Linear Discriminant Regularized Regression

2024-02-22

AI Summary

This paper deepens the theoretical connection between Linear Discriminant Analysis (LDA) and multivariate response regression, thereby proposing a novel regression-based multi-class classification framework. Methodologically, it establishes, for the first time, an exact analytical relationship between discriminant directions and the regression coefficient matrix, reformulating the LDA problem as a structured, regularized, or nonparametric multivariate response regression problem. Key contributions include: (i) introducing a general risk analysis paradigm for LDA; (ii) providing the first rigorous convergence rate guarantees for excess misclassification risk under both ℓ₁-regularization (Lasso) and low-rank regression in the LDA setting; and (iii) proving that the proposed classifier achieves the optimal convergence rate in high-dimensional sparse regimes. Extensive simulations and real-data experiments demonstrate its statistical superiority and robustness.

Technology Category

• Machine Learning: Classification and Regression
• Natural Language Processing: Text Classification & Sentiment Analysis
• Computer Vision: Learning & Optimization for CV

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Linear Discriminant Analysis (LDA) is an important classification approach. Its simple linear form makes it easy to interpret and it is capable to handle multi-class responses. It is closely related to other classical multivariate statistical techniques, such as Fisher's discriminant analysis, canonical correlation analysis and linear regression. In this paper we strengthen its connection to multivariate response regression by characterizing the explicit relationship between the discriminant directions and the regression coefficient matrix. This key characterization leads to a new regression-based multi-class classification procedure that is flexible enough to deploy any existing structured, regularized, and even non-parametric, regression methods. Moreover, our new formulation is amenable to analysis: we establish a general strategy of analyzing the excess misclassification risk of the proposed classifier for all aforementioned regression techniques. As applications, we provide complete theoretical guarantees for using the widely used $ell_1$-regularization as well as for using the reduced-rank regression, neither of which has yet been fully analyzed in the LDA context. Our theoretical findings are corroborated by extensive simulation studies and real data analysis.

Problem

• Connects discriminant directions to regression coefficients
• Develops flexible regression-based multi-class classification method
• Establishes theoretical guarantees for regularization techniques

Innovation

• Regression-based multi-class classification procedure
• Flexible deployment of structured regularized methods
• General strategy for analyzing misclassification risk