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.

We introduce an incremental singular value decomposition (SVD) of incomplete data. The SVD is developed as data arrives, and can handle arbitrary missing/untrusted values, correlated uncertainty across rows or columns of the measurement matrix, and user priors. Since incomplete data does not uniquely specify an SVD, the procedure selects one having minimal rank. For a dense p q matrix of low rank r, the incremental method has time complexity O(pqr) and space complexity O((p + q)r)---better than highly optimized batch algorithms such as MATLAB 's svd(). In cases of missing data, it produces factorings of lower rank and residual than batch SVD algorithms applied to standard missing-data imputations. We show applications in computer vision and audio feature extraction. In computer vision, we use the incremental SVD to develop an efficient and unusually robust subspace-estimating flow-based tracker, and to handle occlusions/missing points in structure-from-motion factorizations.

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Summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure. In this paper we provide an overview of some existing techniques for discovering such embeddings. We then introduce a novel probabilistic interpretation of principal component analysis (PCA) that we term dual probabilistic PCA (DPPCA). The DPPCA model has the additional advantage that the linear mappings from the embedded space can easily be nonlinearised through Gaussian processes. We refer to this model as a Gaussian process latent variable model (GP-LVM). Through analysis of the GP-LVM objective function, we relate the model to popular spectral techniques such as kernel PCA and multidimensional scaling. We then review a practical algorithm for GP-LVMs in the context of large data sets and develop it to also handle discrete valued data and missing attributes. We demonstrate the model on a range of real-world and artificially generated data sets.

Many computer vision, signal processing and statistical problems can be posed as problems of learning low dimensional linear or multi-linear models. These models have been widely used for the representation of shape, appearance, motion, etc, in computer vision applications.

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Networks of linear units are the simplest kind of networks, where the basic questions related to learning, generalization, and self-organisation can sometimes be answered analytically. We survey most of the known results on linear networks, including: (1) back-propagation learning and the structure of the error function landscape; (2) the temporal evolution of generalization; (3) unsupervised learning algorithms and their properties. The connections to classical statistical ideas, such as principal component analysis (PCA), are emphasized as well as several simple but challenging open questions. A few new results are also spread across the paper, including an analysis of the effect of noise on back-propagation networks and a unified view of all unsupervised algorithms. Keywords--- linear networks, supervised and unsupervised learning, Hebbian learning, principal components, generalization, local minima, self-organisation I. Introduction This paper addresses the problems of supervise...

this paper are fast, accurate, and robust in the face of noise and degeneracies. The implementation tracks accurately for thousands of frames in low-res low-quality video, giving results that appear to compare favorably with the state-of-the-art. We are now studying more interesting camera models and the problem of integrating over uncertainty through time

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 higherorder 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.

AbstractÐPrincipal curves and surfaces are nonlinear generalizations of principal components and subspaces, respectively. They can provide insightful summary of high-dimensional data not typically attainable by classical linear methods. Solutions to several problems, such as proof of existence and convergence, faced by the original principal curve formulation have been proposed in the past few years. Nevertheless, these solutions are not generally extensible to principal surfaces, the mere computation of which presents a formidable obstacle. Consequently, relatively few studies of principal surfaces are available. Recently, we proposed the probabilistic principal surface (PPS) to address a number of issues associated with current principal surface algorithms. PPS uses a manifold oriented covariance noise model, based on the generative topographical mapping (GTM), which can be viewed as a parametric formulation of Kohonen's self-organizing map. Building on the PPS, we introduce a unified covariance model that implements PPS … 0< <1†, GTM … ˆ 1†, and the manifold-aligned GTM …>1† by varying the clamping parameter. Then, we comprehensively evaluate the empirical performance (reconstruction error) of PPS, GTM, and the manifold-aligned GTM on three popular benchmark data sets. It is shown in two different comparisons that the PPS outperforms the GTM under identical parameter settings. Convergence of the PPS is found to be identical to that of the GTM and the computational overhead incurred by the PPS decreases to 40 percent or less for more complex manifolds. These results show that the generalized PPS provides a flexible and effective way of obtaining principal surfaces. Index TermsÐPrincipal curve, principal surface, probabilistic, dimensionality reduction, nonlinear manifold, generative topographic mapping. 1