A General Matrix Function Dimensionality Reduction Framework and Extension for Manifold Learning

Many dimensionality reduction methods in the manifold learning field have the so-called small-sample-size (SSS) problem. Starting from solving the SSS problem, we first summarize the existing dimensionality reduction methods and construct a unified criterion function of these methods. Then, combining the unified criterion with the matrix function, we propose a general matrix function dimensionality reduction framework. This framework is configurable, that is, one can select suitable functions to construct such a matrix transformation framework, and then a series of new dimensionality reduction methods can be derived from this framework. In this article, we discuss how to choose suitable functions from two aspects: 1) solving the SSS problem and 2) improving pattern classification ability. As an extension, with the inverse hyperbolic tangent function and linear function, we propose a new matrix function dimensionality reduction framework. Compared with the existing methods to solve the SSS problem, these new methods can obtain better pattern classification ability and have less computational complexity. The experimental results on handwritten digit, letters databases, and two face databases show the superiority of the new methods.