Kernel-based learning algorithms have been successfully applied in various problem domains, given appropriate kernel functions. In this paper, we discuss the problem of designing kernel functions for binary regression and show that using a bell-shaped cosine function as a kernel function is optimal in some sense. The rationale of this result is based on the Karhunen-Loeve expansion, i.e., the optimal approximation to a set of functions is given by the principal component of the correlation operator of the functions.

In general, a two-dimensional display is defined by two orthogonal unit vectors. In developing the display, discriminant analysis has a shortcoming that the extracted axes are not orthogonal in general. First, in order to overcome the shortcoming, we propose discriminant analysis which provides an orthonormal system in the transformed space. The transformation preserves the discriminatory ability in terms of the Fisher criterion. Second, we present a necessary and sufficient condition that discriminant analysis in the original space provides an orthonormal system. Finally, we investigate the relationship between orthogonal discriminant analysis and the Karhunen-Loeve expansion in the original space.

The optimal coding strategy for signal detection in the correlated gaussian noise is established for the distributed sensors system with essentially zero transmission rate constraint. Specifically, we are able to obtain the same performance as in the situation of no restriction on rate from each sensor terminal to the fusion center. This simple result contrasts with the previous ad hoc studies containing many unnatural assumptions such as the independence of noises contaminating received signal at each sensor. For the design of optimal coder, we can use the classical Levinson-Wiggins-Robinson fast algorithm for block Toeplitz matrix to evaluate the necessary weight vector for the maximum-likelihood detection.