# Ballistic Convergence in Hit-and-Run Monte Carlo and a Coordinate-free Randomized Kaczmarz Algorithm

arXiv: https://arxiv.org/abs/2412.07643 · HTML: https://mlsanity.com/paper/2412.07643/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 42 references that are in the map.
- First posted on arXiv 10 Dec 2024.

## Builds on (most cited first, 5 of 5)

- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo](https://mlsanity.com/paper/2105.00887/) (2021)
- [Entropy contraction of the Gibbs sampler under log-concavity](https://mlsanity.com/paper/2410.00858/) (2024)
- [GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo](https://mlsanity.com/paper/2404.15253/) (2024)
- [Stochastic Dual Ascent for Solving Linear Systems](https://mlsanity.com/paper/1512.06890/) (2015)

## Abstract (arXiv)

Hit-and-Run is a coordinate-free Gibbs sampler, yet the quantitative advantages of its coordinate-free property remain largely unexplored beyond empirical studies. In this paper, we prove sharp estimates for the Wasserstein contraction of Hit-and-Run in Gaussian target measures via coupling methods and conclude mixing time bounds. Our results uncover ballistic and superdiffusive convergence rates in certain settings. Furthermore, we extend these insights to a coordinate-free variant of the randomized Kaczmarz algorithm, an iterative method for linear systems, and demonstrate analogous convergence rates. These findings offer new insights into the advantages and limitations of coordinate-free methods for both sampling and optimization.