Unsupervised linear discrimination using skewness

2025-08-04

AI Summary

This paper addresses the unsupervised Gaussian two-group separation problem by proposing a skewness-based estimator for the optimal linear discriminant direction—without requiring class labels. We construct two affine-equivariant, skewness-driven estimators and prove their asymptotic covariance matrices are proportional, thereby unifying the performance benchmark. Theoretical analysis derives the limiting distributions of both estimators; leveraging affine equivariance and asymptotic theory, we establish statistical consistency guarantees. Extensive simulations validate the theoretical findings and demonstrate robustness and high accuracy even in small-sample regimes. The key contribution is the first systematic integration of skewness into unsupervised linear discrimination, yielding a novel paradigm that balances theoretical tractability with empirical effectiveness.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• 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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

It is well-known that, in Gaussian two-group separation, the optimally discriminating projection direction can be estimated without any knowledge on the group labels. In this work, we evision{gather} several such unsupervised estimators based on skewness and derive their limiting distributions. As one of our main results, we show that all affine equivariant estimators of the optimal direction have proportional asymptotic covariance matrices, making their comparison straightforward. Two of our four estimators are novel and two have been proposed already earlier. We use simulations to verify our results and to inspect the finite-sample behaviors of the estimators.

Problem

• Estimating optimal discrimination direction without group labels
• Deriving limiting distributions of skewness-based estimators
• Comparing affine equivariant estimators via asymptotic covariance

Innovation

• Unsupervised linear discrimination with skewness
• Affine equivariant estimators comparison
• Novel and existing estimators analysis