Abstract

Multifactor analysis plays an important role in data analysis since most real-world datasets usually exist with a combination of numerous factors. These factors are usually not independent but interdependent together. Thus, it is a mistake if a method only considers one aspect of the input data while ignoring the others. Although widely used, Multilinear PCA (MPCA), one of the leading multilinear analysis methods, still suffers from three major drawbacks. Firstly, it is very sensitive to outliers and noise and unable to cope with missing values. Secondly, since MPCA deals with huge multidimensional datasets, it is usually computationally expensive. Finally, it loses original local geometry structures due to the averaging process. This thesis sheds new light on the tensor decomposition problem via the ideas of fast low-rank approximation in random projection and tensor completion in compressed sensing. We propose a novel approach called Compressed Submanifold Multifactor Analysis (CSMA) to solve the three problems mentioned above. Our approach is able to deal with the problem of missing values and outliers via our proposed novel sparse Higher-order Singular Value Decomposition approach, named HOSVD-L1 decomposition. The Random Projection method is used to obtain the fast low-rank approximation of a given multifactor dataset.
In addition, our method can preserve geometry of the original data.

In the second part of this thesis, we present a novel pattern classification approach named Sparse Class-dependent Feature Analysis (SCFA), to connect the advantages of sparse representation in an overcomplete dictionary, with a powerful nonlinear classifier. The classifier is based on the estimation of class-specific optimal filters, by solving an L1-norm optimization problem using the Alternating Direction Method of Multipliers. Our method as well as its Reproducing Kernel Hilbert Space (RKHS) version is tolerant to the presence of noise and other variations in an image. Our proposed methods achieve very high classification accuracies in face recognition on two challenging face databases, i.e. the CMU Pose, Illumination and Expression (PIE) database and the Extended YALE-B that exhibit pose and illumination variations; and the AR database that has occluded images. In addition, they also exhibit robustness on other evaluation modalities, such as object classification on the Caltech101 database. Our method outperforms state-of-the-art methods on all these databases and hence they show their applicability to general computer vision and pattern recognition problems.