Spherical Minimum Description Length.

Area of Science:

• Statistics
• Machine Learning
• Geometric Data Analysis

Background:

• Model selection criteria balance goodness of fit with model complexity.
• Traditional methods often inaccurately penalize complexity in curved parameter spaces like hyperspheres.
• Laplace approximation techniques struggle with the geometry of non-Euclidean spaces.

Purpose of the Study:

• To develop a novel model selection criterion, spherical Minimum Description Length (MDL), for distributions with parameters on the hypersphere.
• To address the limitations of current criteria that ignore or misrepresent the geometry of spherical parameter spaces.
• To introduce a more accurate complexity measure that accounts for the shape of the parameter space.

Main Methods:

• Utilizing a constrained Laplace approximation tailored for the hypersphere.
• Developing a new complexity measure that reflects the geometry of spherical parameter spaces.
• Applying the spherical MDL criterion to the problem of bin selection in histogram density estimation.

Main Results:

• The constrained Laplace approximation provides an accurate complexity measure for spherical data.
• The proposed spherical MDL criterion outperforms existing model selection methods.
• Demonstrated favorable performance in histogram density estimation bin selection.

Conclusions:

• Spherical MDL offers a geometrically informed approach to model selection on hyperspheres.
• This method provides a more accurate assessment of model complexity in curved spaces.
• The approach shows practical utility in density estimation tasks.