Estimating densities with non-linear support by using Fisher-Gaussian kernels.

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• Multivariate density estimation is crucial for data analysis.
• Existing methods, particularly those using Gaussian kernels, struggle with data concentrated on non-linear manifolds.
• Current kernels lack the ability to capture data curvature, necessitating large sample sizes relative to data dimensions.

Purpose of the Study:

• To propose a novel generalization of the Gaussian distribution for multivariate density estimation.
• To introduce Fisher-Gaussian kernels capable of modeling data with curvature.
• To demonstrate the effectiveness of Fisher-Gaussian kernels in Bayesian mixture models.

Main Methods:

• Developed a new class of kernels called Fisher-Gaussian kernels.
• Derived the Fisher-Gaussian density by combining von Mises-Fisher sampling on the sphere with Gaussian noise.
• Implemented the Fisher-Gaussian density within Bayesian mixture models using Markov chain Monte Carlo (MCMC) sampling.

Main Results:

• The proposed Fisher-Gaussian kernels can effectively model data with curvature.
• Demonstrated analytic tractability and straightforward implementation of the Fisher-Gaussian density.
• Showcased performance gains over competing methods in both simulated and real-world datasets.

Conclusions:

• Fisher-Gaussian kernels offer a significant advancement in multivariate density estimation, particularly for complex data structures.
• The new kernels provide a flexible and powerful alternative to traditional Gaussian kernels.
• This work facilitates more accurate data modeling in scenarios with non-linear data support.