Learning Markov Random Fields for Combinatorial Structures via Sampling through Lovász Local Lemma

Combinatorial Markov Random Fields

A combinatorial random variable is a discrete random variable defined over a combinatorial set (e.g., a power set of a given set). In this paper we introduce combinatorial Markov random fields (Comrafs), which are Markov random fields where some of the nodes are combinatorial random variables. We argue that Comrafs are powerful models for unsupervised and semi-supervised learning. We put Comraf...

Lovász Local Lemma

Using Lovász Local Lemma in the Space of Random Injections

The Lovász Local Lemma is known to have an extension for cases where independence is missing but negative dependencies are under control. We show that this is often the case for random injections, and we provide easy-to-check conditions for the non-trivial task of verifying a negative dependency graph for random injections. As an application, we prove existence results for hypergraph packing an...

An Improvement of the Lovász Local Lemma via Cluster Expansion

An old result by Shearer relates the Lovász Local Lemma with the independent set polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition function of the hard core lattice gas on graphs. We use this connection and a recent result on the analyticity of the logarithm of the partition function of the abstract polymer gas to get an improved version of the Lovász Lo...

A Constructive Lovász Local Lemma for Permutations

While there has been significant progress on algorithmic aspects of the Lovász Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations. The breakthrough algorithm of Moser & Tardos only works in the setting of independent variables, and does not apply in this context. We resolve this by developing a randomized polynomial-time algor...