditions involving Khatri--Rao products of compound matrices and Kruskal-type conditions. We consider both deterministic and generic uniqueness. We also discuss uniqueness of INDSCAL and other constrained polyadic decompositions.
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on the uniqueness of the canonical polyadic decomposition of third order tensors part i basic results and uniqueness of one factor matrix
arXiv: Spectral Theory, 2013Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical Polyadic Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of Rank-1 tensors. We give an overview of existing results concerning uniqueness. We present new, relaxed, conditions that guarantee uniqueness of one factor matrix. These conditions involve Khatri-Rao products of compound matrices. We make links with existing results involving Ranks and k-Ranks of factor matrices. We give a shorter proof, based on properties of second compound matrices, of existing results concerning overall CPD uniqueness in the case where one factor matrix has Full Column Rank. We develop basic material involving $m$-th compound matrices that will be instrumental in Part II for establishing overall CPD uniqueness in cases where none of the factor matrices has Full Column Rank.
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on the uniqueness of the canonical polyadic decomposition of third order tensors part ii uniqueness of the overall decomposition
arXiv: Spectral Theory, 2013Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical Polyadic (also known as Candecomp/Parafac) Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of Rank-1 tensors. In Part I, we gave an overview of existing results concerning uniqueness and presented new, relaxed, conditions that guarantee uniqueness of one factor matrix. In Part II we use these results for establishing overall CPD uniqueness in cases where none of the factor matrices has Full Column Rank. We obtain uniqueness conditions involving Khatri-Rao products of compound matrices and Kruskal-type conditions.
Ryan J. Tibshirani - One of the best experts on this subject based on the ideXlab platform.
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Efficient implementations of the generalized lasso dual path algorithm. arXiv: 1405.3222
2014Co-Authors: Taylor B. Arnold, Ryan J. TibshiraniAbstract:We consider efficient implementations of the generalized lasso dual path algorithm of Tibshirani & Taylor (2011). We first describe a generic approach that covers any penalty matrix D and any (Full Column Rank) matrix X of predictor variables. We then describe fast implementations for the special cases of trend filtering problems, fused lasso problems, and sparse fused lasso problems, both with X = I and a general matrix X. These specialized implementations offer a considerable improvement over the generic implementation, both in terms of numerical stability and efficiency of the solution path computation. These algorithms are all available for use in the genlasso R package, which can be found in the CRAN repository
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Efficient implementations of the generalized lasso dual path algorithm
2013Co-Authors: Taylor B. Arnold, Ryan J. TibshiraniAbstract:We consider efficient implementations of the generalized lasso dual path algorithm of Tibshirani & Taylor (2011). We first describe a generic approach that covers any penalty matrix D and any (Full Column Rank) matrix X of predictor variables. We then describe fast implementations for the special cases of trend filtering problems, fused lasso problems, and sparse fused lasso problems, both with X = I and a general matrix X. These specialized implementations offer a considerable improvement over the generic implementation, both in terms of numerical stability and efficiency of the solution path computation. These algorithms are all available for use in the genlasso R package, which can be found in the CRAN repository
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degrees of freedom in lasso problems
Annals of Statistics, 2012Co-Authors: Ryan J. Tibshirani, Jonathan TaylorAbstract:We derive the degrees of freedom of the lasso fit, placing no assumptions on the predictor matrix $X$. Like the well-known result of Zou, Hastie and Tibshirani [Ann. Statist. 35 (2007) 2173-2192], which gives the degrees of freedom of the lasso fit when $X$ has Full Column Rank, we express our result in terms of the active set of a lasso solution. We extend this result to cover the degrees of freedom of the generalized lasso fit for an arbitrary predictor matrix $X$ (and an arbitrary penalty matrix $D$). Though our focus is degrees of freedom, we establish some intermediate results on the lasso and generalized lasso that may be interesting on their own.
Wei Hu - One of the best experts on this subject based on the ideXlab platform.
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linear convergence of the primal dual gradient method for convex concave saddle point problems without strong convexity
International Conference on Artificial Intelligence and Statistics, 2019Co-Authors: Simon S Du, Wei HuAbstract:We consider the convex-concave saddle point problem $\min_{x}\max_{y} f(x)+y^\top A x-g(y)$ where $f$ is smooth and convex and $g$ is smooth and strongly convex. We prove that if the coupling matrix $A$ has Full Column Rank, the vanilla primal-dual gradient method can achieve linear convergence even if $f$ is not strongly convex. Our result generalizes previous work which either requires $f$ and $g$ to be quadratic functions or requires proximal mappings for both $f$ and $g$. We adopt a novel analysis technique that in each iteration uses a "ghost" update as a reference, and show that the iterates in the primal-dual gradient method converge to this "ghost" sequence. Using the same technique we further give an analysis for the primal-dual stochastic variance reduced gradient method for convex-concave saddle point problems with a finite-sum structure.
