Adaptive Log-Euclidean Metrics for SPD Matrix Learning.

Area of Science:

• Machine Learning
• Manifold Geometry
• Deep Learning

Background:

• Symmetric Positive Definite (SPD) matrices are crucial for encoding data correlations.
• Existing Riemannian metrics for SPD manifolds are often fixed, limiting adaptability in neural networks.
• This limitation can lead to suboptimal performance in deep SPD neural networks.

Purpose of the Study:

• To introduce Adaptive Log-Euclidean Metrics (ALEMs) for enhanced SPD matrix learning.
• To address the sub-optimal performance of fixed metric tensors in SPD neural networks.
• To improve the adaptability of Riemannian neural networks through learnable metric parameters.

Main Methods:

• Leveraged pullback techniques to extend the Log-Euclidean Metric (LEM).
• Developed ALEMs incorporating learnable parameters for dynamic adaptation.
• Conducted theoretical analysis including algebraic and Riemannian properties.

Main Results:

• ALEMs demonstrated improved performance in SPD neural networks compared to fixed metrics.
• The proposed metrics adapt effectively to complex Riemannian neural network dynamics.
• Minor additional computational cost was observed with ALEMs.

Conclusions:

• ALEMs offer a significant advancement for SPD matrix learning.
• The learnable parameters in ALEMs enhance the adaptability and performance of Riemannian neural networks.
• ALEMs show efficacy across various Riemannian building blocks, including normalization, residual blocks, and classifiers.