Stochastic complexities of reduced rank regression in Bayesian estimation.

Area of Science:

• Statistical Learning Theory
• Machine Learning
• Information Theory

Background:

• Reduced rank regression is a statistical model often viewed as a three-layer neural network.
• It is a non-regular model with a degenerate Fisher information matrix, leading to unknown generalization error.
• Understanding generalization error is crucial for model reliability and performance in statistical estimation.

Purpose of the Study:

• To determine the exact asymptotic form of the generalization error for reduced rank approximation in Bayesian estimation.
• To address the lack of knowledge regarding the generalization error of this non-regular statistical model.
• To introduce a novel method for the complete desingularization of reduced rank approximation.

Main Methods:

• Utilizing Bayesian estimation techniques to analyze the model's generalization error.
• Resolving learning machine singularities through mathematical analysis.
• Calculating the maximum pole of the zeta function relevant to learning theory.
• Developing a new method of recursive blowing-ups for complete desingularization.

Main Results:

• The exact asymptotic form of the generalization error for reduced rank regression in Bayesian estimation has been derived.
• The study successfully resolves singularities in learning machines associated with reduced rank approximation.
• A novel recursive blowing-up technique provides a complete desingularization of the model.

Conclusions:

• This work provides a complete theoretical understanding of the generalization error in reduced rank regression.
• The developed desingularization method offers new analytical tools for non-regular statistical models.
• The findings contribute to advancements in statistical learning theory and Bayesian estimation.