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.

Path-following and semismooth Newton methods for the variational inequality arising from two membranes problem.

A semismooth Newton method, based on variational inequalities and generalized derivative, is designed and analysed for unilateral contact problem between two membranes. The problem is first formulated as a corresponding regularized problem with a nonlinear function, which can be solved by the semismooth Newton method. We prove the convergence of the method in the function space.