Traditional methods like Gibbs sampling or randomized Kaczmarz rely heavily on specific coordinate systems, which can limit their efficiency—especially in ill-conditioned settings. But what happens when we step away from these constraints and adopt a coordinate-free approach?
Our latest research examines this question, rigorously exploring the advantages and limitations of coordinate-free methods for both sampling and optimization. While Hit-and-Run has long been recognized as a coordinate-free Gibbs sampler, its quantitative benefits over coordinate-bound methods have remained largely unexplored—until now.
In collaboration with Andreas Eberle and Stefan Oberdörster (both from the University of Bonn), we rigorously analyze and quantify the performance of coordinate-free methods like Hit-and-Run and a coordinate-free randomized Kaczmarz algorithm, identifying the settings where these methods excel.
✨ Mathematical Insights
- Ballistic and superdiffusive convergence rates:
In certain situations, coordinate-free methods achieve ballistic and superdiffusive rates, far outperforming the diffusive rates typical of coordinate-bound approaches. However, their performance is context-dependent, and they may face limitations in high-dimensional settings. - Hit-and-Run performance:
We establish tight upper and lower bounds on Wasserstein contraction rates for Hit-and-Run and derive sharp total variation mixing time upper bounds, precisely characterizing its behavior as a function of the target distribution’s condition number. - Extension to randomized Kaczmarz:
These insights extend to a coordinate-free variant of the randomized Kaczmarz algorithm, a widely-used iterative method for solving linear systems. Similar convergence behavior is observed, highlighting the geometric advantages of these methods.
Our findings offer a nuanced view of coordinate-free methods in sampling and optimization. While these methods demonstrate remarkable convergence rates—ballistic and superdiffusive—in certain scenarios, they also face limitations, particularly in high-dimensional spaces. Quantitatively understanding these trade-offs provides a clearer framework for leveraging coordinate-free approaches where they are most effective.
🔗 Read the full paper here: arXiv:2412.07643.