Convergence of Kinetic Langevin Monte Carlo on Lie groups: Sampling Dynamics on Lie Groups

Conceitos Básicos

Optimizing functions on Lie groups using momentum-based dynamics with tractable noise for sampling.

Resumo

The article introduces a novel approach to sampling dynamics on Lie groups, preserving the Lie group structure. It proves exponential convergence under W2 distance without requiring convexity or isoperimetry explicitly. The proposed algorithm is efficient and fully implementable, providing rigorous and quantitative analysis of geometric ergodicities. The discretization technique ensures structure preservation, eliminating the need for additional corrections to maintain points on the curved space. The error bounds for both continuous and discrete samplers are quantified nonasymptotically, showcasing the benefits of momentum in sampling algorithms.

Estatísticas

Exponential convergence with explicit convergence rate is proved under W2 distance.
Only compactness of the Lie group and geodesically L-smoothness of the potential function are needed.

Citações

"Sampling dynamics we construct adds tractable noise to optimization dynamics for efficient sampling."
"Lie-group MCMC sampler preserves exact Lie group structure through delicate discretization."
"First convergence result for kinetic Langevin on curved spaces without requiring convexity."

Principais Insights Extraídos De

Convergence of Kinetic Langevin Monte Carlo on Lie groups

by Lingkai Kong... às arxiv.org 03-19-2024

https://arxiv.org/pdf/2403.12012.pdf

Perguntas Mais Profundas

How does the proposed algorithm compare to traditional sampling methods

The proposed algorithm, the Kinetic Langevin Monte Carlo (KLMC) sampler on Lie groups, offers several advantages over traditional sampling methods. Firstly, it leverages momentum-based dynamics to optimize functions defined on Lie groups efficiently. This approach is inspired by techniques such as variational optimization and left trivialization, allowing for more effective sampling in curved spaces like Lie groups. The use of tractable noise in the optimization dynamics transforms it into a sampling dynamics with a Euclidean momentum variable despite living on a manifold.
Additionally, the discretization of the kinetic Langevin-type sampling dynamics preserves the Lie group structure exactly. This ensures that the algorithm maintains accuracy while converging towards the target distribution exponentially under Wasserstein-2 distance metrics. The algorithm's convergence rate is explicitly quantified without requiring convexity or isoperimetry assumptions.
In comparison to traditional methods like Langevin Monte Carlo (LMC), which are gradient-based and widely used but may face challenges in analyzing samplers based on kinetic Langevin due to noise degeneracy issues, KLMC provides a robust solution for efficient and accurate sampling on curved spaces like Lie groups.

What are the implications of not requiring convexity or isoperimetry in sampling dynamics

The implications of not requiring convexity or isoperimetry in sampling dynamics are significant as they relax some common constraints typically associated with convergence guarantees in optimization and sampling algorithms. In this research context, where momentum-accelerated optimization is applied to Riemannian manifolds such as Lie groups using kinetic Langevin dynamics, these relaxed requirements open up new possibilities for practical applications.
By demonstrating exponential convergence without explicit convexity or isoperimetric inequalities needed for both continuous time dynamics and discrete samplers under W2 distance metrics, this research showcases a novel approach that broadens the scope of applicable problems beyond traditional constraints. It highlights that complex geometric structures can be effectively navigated without strict mathematical conditions usually imposed by conventional methodologies.
This flexibility allows for more versatile applications across various domains where non-convex or non-isoperimetric scenarios exist but still require efficient and accurate sampling techniques tailored to specific geometries.

How can this research be applied to real-world problems beyond mathematical simulations

The research on Kinetic Langevin Monte Carlo (KLMC) samplers applied to Lie groups has significant real-world implications beyond mathematical simulations:

Machine Learning: The findings can be utilized in machine learning applications involving complex data sets represented by nonlinear manifolds like those found in natural language processing or image recognition tasks.

Robotics: Implementing these advanced algorithms can enhance robotic motion planning strategies by optimizing trajectories along curved surfaces efficiently.

Biomedical Research: Applications include modeling biological processes influenced by intricate spatial configurations within cellular structures or protein folding studies.

Financial Modeling: Improved stochastic simulation techniques could benefit risk assessment models considering non-linear market behaviors accurately.

Climate Science: Analyzing climate data represented through high-dimensional manifolds could benefit from optimized sampling methods provided by KLMC algorithms tailored for curved spaces.

Overall, this research opens up avenues for solving diverse real-world problems that involve complex geometric structures where traditional methods may fall short due to stringent assumptions around convexity and isoperimetry requirements during dynamic optimizations and samplings operations on specialized spaces like Lie groups."