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11
Decoupling multivariate polynomials using firstorder information and tensor decompositions
 SIAM J. Matrix Anal. Appl
, 2015
"... Abstract. We present a method to decompose a set of multivariate real polynomials into linear combinations of univariate polynomials in linear forms of the input variables. The method proceeds by collecting the firstorder information of the polynomials in a set of sampling points, which is captured ..."
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Abstract. We present a method to decompose a set of multivariate real polynomials into linear combinations of univariate polynomials in linear forms of the input variables. The method proceeds by collecting the firstorder information of the polynomials in a set of sampling points, which is captured by the Jacobian matrix evaluated at the sampling points. The canonical polyadic decomposition of the threeway tensor of Jacobian matrices directly returns the unknown linear relations as well as the necessary information to reconstruct the univariate polynomials. The conditions under which this decoupling procedure works are discussed, and the method is illustrated on several numerical examples.
Tensor Decomposition via Joint Matrix Schur Decomposition Nicolò Colombo
"... Abstract We describe an approach to tensor decomposition that involves extracting a set of observable matrices from the tensor and applying an approximate joint Schur decomposition on those matrices, and we establish the corresponding firstorder perturbation bounds. We develop a novel iterative Gau ..."
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Abstract We describe an approach to tensor decomposition that involves extracting a set of observable matrices from the tensor and applying an approximate joint Schur decomposition on those matrices, and we establish the corresponding firstorder perturbation bounds. We develop a novel iterative GaussNewton algorithm for joint matrix Schur decomposition, which minimizes a nonconvex objective over the manifold of orthogonal matrices, and which is guaranteed to converge to a global optimum under certain conditions. We empirically demonstrate that our algorithm is faster and at least as accurate and robust than stateoftheart algorithms for this problem.
(article begins on next page) Breaking the Curse of Dimensionality using Decompositions of Incomplete Tensors
"... scientific computing in big data analysis ..."
Dictionary Learning and Sparse Coding for Thirdorder Supersymmetric Tensors
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Tensors and latent variable models
"... Abstract. In this paper we discuss existing and new connections between latent variable models from machine learning and tensors (multiway arrays) from multilinear algebra. A few ideas have been developed independently in the two communities. However, there are still many useful but unexplored lin ..."
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Abstract. In this paper we discuss existing and new connections between latent variable models from machine learning and tensors (multiway arrays) from multilinear algebra. A few ideas have been developed independently in the two communities. However, there are still many useful but unexplored links and ideas that could be borrowed from one of the communities and used in the other. We will start our discussion from simple concepts such as independent variables and rank1 matrices and gradually increase the difficulty. The final goal is to connect discrete latent tree graphical models to state of the art tensor decompositions in order to find tractable representations of probability tables of many variables.
1Uniqueness of Nonnegative Tensor Approximations
"... We show that a best nonnegative rankr approximation of a nonnegative tensor is almost always unique and that nonnegative tensors with nonunique best nonnegative rankr approximation form a semialgebraic set contained in an algebraic hypersurface. We then establish a singular vector variant of the P ..."
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We show that a best nonnegative rankr approximation of a nonnegative tensor is almost always unique and that nonnegative tensors with nonunique best nonnegative rankr approximation form a semialgebraic set contained in an algebraic hypersurface. We then establish a singular vector variant of the Perron–Frobenius Theorem for positive tensors and apply it to show that a best nonnegative rankr approximation of a positive tensor can almost never be obtained by deflation. We show the subset of real tensors which admit more than one best rank one approximations is a hypersurface, and give a polynomial equation to ensure a tensor without satisfying this equation to have a unique best rank one approximation. I.
PERFORMANCE ESTIMATION FOR TENSOR CP DECOMPOSITION WITH STRUCTURED FACTORS
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
ERROR ANALYSIS OF LOWRANK THREEWAY TENSOR FACTORIZATION APPROACH TO BLIND SOURCE SEPARATION
"... In tensor factorization approach to blind separation of multidimensional sources two formulas for calculating the source tensor have emerged. In practice, it is observed that these two schemes exhibit different levels of robustness against perturbations of the factors involved in the tensor model. M ..."
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In tensor factorization approach to blind separation of multidimensional sources two formulas for calculating the source tensor have emerged. In practice, it is observed that these two schemes exhibit different levels of robustness against perturbations of the factors involved in the tensor model. Motivated by both practical reasons and the will to better figure this out, we present error analyses in source tensor estimation performed by lowrank factorization of threeway tensors. To that aim, computer simulations as well as the analytical calculation of the theoretical error are carried out. The conclusions drawn from these numerical and analytical error analyses are supported by the results obtained thanks to the tensor factorization based blind decomposition of an experimental multispectral image of a skin tumor.
Joint Source Estimation and Localization
"... The estimation of directions of arrival is formulated as the decomposition of a 3way array into a sum of rankone terms, which is possible when the receive array enjoys some geometrical structure. The main advantage is that this decomposition is essentially unique under mild assumptions, if compute ..."
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The estimation of directions of arrival is formulated as the decomposition of a 3way array into a sum of rankone terms, which is possible when the receive array enjoys some geometrical structure. The main advantage is that this decomposition is essentially unique under mild assumptions, if computed exactly. The drawback is that a lowrank approximation does not always exist. Therefore, a constraint is first introduced that ensures the existence of the latter best approximate. Then CramérRao bounds are derived for localization parameters and source signals, assuming the others are nuisance parameters; some inaccuracies found in the literature are pointed out. Performances are eventually compared with reference algorithms such as ESPRIT, in the presence of additive Gaussian noise, with possibly non circular distribution. Index Terms multiway array, tensor decomposition, source localization, antenna array processing, lowrank approximation, complex CramerRao bounds I.
NONNEGATIVE TENSOR APPROXIMATIONS
"... Abstract. Necessary conditions are derived for a rankr tensor to be a best rankr approximation of a given tensor. It is shown that a positive tensor with rank> 1 has a unique rank one approximation, and that a non negative tensor generally has a unique lowrank nonnegative approximate. We discu ..."
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Abstract. Necessary conditions are derived for a rankr tensor to be a best rankr approximation of a given tensor. It is shown that a positive tensor with rank> 1 has a unique rank one approximation, and that a non negative tensor generally has a unique lowrank nonnegative approximate. We discuss the notion of rsingular values and their corresponding rsingular vector tuples, which is closely related to best rankr approximations. We then show that a generic tensor has a finite number of rsingular vector tuples for some r. 1.