Sparse Density Estimation on the Multinomial Manifold.

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• Kernel density estimation is crucial for non-parametric data analysis.
• Finite mixture models offer flexibility in capturing complex data distributions.
• Existing methods may lack sparsity or struggle with constraints on mixing coefficients.

Purpose of the Study:

• To introduce a new sparse kernel density estimator for finite mixture models.
• To address the challenge of constrained mixing coefficients on the multinomial manifold.
• To enhance accuracy and sparsity in density estimation.

Main Methods:

• Developed a sparse kernel density estimator based on the minimum integrated square error criterion.
• Utilized the Riemannian trust-region (RTR) algorithm to handle constraints on the multinomial manifold.
• Derived and applied first- and second-order Riemannian geometry within the RTR algorithm.

Main Results:

• The proposed approach effectively constructs sparse kernel density estimators.
• Numerical examples demonstrate competitive accuracy compared to existing estimators.
• The RTR algorithm successfully handles the multinomial manifold constraints.

Conclusions:

• The new sparse kernel density estimator is effective for finite mixture models.
• The integration of Riemannian geometry with the RTR algorithm provides a robust solution.
• This method offers a promising alternative for accurate and sparse density estimation.