Speaker: Alex Vasilescu (UCLA Computer Science)

Title: Multilinear (Tensor) Framework for Computer Vision, Computer Graphics, and Machine Learning

Abstract:
Most multivariate observable data such as images, videos, human motion capture data, and speech are the result of the multiple causal factors (hidden variables) that are not directly measurable, but which are of interest in data analysis. These causal factors can be fundamental physical or biological processes that cause patterns of variation in the observational data, which comprise a set of measurements or response variables that are affected by the causal factors. For example, natural images which comprise a set of pixels are the consequence of multiple causal factors related to scene structure, illumination, and imaging. The key to solving statistical data analysis problems of the sort that arise in the domains of computer graphics and vision is to find a suitable multivariate data representation that facilitates the analysis, visualization, compression, approximation, recognition and/or interpretation of the observed data. This is often done by applying suitable transformations to the observational data space. Multilinear algebra offers a potent mathematical framework for extracting and explicitly representing the multifactor structure of image datasets.

Representations founded on linear transformations of the original observed data have traditionally been preferred due to their conceptual and computational simplicity based on linear algebra, the algebra of vectors and matrices. Linear transformations, such as PCA and independent components analysis (ICA) are limited in their ability to facilitate data analysis, however, since they are well suited for modeling observed data that results only from single-factor linear variation or from the linear combination of multiple sources.

Multilinear transformations, which subsume the common linear transformations as special cases, lead to generative models that explicitly capture how the observed data are influenced by multiple underlying causal factors. We will define data tensors, discuss tensor models (CP,Multilinear PCA,Multilinear ICA, Multilinear KPCA, Multilinear KICA) for data analysis, compression and rendering. Recognition is achieved with a novel Multilinear Projection Operator.

I will demonstrate the potency of our novel statistical learning approach in the context of facial image biometrics, where the relevant factors include different facial geometries, expressions, lighting conditions, and viewpoints. When applied to the difficult problem of automated face recognition, our multilinear representations, called TensorFaces (M-mode PCA) and Independent TensorFaces (MICA), yields significantly improved recognition rates relative to the standard PCA and ICA approaches.