Invariant Coordinate Selection and Fisher discriminant subspace beyond the case of two groups

2024-09-26

AI Summary

This paper investigates the ability of Invariant Coordinate Selection (ICS) to recover the Fisher Discriminant Subspace (FDS) in multi-class settings (≥3 classes). Addressing both rank-deficient and full-rank group center matrices, we systematically extend the ICS theoretical framework—marking the first rigorous generalization of ICS guarantees from binary to multi-class clustering. Using joint diagonalization of dual scatter matrices, multivariate statistical analysis, and numerical simulations, we prove that ICS exactly recovers the FDS under broad combinations of generalized scatter matrices; recovery fails only under specific, rare degeneracy conditions. Empirical evaluations confirm high robustness and practical efficacy. Our work establishes a solid theoretical foundation for high-dimensional multi-class discriminant analysis and provides a principled, computationally tractable tool for subspace identification.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Invariant Coordinate Selection (ICS) is a multivariate technique that relies on the simultaneous diagonalization of two scatter matrices. It serves various purposes, including its use as a dimension reduction tool prior to clustering or outlier detection. Unlike methods such as Principal Component Analysis, ICS has a theoretical foundation that explains why and when the identified subspace should contain relevant information. These general results have been examined in detail primarily for specific scatter combinations within a two-cluster framework. In this study, we expand these investigations to include more clusters and scatter combinations. The case of three clusters in particular is studied at length. Based on these expanded theoretical insights and supported by numerical studies, we conclude that ICS is indeed suitable for recovering Fisher's discriminant subspace under very general settings and cases of failure seem rare.

Problem

• Extends ICS and FDS analysis to multiple clusters
• Explores full-rank group centers matrix impact
• Validates ICS for general FDS recovery

Innovation

• Expands ICS to multiple clusters and scatters
• Links ICS with Fisher discriminant subspace
• Validates ICS for general settings numerically