• Corpus ID: 189928602

A New Family of Tractable Ising Models

@article{Likhosherstov2019ANF,
  title={A New Family of Tractable Ising Models},
  author={Valerii Likhosherstov and Yury Maximov and Michael Chertkov},
  journal={ArXiv},
  year={2019},
  volume={abs/1906.06431}
}
We present a new family of zero-field Ising models over N binary variables/spins obtained by consecutive "gluing" of planar and $O(1)$-sized components along with subsets of at most three vertices into a tree. The polynomial time algorithm of the dynamic programming type for solving exact inference (partition function computation) and sampling consists of a sequential application of an efficient (for planar) or brute-force (for $O(1)$-sized) inference and sampling to the components as a black… 
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References

SHOWING 1-10 OF 34 REFERENCES
Inference and Sampling of K33-free Ising Models
TLDR
This work calls an Ising model tractable when it is possible to compute its partition function value in polynomial time, and extends the basic case of planar zero-field Ising models to models, whose triconnected components are either planar or graphs of $O(1)$ size.
Exact algorithm for sampling the two-dimensional Ising spin glass.
A sampling algorithm is presented that generates spin-glass configurations of the two-dimensional Edwards-Anderson Ising spin glass at finite temperature with probabilities proportional to their
On the computational complexity of Ising spin glass models
TLDR
In a spin glass with Ising spins, the problems of computing the magnetic partition function and finding a ground state are studied and are shown to belong to the class of NP-hard problems, both in the two-dimensional case within a magnetic field, and in the three-dimensional cases.
Crystal statistics. I. A two-dimensional model with an order-disorder transition
The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the
Efficient Exact Inference in Planar Ising Models
TLDR
The approach provides an interesting alternative to the well-known graph cut paradigm in that it does not impose any submodularity constraints; instead it requires planarity to establish a correspondence with perfect matchings in an expanded dual graph.
A new class of upper bounds on the log partition function
TLDR
A new class of upper bounds on the log partition function of a Markov random field (MRF) is introduced, based on concepts from convex duality and information geometry, and the Legendre mapping between exponential and mean parameters is exploited.
Approximate inference using planar graph decomposition
TLDR
This work base the approximation on a different tractable model, planar graphs with binary variables and pure interaction potentials (no external field), and shows how such tractable planar models can be used in a decomposition to derive upper bounds on the partition function of non-planar models.
Dimer Statistics and Phase Transitions
After the introduction of the concept of lattice graph and a brief discussion of its role in the theory of the Ising model, a related combinatorial problem is discussed, namely that of the statistics
Counting perfect matchings in graphs that exclude a single-crossing minor
TLDR
The algorithm uses black-boxes for counting perfect matchings in planar graphs and for computing certain graph decompositions and is one of the first nontrivial algorithms to not inherently rely on Pfaffian orientations.
On the ground states of the frustration model of a spin glass by a matching method of graph theory
The ground states of a quenched random Ising spin system with variable concentration of mixed nearest-neighbour exchange couplings +or-J on a square lattice (frustration model) are studied by a new
...
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