[Paper Review] Projecting Markov Random Field Parameters for Fast Mixing

Xianghang Liu, Justin Domke|arXiv (Cornell University)|Nov 5, 2014

Bayesian Methods and Mixture Models14 references3 citations

TL;DR

This paper proposes sufficient conditions to guarantee fast mixing in univariate Gibbs sampling for Markov Random Fields (MRFs), along with an efficient Euclidean projection algorithm onto the set of fast-mixing parameters. The method enables faster convergence in MCMC sampling, outperforming standard Gibbs sampling and variational methods in accuracy with reduced computation time, particularly when using the $∞$-norm projection for improved efficiency.

ABSTRACT

Markov chain Monte Carlo (MCMC) algorithms are simple and extremely powerful techniques to sample from almost arbitrary distributions. The flaw in practice is that it can take a large and/or unknown amount of time to converge to the stationary distribution. This paper gives sufficient conditions to guarantee that univariate Gibbs sampling on Markov Random Fields (MRFs) will be fast mixing, in a precise sense. Further, an algorithm is given to project onto this set of fast-mixing parameters in the Euclidean norm. Following recent work, we give an example use of this to project in various divergence measures, comparing univariate marginals obtained by sampling after projection to common variational methods and Gibbs sampling on the original parameters.

Motivation & Objective

• To identify sufficient conditions under which univariate Gibbs sampling on MRFs achieves fast mixing.
• To develop an efficient algorithm to project MRF parameters onto the set of fast-mixing parameters in the Euclidean norm.
• To evaluate the performance of projected parameters in improving sampling accuracy and convergence speed compared to standard Gibbs sampling and variational inference.
• To explore the use of alternative matrix norms (e.g., $∞$-norm) as a computationally efficient alternative to spectral norm-based projections.
• To demonstrate practical utility through applications in image denoising and grid models, showing improved trade-offs between time and accuracy.

Proposed method

• Derives sufficient conditions for fast mixing in univariate Gibbs sampling based on the $∞$-norm of a matrix derived from MRF interaction strengths.
• Proposes a Euclidean projection algorithm to map arbitrary MRF parameters onto the fast-mixing parameter set, ensuring rapid mixing.
• Uses projected gradient descent to minimize divergences (e.g., reversed KL) between the true and projected distributions, improving sample quality.
• Employs the $∞$-norm as a looser but more computationally efficient alternative to the spectral norm for parameter projection.
• Validates the method via sampling on Ising models and Potts models, comparing marginal error and convergence speed.
• Applies the framework to binary image denoising, using Swendsen-Wang samples as ground truth for evaluation.

Experimental results

Research questions

• RQ1 Can sufficient conditions be derived to guarantee fast mixing in univariate Gibbs sampling for general MRFs?
• RQ2 How efficiently can MRF parameters be projected onto the set of fast-mixing parameters in the Euclidean norm?
• RQ3 Does projecting parameters to ensure fast mixing lead to better sampling accuracy and faster convergence than standard Gibbs sampling?
• RQ4 How does the $∞$-norm projection compare to spectral norm projection in terms of computational cost and accuracy?
• RQ5 Can this framework improve inference performance in real-world applications like image denoising?

Key findings

• The $∞$-norm projection method achieves faster convergence than standard Gibbs sampling, with marginal error decreasing below 0.02 after 10,000 samples on 16×16 grids.
• For $d_e = 1.5$, the $∞$-norm projection method required only 1.50s for 10^6 Gibbs steps, compared to 25.63s for spectral norm projection.
• On 16×16 grids with $d_e = 3.0$, the $∞$-norm projection reduced time from 164.34s (spectral norm) to 3.26s for 10^6 Gibbs steps.
• In image denoising, projected parameters enabled convergence to low marginal error in under 10^4 samples, while original Gibbs sampling remained slow.
• The reversed KL divergence minimization with projected parameters achieved accuracy competitive with variational methods, with 13.13s runtime for 10^6 steps using $∞$-norm projection.
• Using a larger constraint $c=2.5$ than the theoretical bound $c=1$ improved performance, indicating looseness in the theoretical bound.