Spectral Regression : a Regression Framework for Efficient Regularized Subspace Learning

Spectral methods have recently emerged as a powerful tool for dimensionality reduction and manifold learning. These methods use information contained in the eigenvectors of a data affinity (i.e., item-item similarity) matrix to reveal the low dimensional structure in the high dimensional data. The most popular manifold learning algorithms include Locally Linear Embedding, ISOMAP, and Laplacian Eigenmap. However, these algorithms only provide the embedding results of training samples. There are many extensions of these approaches which try to solve the out-of-sample extension problem by seeking an embedding function in reproducing kernel Hilbert space. However, a disadvantage of all these approaches is that their computations usually involve eigen-decomposition of dense matrices which is expensive in both time and memory. In this thesis, we introduce a novel dimensionality reduction framework, called Spectral Regression (SR). SR casts the problem of learning an embedding function into a regression framework, which avoids eigendecomposition of dense matrices. Also, with the regression as a building block, different kinds of regularizers can be naturally incorporated into our framework which makes it more flexible. SR can be performed in supervised, unsupervised and semi-supervised situation. It can make efficient use of both labeled and unlabeled points to discover the intrinsic discriminant structure in the data. We have applied our algorithms to several real world applications, e.g. face analysis, document representation and content-based image retrieval.