De Lathauwer Lieven - One of the best experts on this subject based on the ideXlab platform.
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Canonical polyadic decomposition of third-order tensors: reduction to generalized eigenvalue decomposition
'Society for Industrial & Applied Mathematics (SIAM)', 2014Co-Authors: Domanov Ignat, De Lathauwer LievenAbstract:Canonical polyadic decomposition (CPD) of a third-order tensor is decomposition in a minimal number of Rank-1 tensors. We call an algorithm algebraic if it is guaranteed to find the decomposition when it is exact and if it relies only on standard linear algebra (essentially sets of linear equations and matrix factorizations). The known algebraic algorithms for the computation of the CPD are limited to cases where at least one of the factor matrices has Full Column Rank. In this paper we present an algebraic algorithm for the computation of the CPD in cases where none of the factor matrices has Full Column Rank. In particular, we show that if the famous Kruskal condition holds, then the CPD can be found algebraically. © 2014 Society for Industrial and Applied Mathematics.32 pages = 25 pages of paper itself + 7 pages of supplementary materialsstatus: publishe
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On the uniqueness of the canonical polyadic decomposition of third-order tensors - Part I : Basic results and uniqueness of one factor matrix
'Society for Industrial & Applied Mathematics (SIAM)', 2013Co-Authors: Domanov Ignat, De Lathauwer LievenAbstract:Canonical polyadic decomposition (CPD) of a higher-order tensor is decomposition into a minimal number of Rank-1 tensors. We give an overview of existing results concerning uniqueness. We present new, relaxed, conditions that guarantee uniqueness of one factor matrix. These conditions involve Khatri-Rao products of compound matrices. We make links with existing results involving Ranks and k-Ranks of factor matrices. We give a shorter proof, based on properties of second compound matrices, of existing results concerning overall CPD uniqueness in the case where one factor matrix has Full Column Rank. We develop basic material involving mth compound matrices that will be instrumental in Part II for establishing overall CPD uniqueness in cases where none of the factor matrices has Full Column Rank. Copyright © 2012 by SIAM.28 pagesstatus: publishe
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Canonical polyadic decomposition of third-order tensors: reduction to generalized eigenvalue decomposition
'Society for Industrial & Applied Mathematics (SIAM)', 2013Co-Authors: Domanov Ignat, De Lathauwer LievenAbstract:Canonical Polyadic Decomposition (CPD) of a third-order tensor is decomposition in a minimal number of Rank-$1$ tensors. We call an algorithm algebraic if it is guaranteed to find the decomposition when it is exact and if it only relies on standard linear algebra (essentially sets of linear equations and matrix factorizations). The known algebraic algorithms for the computation of the CPD are limited to cases where at least one of the factor matrices has Full Column Rank. In the paper we present an algebraic algorithm for the computation of the CPD in cases where none of the factor matrices has Full Column Rank. In particular, we show that if the famous Kruskal condition holds, then the CPD can be found algebraically.Comment: 32 pages = 25 pages of paper itself + 7 pages of supplementary material
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On the uniqueness of the canonical polyadic decomposition of third-order tensors - Part II : Uniqueness of the overall decomposition
'Society for Industrial & Applied Mathematics (SIAM)', 2013Co-Authors: Domanov Ignat, De Lathauwer LievenAbstract:Canonical Polyadic (also known as Candecomp/Parafac) Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of Rank-1 tensors. In Part I, we gave an overview of existing results concerning uniqueness and presented new, relaxed, conditions that guarantee uniqueness of one factor matrix. In Part II we use these results for establishing overall CPD uniqueness in cases where none of the factor matrices has Full Column Rank. We obtain uniqueness conditions involving Khatri-Rao products of compound matrices and Kruskal-type conditions.28 pagesstatus: publishe
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On the Uniqueness of the Canonical Polyadic Decomposition of third-order tensors --- Part I: Basic Results and Uniqueness of One Factor Matrix
'Society for Industrial & Applied Mathematics (SIAM)', 2013Co-Authors: Domanov Ignat, De Lathauwer LievenAbstract:Canonical Polyadic Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of Rank-1 tensors. We give an overview of existing results concerning uniqueness. We present new, relaxed, conditions that guarantee uniqueness of one factor matrix. These conditions involve Khatri-Rao products of compound matrices. We make links with existing results involving Ranks and k-Ranks of factor matrices. We give a shorter proof, based on properties of second compound matrices, of existing results concerning overall CPD uniqueness in the case where one factor matrix has Full Column Rank. We develop basic material involving $m$-th compound matrices that will be instrumental in Part II for establishing overall CPD uniqueness in cases where none of the factor matrices has Full Column Rank.Comment: 28 page
Ignat Domanov - One of the best experts on this subject based on the ideXlab platform.
