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Kernel Stein Discrepancy on Lie Groups: Theory and Applications.

Xiaoda Qu¹, Xiran Fan², Baba C Vemuri³

¹Department of Statistics, University of Florida, Gainesville, FL 32611 USA.

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|December 20, 2024

Summary

This summary is machine-generated.

This study introduces a novel normalization-free loss function for distributional approximation on Lie groups, crucial for machine learning in science and engineering. The new method, minimum kernel Stein discrepancy estimator (MKSDE), offers advantages over traditional techniques.

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Area of Science:

Machine Learning
Computational Mathematics
Data Science

Background:

Distributional approximation is vital in machine learning and scientific applications.
Intractable normalization constants pose challenges, especially for manifold-valued data like rotation matrices.
Lie groups are frequently used in computer vision, robotics, and medical imaging.

Purpose of the Study:

To address the distributional approximation problem on Lie groups.
To develop a novel, normalization-free loss function for these complex distributions.
To introduce and analyze a new estimator based on this loss function.

Main Methods:

Developed a novel Stein's operator tailored for Lie groups.
Introduced a kernel Stein discrepancy (KSD) as a normalization-free loss function.
Derived and analyzed the minimum KSD estimator (MKSDE).

Main Results:

Established theoretical properties of the new KSD on Lie groups.
Proved strong consistency and central limit theorem (CLT) for the MKSDE.
Derived a closed-form solution for MKSDE for specific distributions on Lie groups.

Conclusions:

The novel KSD and MKSDE provide an effective approach for distributional approximation on Lie groups.
MKSDE demonstrates advantages over maximum likelihood estimation in experimental results.
This work offers a powerful tool for machine learning applications involving manifold-valued data.