# Efficient Mass Matrix Estimation with Gaussian Cooling

arXiv: https://arxiv.org/abs/2610.04806 · HTML: https://mlsanity.com/paper/2610.04806/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 40 references that are in the map.
- First posted on arXiv 3 Oct 2026.

## Builds on (most cited first, 9 of 9)

- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [New affine invariant ensemble samplers and their dimensional scaling](https://mlsanity.com/paper/2505.02987/) (2025)
- [Exact simulation of diffusions and improved algorithms for log-concave sampling](https://mlsanity.com/paper/2608.05022/) (2026)
- [Shifted Composition IV: Toward Ballistic Acceleration for Log-Concave Sampling](https://mlsanity.com/paper/2506.23062/) (2025)
- [A Non-asymptotic Analysis for Learning and Applying a Preconditioner in MCMC](https://mlsanity.com/paper/2602.10714/) (2026)
- [Algorithmic warm starts for Hamiltonian Monte Carlo](https://mlsanity.com/paper/2603.22741/) (2026)
- [Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler](https://mlsanity.com/paper/2609.06905/) (2026)
- [Smoothed Picard Hamiltonian Monte Carlo](https://mlsanity.com/paper/2609.06906/) (2026)

## Abstract (arXiv)

We introduce an algorithm for efficiently preconditioning log-concave and log-smooth distributions that scales logarithmically with the condition number of the underlying distribution. Based on Gaussian cooling, our multistage method approximately samples from a sequence of well-conditioned distributions to construct a preconditioner for each subsequent stage of the cooling schedule. This method is motivated by existing practical approaches for mass matrix estimation in Markov chain Monte Carlo (MCMC), in which an effective preconditioner for the target distribution is estimated from samples produced during a warm-up period. However, our approach is conceptually distinct from existing approaches and moreover permits theoretical analysis. Our algorithmic framework is flexible, allowing essentially any log-concave sampling algorithm to act as the subroutine within each cooling stage. For instance, using first-order rejection sampling (FORS) as the sampler, the first-order query complexity of our method is tildeO(d4/3 log kappa) for a large class of log-concave and log-smooth distributions with condition number kappa and tildeO(d1/3 log kappa) for translation-invariant distributions. In the translation-invariant setting, we provide numerical experiments for the lattice φ4 model, a simple lattice field theory.