Hierarchical Linear Discriminant Analysis for Beamforming

Abstract

This paper demonstrates the applicability of the recently proposed supervised dimension reduction, hierarchical linear discriminant analysis (h-LDA) to a well-known spatial localization technique in signal processing, beamforming. The main motivation of h-LDA is to overcome the drawback of LDA that each cluster is modeled as a unimodal Gaussian distribution. For this purpose, h-LDA extends the variance decomposition in LDA to the subcluster level, and modifies the definition of the within-cluster scatter matrix. In this paper, we present an efficient h-LDA algorithm for oversampled data, where the data dimension is larger than the dimension of the data vectors. The new algorithm utilizes the Cholesky decomposition based on a generalized singular value decomposition framework. Furthermore, we analyze the data model of h-LDA by relating it to the two-way multivariate analysis of variance (MANOVA), which fits well within the context of beamforming applications. Although beamforming has been generally dealt with as a regression problem, we propose a novel way of viewing beamforming as a classification problem, and apply a supervised dimension reduction, which allows the classifier to achieve better accuracy. Our experimental results show that h-LDA outperforms several dimension reduction methods such as LDA and kernel discriminant analysis, and regression approaches such as the regularized least squares and kernelized support vector regression.

Robust Linear Discriminant Analysis Using Ratio Minimization of L1,2-Norms

“…Usually the Principal Component Analysis (PCA) is employed as pre-processing step to discard the null space of the overall scatter matrix before the LDA is used. To solve the null space problem, many variations of LDA methods have been proposed [1,2,3,4,5,6,7,8] in machine learning communities. More recently, the trace-ratio LDA objective has been studied and shown with promising results [9,10,11].…”