# Class Mean Vector Component and Discriminant Analysis

## TL;DR

This paper introduces a kernel-based component analysis method that optimally preserves class mean distances for improved dimensionality reduction, with connections to kernel PCA and discriminant analysis.

## Contribution

It proposes a novel kernel component analysis technique focused on class mean distances, enhancing the understanding of feature space properties.

## Key findings

- The method effectively preserves class mean distances in the feature space.
- Connections established between the proposed method and existing kernel PCA and discriminant analysis.
- Provides insights into the properties of feature spaces obtained by kernel methods.

## Abstract

The kernel matrix used in kernel methods encodes all the information required for solving complex nonlinear problems defined on data representations in the input space using simple, but implicitly defined, solutions. Spectral analysis on the kernel matrix defines an explicit nonlinear mapping of the input data representations to a subspace of the kernel space, which can be used for directly applying linear methods. However, the selection of the kernel subspace is crucial for the performance of the proceeding processing steps. In this paper, we propose a component analysis method for kernel-based dimensionality reduction that optimally preserves the pair-wise distances of the class means in the feature space. We provide extensive analysis on the connection of the proposed criterion to those used in kernel principal component analysis and kernel discriminant analysis, leading to a discriminant analysis version of the proposed method. Our analysis also provides more insights on the properties of the feature spaces obtained by applying these methods.