Journal of the ACM

Counting, sampling and integrating: Algorithms and complexity

We show that the mixing time of Glauber (single edge update) dynamics for the random cluster model at $q=2$ is bounded by a polynomial in the size of the underlying graph. As a consequence, the Swendsen-Wang algorithm for the ferromagnetic Ising model at any temperature has the same polynomial mixing time bound.

Random cluster dynamics for the Ising model is rapidly mixing

We show for the first time that the mixing time of Glauber (single edge update) dynamics for the random cluster model at q = 2 is bounded by a polynomial in the size of the underlying graph. As a consequence, the Swendsen-Wang algorithm for the ferromagnetic Ising model at any temperature has the same polynomial mixing time bound.

Random cluster dynamics for the ising model is rapidly mixing

We study the identity testing problem in the context of spin systems or undirected graphical models, where it takes the following form: given the parameter specification of the model $M$ and a sampling oracle for the distribution $\mu_{\hat{M}}$ of an unknown model $\hat{M}$, can we efficiently determine if the two models $M$ and $\hat{M}$ are the same? We consider identity testing for both soft-constraint and hard-constraint systems. In particular, we prove hardness results in two prototypical cases, the Ising model and proper colorings, and explore whether identity testing is any easier than structure learning. 
For the ferromagnetic (attractive) Ising model, Daskalakis et al. (2018) presented a polynomial time algorithm for identity testing. We prove hardness results in the antiferromagnetic (repulsive) setting in the same regime of parameters where structure learning is known to require a super-polynomial number of samples. In particular, for $n$-vertex graphs of maximum degree $d$, we prove that if $|\beta| d = \omega(\log{n})$ (where $\beta$ is the inverse temperature parameter), then there is no polynomial running time identity testing algorithm unless $RP=NP$. We also establish computational lower bounds for a broader set of parameters under the (randomized) exponential time hypothesis. Our proofs utilize insights into the design of gadgets using random graphs in recent works concerning the hardness of approximate counting by Sly (2010). In the hard-constraint setting, we present hardness results for identity testing for proper colorings. Our results are based on the presumed hardness of #BIS, the problem of (approximately) counting independent sets in bipartite graphs. In particular, we prove that identity testing is hard in the same range of parameters where structure learning is known to be hard.

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Lower bounds for testing graphical models: colorings and antiferromagnetic Ising models

We prove rapid mixing of the worm process for the zero-field ferromagnetic Ising model, on all finite connected graphs, and at all temperatures. As a corollary, we obtain a fully-polynomial randomized approximation scheme for the Ising susceptibility, and for a certain restriction of the two-point correlation function

The worm process for the Ising model is rapidly mixing

We prove rapid mixing of the worm process for the zero-field ferromagnetic Ising model, on all finite connected graphs, and at all temperatures. As a corollary, we obtain a fully-polynomial randomized approximation scheme for the Ising susceptibility, and for a certain restriction of the two-point correlation function.

The Worm Process for the Ising Model is Rapidly Mixing

Corresponding to $n$ independent non-negative random variables $X_1,...,X_n$, are values $M_1,...,M_n$, where each $M_i$ is the expected value of the maximum of $n$ independent copies of $X_i$. We obtain an upper bound to the expected value of the maximum of $X_1,...,X_n$ in terms of $M_1,...,M_n$. This inequality is sharp in the sense that the quantity and its bound can be made as close to each other as we want. We also present related comparison results.

The mixing advantage is less than 2

We prove rapid mixing of the Prokofiev-Svistunov (or worm) algorithm for the zero-field ferromagnetic Ising model, on all finite graphs and at all temperatures As a corollary, we show how to rigorously construct simple and efficient approximation schemes for the Ising susceptibility and two-point correlation function

The worm algorithm for the Ising model is rapidly mixing.

Corresponding to n independent non-negative random variables X_1,...,X_n , are values M_1,...,M_n , where each M_i is the expected value of the maximum of n independent copies of X_i. We obtain an upper bound for the expected value of the maximum of X_1,...,X_n in terms of M_1,...,M_n . This inequality is sharp in the sense that the random variables can be chosen so that the bound is approached arbitrarily closely. We also present related comparison results.

Daniel Tokarev

Papers

The worm process for the Ising model is rapidly mixing

The Worm Process for the Ising Model is Rapidly Mixing

The mixing advantage is less than 2

The worm algorithm for the Ising model is rapidly mixing.

The mixing advantage is less than 2