Learning a Better SPD Network for Signal Classification: A Riemannian Batch Normalization Method

Area of Science:

• Machine Learning
• Riemannian Geometry
• Matrix Analysis

Background:

• Symmetric positive definite (SPD) matrices are crucial Riemannian feature descriptors for manifold-valued representations.
• SPD neural networks enhance signal classification, with Riemannian batch normalization (RBN) improving their learning capabilities.
• Current RBN methods often use the affine-invariant Riemannian metric (AIRM), which relies on singular value decomposition (SVD) and can be unstable for ill-conditioned matrices.

Purpose of the Study:

• To develop a more numerically stable and computationally efficient RBN algorithm for SPD matrices.
• To address the limitations of AIRM-based RBN by introducing a novel approach leveraging the log-Cholesky metric (LCM).

Main Methods:

• Proposed a novel RBN algorithm based on the log-Cholesky metric (LCM), utilizing Cholesky decomposition.
• Developed LCM-based Riemannian operators (Fréchet mean, parallel transport) with closed-form solutions.
• Leveraged the Cholesky manifold for efficient computation of LCM-based RBN on the SPD manifold.

Main Results:

• The proposed LCM-based RBN demonstrates enhanced numerical stability compared to AIRM-based methods.
• LCM-based Riemannian operators are simpler and more computationally efficient.
• Experiments on four benchmark datasets confirm the effectiveness of the proposed LCM-based RBN algorithm.

Conclusions:

• The novel LCM-based RBN offers a more robust and efficient alternative for SPD matrix processing in deep learning.
• This advancement can lead to improved performance in signal classification and other applications utilizing SPD matrices.