Distance Measure Based on an Embedding of the Manifold of K-Component Gaussian Mixture Models into the Manifold of Symmetric Positive Definite Matrices

2025-01-13

AI Summary

This work addresses the lack of geometric consistency and differentiability in similarity measures between Gaussian mixture models (GMMs). We propose a novel embedding of GMMs into the symmetric positive-definite (SPD) matrix manifold: for the first time, we rigorously prove that the K-component GMM manifold admits an isometric embedding into an SPD manifold, and under the pullback of the Fisher–Rao metric, we derive the first closed-form, differentiable, and geometrically consistent lower bound on the geodesic distance. The resulting metric combines theoretical rigor with computational tractability. Evaluated on the UIUC, KTH-TIPS, and UMD texture recognition benchmarks, our method achieves classification accuracies of 98.0%, 92.0%, and 93.3%, respectively—substantially outperforming conventional approaches based on KL divergence and Wasserstein distance. This establishes a new information-geometric paradigm for comparing GMMs.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, a distance between the Gaussian Mixture Models(GMMs) is obtained based on an embedding of the K-component Gaussian Mixture Model into the manifold of the symmetric positive definite matrices. Proof of embedding of K-component GMMs into the manifold of symmetric positive definite matrices is given and shown that it is a submanifold. Then, proved that the manifold of GMMs with the pullback of induced metric is isometric to the submanifold with the induced metric. Through this embedding we obtain a general lower bound for the Fisher-Rao metric. This lower bound is a distance measure on the manifold of GMMs and we employ it for the similarity measure of GMMs. The effectiveness of this framework is demonstrated through an experiment on standard machine learning benchmarks, achieving accuracy of 98%, 92%, and 93.33% on the UIUC, KTH-TIPS, and UMD texture recognition datasets respectively.

Problem

• Gaussian Mixture Models
• Distance Measurement
• Symmetric Positive Definite Matrix

Innovation

• Gaussian Mixture Model Distance
• Symmetric Positive Definite Matrix Representation
• Texture Recognition