Patch Alignment for Dimensionality Reduction

Spectral analysis-based dimensionality reduction algorithms are important and have been popularly applied in data mining and computer vision applications. To date many algorithms have been developed, e. g., principal component analysis, locally linear embedding, Laplacian eigenmaps, and local tangent space alignment. All of these algorithms have been designed intuitively and pragmatically, i.e., on the basis of the experience and knowledge of experts for their own purposes. Therefore, it will be more informative to provide a systematic framework for understanding the common properties and intrinsic difference in different algorithms. In this paper, we propose such a framework, named "patch alignment," which consists of two stages: part optimization and whole alignment. The framework reveals that 1) algorithms are intrinsically different in the patch optimization stage and 2) all algorithms share an almost identical whole alignment stage. As an application of this framework, we develop a new dimensionality reduction algorithm, termed Discriminative Locality Alignment (DLA), by imposing discriminative information in the part optimization stage. DLA can 1) attack the distribution nonlinearity of measurements; 2) preserve the discriminative ability; and 3) avoid the small-sample-size problem. Thorough empirical studies demonstrate the effectiveness of DLA compared with representative dimensionality reduction algorithms.