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Tensor Decompositions and Applications
 SIAM REVIEW
, 2009
"... This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal proce ..."
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Cited by 723 (18 self)
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This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo
sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rankone tensors, and the Tucker decomposition is a higherorder form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The Nway Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.
Unsupervised multiway data analysis: A literature survey
 IEEE Transactions on Knowledge and Data Engineering
, 2008
"... Multiway data analysis captures multilinear structures in higherorder datasets, where data have more than two modes. Standard twoway methods commonly applied on matrices often fail to find the underlying structures in multiway arrays. With increasing number of application areas, multiway data anal ..."
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Cited by 82 (10 self)
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Multiway data analysis captures multilinear structures in higherorder datasets, where data have more than two modes. Standard twoway methods commonly applied on matrices often fail to find the underlying structures in multiway arrays. With increasing number of application areas, multiway data analysis has become popular as an exploratory analysis tool. We provide a review of significant contributions in literature on multiway models, algorithms as well as their applications in diverse disciplines including chemometrics, neuroscience, computer vision, and social network analysis. 1.
RankR Approximation of Tensors Using ImageasMatrix Representation
 in Proc. CVPR’05
, 2005
"... We present a novel multilinear algebra based approach for reduced dimensionality representation of image ensembles. We treat an image as a matrix, instead of a vector as in traditional dimensionality reduction techniques like PCA, and higherdimensional data as a tensor. This helps exploit spatiote ..."
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Cited by 22 (1 self)
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We present a novel multilinear algebra based approach for reduced dimensionality representation of image ensembles. We treat an image as a matrix, instead of a vector as in traditional dimensionality reduction techniques like PCA, and higherdimensional data as a tensor. This helps exploit spatiotemporal redundancies with less information loss than imageasvector methods. The challenges lie in the computational and memory requirements for large ensembles. Currently, there exists a rankR approximation algorithm which, although applicable to any number of dimensions, is efficient for only lowrank approximations. For larger dimensionality reductions, the memory and time costs of this algorithm become prohibitive. We propose a novel algorithm for rankR approximations of thirdorder tensors, which is e#cient for arbitrary R but for the important special case of 2D image ensembles, e.g. video. Both of these algorithms reduce redundancies present in all dimensions. RankR tensor approximation yields the most compact data representation among all known imageasmatrix methods. We evaluated the performance of our algorithm vs. other approaches on a number of datasets with the following two main results. First, for a fixed compression ratio, the proposed algorithm yields the best representation of image ensembles visually as well as in the least squares sense. Second, proposed representation gives the best performance for object classification.
Minimal subspaces in tensor representations
, 2011
"... In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximat ..."
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Cited by 17 (6 self)
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In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means a tensor given in a typical representation format (Tucker, hierarchical or tensor train). We show that this result holds in a tensor Banach space with a norm stronger that the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples by using topological tensor products of standard Sobolev spaces are given.
A Tensor Approximation Approach to Dimensionality Reduction
"... Abstract Dimensionality reduction has recently been extensively studied for computer vision applications. We present a novel multilinear algebra based approach to reduced dimensionality representation of multidimensional data, such as image ensembles, video sequences and volume data. Before reducing ..."
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Cited by 14 (0 self)
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Abstract Dimensionality reduction has recently been extensively studied for computer vision applications. We present a novel multilinear algebra based approach to reduced dimensionality representation of multidimensional data, such as image ensembles, video sequences and volume data. Before reducing the dimensionality we do not convert it into a vector as is done by traditional dimensionality reduction techniques like PCA. Our approach works directly on the multidimensional form of the data (matrix in 2D and tensor in higher dimensions) to yield what we call a DatumasIs representation. This helps exploit spatiotemporal redundancies with less information loss than imageasvector methods. An efficient rankR tensor approximation algorithm is presented to approximate higherorder tensors. We show that rankR tensor approximation using DatumasIs representation generalizes many existing approaches that use imageasmatrix representation, such as generalized low rank approximation of matrices (GLRAM) (Ye, Y. in Mach. Learn. 61:167–191, 2005), rankone decomposition of matrices (RODM) (Shashua, A., Levin, A. in CVPR’01:
Tensor Principal Component Analysis via Convex Optimization
, 2012
"... This paper is concerned with the computation of the principal components for a general tensor, known as the tensor principal component analysis (PCA) problem. We show that the general tensor PCA problem is reducible to its special case where the tensor in question is supersymmetric with an even degr ..."
