Singularities affect dynamics of learning in neuromanifolds

Area of Science:

• Computational Neuroscience
• Statistical Machine Learning
• Information Geometry

Background:

• Hierarchical systems like multilayer perceptrons possess singular parameter spaces due to hidden unit symmetry and degeneration.
• These parameter spaces form geometric manifolds (neuromanifolds), where the Fisher information matrix, acting as a Riemannian metric, degenerates at singularities.
• Singular structures are common in various statistical models, including Gaussian mixture densities and ARMA time-series models.

Purpose of the Study:

• To provide an overview of phenomena arising from singularities in statistical manifolds relevant to multilayer perceptrons and Gaussian mixtures.
• To demonstrate recent research findings and use toy models for explicit solutions.
• To explain the impact of singularities on statistical inference, model selection, and learning dynamics.

Main Methods:

• Analysis of parameter spaces as geometric manifolds (neuromanifolds).
• Investigation of the Fisher information matrix and its degeneration at singularities.
• Application of toy models to illustrate explicit solutions and phenomena.
• Study of learning dynamics and the performance of the natural gradient method.

Main Results:

• Maximum likelihood estimators deviate from Gaussian distribution asymptotically due to Fisher information matrix degeneration.
• Standard model selection criteria (AIC, BIC, MDL) fail in singular models.
• Bayesian priors become singular, and learning dynamics exhibit plateaus or slow manifolds.
• The natural gradient method shows superior performance by accounting for the singular geometry.

Conclusions:

• Singularities in statistical manifolds present significant challenges to conventional statistical theories and machine learning practices.
• Understanding and addressing these singularities is crucial for robust parameter estimation, inference, and efficient learning.
• The natural gradient method offers a promising approach to navigate and leverage the geometric structure of singular models.