# On the mixing time of coordinate Hit-and-Run

arXiv: https://arxiv.org/abs/2009.14004 · HTML: https://mlsanity.com/paper/2009.14004/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2020.

## Built on by (most cited first, 5 of 5)

- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [On Sampling with Approximate Transport Maps](https://mlsanity.com/paper/2302.04763/) (2023)
- [Oracle-based Uniform Sampling from Convex Bodies](https://mlsanity.com/paper/2510.02983/) (2025)
- [Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run](https://mlsanity.com/paper/2608.16878/) (2026)
- [Truncated Log-concave Sampling with Reflective Hamiltonian Monte Carlo](https://mlsanity.com/paper/2102.13068/) (2021)

## Abstract (arXiv)

We obtain a polynomial upper bound on the mixing time TCHR(ε) of the coordinate Hit-and-Run random walk on an n-dimensional convex body, where TCHR(ε) is the number of steps needed in order to reach within ε of the uniform distribution with respect to the total variation distance, starting from a warm start (i.e., a distribution which has a density with respect to the uniform distribution on the convex body that is bounded above by a constant). Our upper bound is polynomial in n, R and frac1ε, where we assume that the convex body contains the unit Vert·Vert_∞-unit ball B_∞ and is contained in its R-dilation R· B_∞. Whether coordinate Hit-and-Run has a polynomial mixing time has been an open question.