Generalized Learning Vector Quantization With Log-Euclidean Metric Learning on Symmetric Positive-Definite Manifold

Area of Science:

• Machine Learning
• Differential Geometry
• Data Science

Background:

• Classification tasks often involve data residing on Riemannian manifolds, specifically symmetric positive-definite (SPD) matrices.
• Standard Euclidean algorithms struggle with the non-Euclidean geometry inherent in SPD matrix data.
• Generalized Learning Vector Quantization (GLVQ) is a successful Euclidean-based classification method.

Purpose of the Study:

• To adapt the Generalized Learning Vector Quantization (GLVQ) algorithm for data on the manifold of symmetric positive-definite (SPD) matrices.
• To incorporate the nonlinear Riemannian geometry of SPD matrices using the log-Euclidean metric (LEM).
• To develop and evaluate metric learning extensions for improved classification performance.

Main Methods:

• Generalized GLVQ to SPD matrices using LEM-induced geodesic distance (GLVQ-LEM).
• Extended GLVQ-LEM with metric learning, considering both vectorized log-transformed SPD matrices and full tensor structures.
• Proposed two algorithms for full LEM learning (LEML): GLVQ-LEML-LEM (full-rank metric tensor) and GLVQ-LEML-FM (fixed-rank positive semidefinite metric tensor).

Main Results:

• The proposed GLVQ-LEM and its metric learning extensions (GLVQ-LEML-LEM, GLVQ-LEML-FM) demonstrate strong performance across diverse datasets.
• Accounting for the Riemannian geometry of SPD matrices leads to superior classification results compared to Euclidean methods.
• The metric learning extensions further improve the adaptability and accuracy of the classification models.

Conclusions:

• The reformulated GLVQ using log-Euclidean metrics provides a principled and effective approach for classification on SPD manifolds.
• The developed metric learning strategies enhance the robustness and performance of these methods.
• These advancements offer significant potential for applications involving complex, non-Euclidean data structures.