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This paper is concerned with the computation of the principal components for a general tensor, known as the tensor principal component analysis (PCA) problem. We show that the general tensor PCA problem is reducible to its special case where the tensor in question is supersymmetric with an even degree. In that case, the tensor can be embedded into a symmetric matrix. We prove that if the tensor is rankone, then the embedded matrix must be rankone too, and vice versa. The tensor PCA problem can thus be solved by means of matrix optimization under a rankone constraint, for which we propose two solution methods: (1) imposing a nuclear norm penalty in the objective to enforce a lowrank solution; (2) relaxing the rankone constraint by Semidefinite Programming. Interestingly, our experiments show that both methods yield a rankone solution with high probability, thereby solving the original tensor PCA problem to optimality with high probability. To further cope with the size of the resulting convex optimization models, we propose to use the alternating direction method of multipliers, which reduces significantly the computational efforts. Various extensions of the model are considered as well.
Fast Learning for Customizable Head Pose Recognition in Robotic Wheelchair Control
"... In the PLAYBOT project, we aim at assisting disabled children at play. To this end, we are developing a semi autonomous robotic wheelchair. It is equipped with several visual sensors and a robotic manipulator and thus conveniently enhances the innate capabilities of a disabled child. In addition to ..."
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In the PLAYBOT project, we aim at assisting disabled children at play. To this end, we are developing a semi autonomous robotic wheelchair. It is equipped with several visual sensors and a robotic manipulator and thus conveniently enhances the innate capabilities of a disabled child. In addition to a touch screen, the child may control the wheelchair using simple head movements. As control based on head posture requires reliable face detection and head pose recognition, we are in need of a robust technique that may effortlessly be tailored to individual users. In this paper, we present a multilinear classification algorithm for fast and reliable face detection. It trains within seconds and thus can easily be customized to the home environment of a disabled child. Subsequent head pose recognition is done using support vector machines. Experimental results show that this two stage approach to head posebased robotic wheelchair control performs fast and very robust. 1
Fast, Illumination Insensitive Face Detection Based on Multilinear Techniques and Curvature Features
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Pattern Analysis and Applications manuscript No. (will be inserted by the editor) Dimensionality Reduction and Topographic Mapping of Binary
"... Abstract In this paper, a decomposition method for binary tensors, generalized multilinear model for principal component analysis (GMLPCA) is proposed. To the best of our knowledge at present there is no other principled systematic framework for decomposition or topographic mapping of binary tensor ..."
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Abstract In this paper, a decomposition method for binary tensors, generalized multilinear model for principal component analysis (GMLPCA) is proposed. To the best of our knowledge at present there is no other principled systematic framework for decomposition or topographic mapping of binary tensors. In the model formulation, we constrain the natural parameters of the Bernoulli distributions for each tensor element to lie in a subspace spanned by a reduced set of basis (principal) tensors. We evaluate and compare the proposed GMLPCA technique with existing realvalued tensor decomposition methods in two scenarios: (1) in a series of controlled experiments involving synthetic data; (2) on a real world biological dataset of DNA subsequences from different functional regions, with sequences represented by binary tensors. The experiments suggest that the GMLPCA model is better suited for modeling bi
Analysis and Synthesis of Behavioural Specific Facial Motion
, 2007
"... This thesis presents work concerning the analysis of behaviour specific facial motion and its automatic synthesis. Psychology research has shown that facial motion provides important cues to the human visual system for recognition of emotion, identity and gender. Similarly, in Computer Vision facial ..."
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This thesis presents work concerning the analysis of behaviour specific facial motion and its automatic synthesis. Psychology research has shown that facial motion provides important cues to the human visual system for recognition of emotion, identity and gender. Similarly, in Computer Vision facial motion information has been used in face and facial expression recognition. However, the fact that facial motion is behaviour specific has not yet been exploited in facial animation systems. Two parametric modelling techniques have been evaluated, MultiVariate AutoRegressive (VAR) temporal modelling and a tensor framework for modelling facial motion dynamics. Both modelling techniques adopt a ‘black box ’ approach to facial motion modelling, where the emphasis of modelling is on motion information and not on textural information. Of these methods, VAR modelling is found to be more suitable for motion synthesis. Nevertheless, it is found that the tensor framework is more suited than VAR modelling as a potential tool for facial motion analysis. It is found that the VAR modelling technique encapsulated the temporal and motion dynamics