Dimensionality reduction is one of the important preprocessing steps in practical pattern recognition. SEmi-supervised Local Fisher discriminant analysis (SELF)--which is a semi-supervised and local extension of Fisher discriminant analysis--was shown to work excellently in experiments. However, when data dimensionality is very high, a naive use of SELF is prohibitive due to high computational costs and large memory requirement. In this paper, we introduce computational tricks for making SELF applicable to large-scale problems.

Using the recursive generalized eigendecomposition method, we develop a recursive form solution to the data least squares (DLS) problem in which the error is assumed to lie in the data matrix only. We apply it to a linear channel equalizer. Simulations shows that the DLS-based equalizer outperforms the ordinary least squares-based one in a channel equalization problem.

In this letter we present a new way for computing generalized eigenvalue problems in engineering applications. To transform a generalized eigenvalue problem into an associated problem for solving nonlinear dynamic equations by using optimization techniques, we can determine all eigenvalues and their eigenvectors for general complex matrices. Numerical examples are given to verify the formula of dynamic equations.

A stochastic system represented by an input-output model can be described by mainly two different types of state space representation. Corresponding to state space representations innovation models are examined. The relationship between both representations is made clear systematically. An easy transformation between them is presented. Zeros of innovation models are the same as those of an ARMA model which is stochastically equivalent to innovation models, and related to stable eigenvalues of generalized eigenvalue problem of matrix Riccati equation.