Unsupervised neural learning on lie group

Area of Science:

• Machine Learning
• Artificial Intelligence
• Computational Neuroscience

Background:

• Neural networks often use matrices for learnable parameters, creating Euclidean parameter spaces.
• Orthonormality constraints on these matrices restrict parameter spaces to differential manifolds like orthogonal and Stiefel manifolds.
• Analyzing learning dynamics on these restricted spaces requires advanced mathematical tools.

Purpose of the Study:

• Introduce unsupervised neural learning concepts with orthonormality constraints.
• Detail the mathematical framework using differential geometry and Lie groups.
• Investigate learning dynamics within these constrained parameter spaces.

Main Methods:

• Employ differential geometry and Lie group theory to characterize learning equations.
• Analyze single non-linear layers with connection matrices.
• Study first-order (gradient-based) and second-order (non-gradient-based) learning theories.

Main Results:

• Demonstrate how orthonormality constraints define specific differential manifolds for parameter spaces.
• Provide mathematical instruments derived from Lie group theory for analyzing learning dynamics.
• Offer a detailed study of two subclasses of Lie-group learning theories.

Conclusions:

• Unsupervised learning with orthonormality constraints can be rigorously studied using Lie group theory.
• The geometric structure of parameter spaces significantly influences learning dynamics.
• This framework offers a general approach applicable to various unsupervised learning paradigms.