Kernel Stein Discrepancy on Lie Groups: Theory and Applications

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.