This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component 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 N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

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Tensor Decompositions and Applications

Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA) – a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is defined everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets.

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Locality Preserving Projections

https://www.cs.ucla.edu/~dt/papers/mia96/mia96.pdf

Deformable models in medical image analysis: a survey

A morphable model for the synthesis of 3D faces

We introduce the SCAPE method (Shape Completion and Animation for PEople)---a data-driven method for building a human shape model that spans variation in both subject shape and pose. The method is based on a representation that incorporates both articulated and non-rigid deformations. We learn a pose deformation model that derives the non-rigid surface deformation as a function of the pose of the articulated skeleton. We also learn a separate model of variation based on body shape. Our two models can be combined to produce 3D surface models with realistic muscle deformation for different people in different poses, when neither appear in the training set. We show how the model can be used for shape completion --- generating a complete surface mesh given a limited set of markers specifying the target shape. We present applications of shape completion to partial view completion and motion capture animation. In particular, our method is capable of constructing a high-quality animated surface model of a moving person, with realistic muscle deformation, using just a single static scan and a marker motion capture sequence of the person.

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SCAPE: shape completion and animation of people

Natural images are the composite consequence of multiple factors related to scene structure, illumination, and imaging. Multilinear algebra, the algebra of higher-order tensors, offers a potent mathematical framework for analyzing the multifactor structure of image ensembles and for addressing the difficult problem of disentangling the constituent factors or modes. Our multilinear modeling technique employs a tensor extension of the conventional matrix singular value decomposition (SVD), known as the N-mode SVD. As a concrete example, we consider the multilinear analysis of ensembles of facial images that combine several modes, including different facial geometries (people), expressions, head poses, and lighting conditions. Our resulting "TensorFaces" representation has several advantages over conventional eigenfaces. More generally, multilinear analysis shows promise as a unifying framework for a variety of computer vision problems.

/pdf/multilinear-analysis-of-image-ensembles-tensorfaces-1a5mqid75n.pdf

Multilinear Analysis of Image Ensembles: TensorFaces

Multilinear algebra, the algebra of higher-order tensors, offers a potent mathematical framework for analyzing ensembles of images resulting from the interaction of any number of underlying factors. We present a dimensionality reduction algorithm that enables subspace analysis within the multilinear framework. This N-mode orthogonal iteration algorithm is based on a tensor decomposition known as the N-mode SVD, the natural extension to tensors of the conventional matrix singular value decomposition (SVD). We demonstrate the power of multilinear subspace analysis in the context of facial image ensembles, where the relevant factors include different faces, expressions, viewpoints, and illuminations. In prior work we showed that our multilinear representation, called TensorFaces, yields superior facial recognition rates relative to standard, linear (PCA/eigenfaces) approaches. We demonstrate factor-specific dimensionality reduction of facial image ensembles. For example, we can suppress illumination effects (shadows, highlights) while preserving detailed facial features, yielding a low perceptual error.

/pdf/multilinear-subspace-analysis-of-image-ensembles-36hhmralue.pdf

Multilinear subspace analysis of image ensembles

Natural images are the composite consequence of multiple factors related to scene structure, illumination, and imaging. For facial images, the factors include different facial geometries, expressions, head poses, and lighting conditions. We apply, multilinear algebra, the algebra of higher-order tensors, to obtain a parsimonious representation of facial image ensembles which separates these factors. Our representation, called TensorFaces, yields improved facial recognition rates relative to standard eigenfaces.

/pdf/multilinear-image-analysis-for-facial-recognition-3hvqozdaqv.pdf

Multilinear image analysis for facial recognition

An approach to visual sampling and reconstruction motivated by concepts from numerical grid generation is presented. Adaptive meshes that can nonuniformly sample and reconstruct intensity and range data are presented. These meshes are dynamic models which are assembled by interconnecting nodal masses with adjustable springs. Acting as mobile sampling sites, the nodes observe properties of the input data, such as intensities, depths, gradients, and curvatures. Based on these nodal observations, the springs automatically adjust their stiffnesses so as to distribute the available degrees of freedom of the reconstructed model in accordance with the local complexity of the input data. The adaptive mesh algorithm runs at interactive rates with continuous 3-D display on a graphics workstation It is applied to the adaptive sampling and reconstruction of images and surfaces. >

/pdf/sampling-and-reconstruction-with-adaptive-meshes-5apkmcv8cm.pdf

Sampling and reconstruction with adaptive meshes

Independent components analysis (ICA) maximizes the statistical independence of the representational components of a training image ensemble, but it cannot distinguish between the different factors, or modes, inherent to image formation, including scene structure, illumination, and imaging. We introduce a nonlinear, multifactor model that generalizes ICA. Our multilinear ICA (MICA) model of image ensembles learns the statistically independent components of multiple factors. Whereas ICA employs linear (matrix) algebra, MICA exploits multilinear (tensor) algebra. We furthermore introduce a multilinear projection algorithm which projects an unlabeled test image into the N constituent mode spaces to simultaneously infer its mode labels. In the context of facial image ensembles, where the mode labels are person, viewpoint, illumination, expression, etc., we demonstrate that the statistical regularities learned by MICA capture information that, in conjunction with our multilinear projection algorithm, improves automatic face recognition.

/pdf/multilinear-independent-components-analysis-22l18i9cs7.pdf

M.A.O. Vasilescu

Papers

Multilinear Analysis of Image Ensembles: TensorFaces

Multilinear subspace analysis of image ensembles

Multilinear image analysis for facial recognition

Sampling and reconstruction with adaptive meshes

Multilinear independent components analysis