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canonical polyadic decomposition of third order tensors reduction to generalized eigenvalue decomposition
SIAM Journal on Matrix Analysis and Applications, 2014Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical polyadic decomposition (CPD) of a third-order tensor is decomposition in a minimal number of Rank-1 tensors. We call an algorithm algebraic if it is guaranteed to find the decomposition when it is exact and if it relies only on standard linear algebra (essentially sets of linear equations and matrix factorizations). The known algebraic algorithms for the computation of the CPD are limited to cases where at least one of the factor matrices has Full Column Rank. In this paper we present an algebraic algorithm for the computation of the CPD in cases where none of the factor matrices has Full Column Rank. In particular, we show that if the famous Kruskal condition holds, then the CPD can be found algebraically.
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canonical polyadic decomposition of third order tensors reduction to generalized eigenvalue decomposition
arXiv: Spectral Theory, 2013Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical Polyadic Decomposition (CPD) of a third-order tensor is decomposition in a minimal number of Rank-$1$ tensors. We call an algorithm algebraic if it is guaranteed to find the decomposition when it is exact and if it only relies on standard linear algebra (essentially sets of linear equations and matrix factorizations). The known algebraic algorithms for the computation of the CPD are limited to cases where at least one of the factor matrices has Full Column Rank. In the paper we present an algebraic algorithm for the computation of the CPD in cases where none of the factor matrices has Full Column Rank. In particular, we show that if the famous Kruskal condition holds, then the CPD can be found algebraically.
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on the uniqueness of the canonical polyadic decomposition of third order tensors part ii uniqueness of the overall decomposition
SIAM Journal on Matrix Analysis and Applications, 2013Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical polyadic (also known as Candecomp/Parafac) decomposition (CPD) of a higher-order tensor is decomposition into a minimal number of Rank-$1$ tensors. In Part I, we gave an overview of existing results concerning uniqueness and presented new, relaxed, conditions that guarantee uniqueness of one factor matrix. In Part II we use these results for establishing overall CPD uniqueness in cases where none of the factor matrices has Full Column Rank. We obtain uniqueness conditions involving Khatri--Rao products of compound matrices and Kruskal-type conditions. We consider both deterministic and generic uniqueness. We also discuss uniqueness of INDSCAL and other constrained polyadic decompositions.
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on the uniqueness of the canonical polyadic decomposition of third order tensors part i basic results and uniqueness of one factor matrix
arXiv: Spectral Theory, 2013Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical Polyadic Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of Rank-1 tensors. We give an overview of existing results concerning uniqueness. We present new, relaxed, conditions that guarantee uniqueness of one factor matrix. These conditions involve Khatri-Rao products of compound matrices. We make links with existing results involving Ranks and k-Ranks of factor matrices. We give a shorter proof, based on properties of second compound matrices, of existing results concerning overall CPD uniqueness in the case where one factor matrix has Full Column Rank. We develop basic material involving $m$-th compound matrices that will be instrumental in Part II for establishing overall CPD uniqueness in cases where none of the factor matrices has Full Column Rank.
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on the uniqueness of the canonical polyadic decomposition of third order tensors part ii uniqueness of the overall decomposition
arXiv: Spectral Theory, 2013Co-Authors: Ignat Domanov, Lieven De LathauwerAbstract:Canonical Polyadic (also known as Candecomp/Parafac) Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of Rank-1 tensors. In Part I, we gave an overview of existing results concerning uniqueness and presented new, relaxed, conditions that guarantee uniqueness of one factor matrix. In Part II we use these results for establishing overall CPD uniqueness in cases where none of the factor matrices has Full Column Rank. We obtain uniqueness conditions involving Khatri-Rao products of compound matrices and Kruskal-type conditions.
