Multiple Manifold Data Modeling And Its Applications

Many common data sets can be modeled by a mixture of simple geometric objects, e.g.,manifolds. A well known example of such data is the NIST images of hand-written digitsand face images. The effective modeling of such data together with its careful analysis isa challenging mathematical problem. Earlier work at the beginning of the current centuryrevealed effective methods for modeling data by a single manifold. Later work modeled databy an arrangement of affine subspaces. The generalization of these works to multi-manifolddata modeling is currently being developed. Unlike single manifold learning, multi-manifoldlearning aims at partitioning a set of unorganized data into several different clusters each ofwhich corresponds to a separate, simple low dimensional manifold. Given a set of data frommulti-manifold, multi-manifold data modeling is: to analyze the number of sub-manifoldsand their respective dimensions, data division (data belonging to different low-dimensionalmanifolds) and the low-dimensional embedding.Multi-manifold data modeling can reveal the potential spatial distribution of the data,to address the complex multi-manifold structure of data, this paper proposed several multi-manifold data modeling methods for head pose estimation and face recognition problems.the main contributions of this dissertation are summarized as follows:1. Supervised Manifold Embedding (SME) algorithm for head pose estimation: Manydimensionality reduction methods can be used to find the potential pose manifold em-bedding, but the difficulties is how to find an efficient embedding, retain the posefeatures, and ignore the pose-independent features, such as identity information. Wepropose a supervised manifold embedding algorithm to learning features of head posewhile removing the impact of identity and other information. Finally, experiment'sresults show that: the embeddings in low-dimensional space have good intra-classcompactness and inter-class separability.2. A multi-manifold data modeling method based on affinity propagation: In order tolearn the identity-independent pose features, we model multi-manifold for head posedata, the reasons are that: The appearance variations caused by identity lead to transla-tion, rotation and warp changes of the subject's embeddings, and it is difficult to make sure that the pose data lie on a single continuous manifold for the individual variations.Thus, we proposed a Multi-Manifold Embedding (MME) algorithm based on affinitypropagation, we consider the pose data space as multiple manifolds in which eachmanifold characterizes the underlying subspace of subjects with similar appearance.3. A multi-manifold data modeling method based on projected clustering: When we clus-ter the subjects by affinity propagation, the large dimension of the subjects affects theperformance of clustering. Thus, we use the idea of projected clustering, and pro-posed the projected clustering based multi-manifold embedding for head pose estima-tion, including the three stages (Constructing affinity simplex, manifold embeddingand K-manifold clustering). Experiment's results show that the method can improvethe intra-class compactness and inter-class separability.4. A multi-manifold data modeling method based on group sparsity and non-negativematrix factorization: When the L1/L2 regularizer is used on the column of coefficientmatrix H, multiple manifold (in W) can be obtained. Thus, we proposed the GroupSparse Non-negative Matrix Factorization (GSNMF) algorithm. Via the group sparsityconstraint imposed on the column vectors of the coefficient matrix, we obtain multiplemanifolds each of them belongs to a particular class. For a test image, we representit as a linear combination of the learned multiple manifolds, and then the representa-tion is naturally group sparse: only the coefficients corresponding to the same classare nonzero. We apply GSNMF algorithm to face recognition, and obtain good facerecognition accuracy.