Spectral Regression for Dimensionality Reduction∗

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 low dimensional structure in 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 paper, we propose a novel dimensionality reduction method, called Spectral Regression (SR). SR casts the problem of learning an embedding function into a regression framework, which avoids eigen-decomposition of dense matrices. Also, with the regression based framework, different kinds of regularizers can be naturally incorporated into our algorithm which makes it more flexible. SR can be performed in supervised, unsupervised and semisupervised situation. It can make efficient use of both labeled and unlabeled points to discover the intrinsic discriminant structure in the data. Experimental results on classification and semi-supervised classification demonstrate the effectiveness and efficiency of our algorithm.