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Inference and Sampling of K 33 -free Ising Models Valerii Likhosherstov 1 , Yury Maximov 1,2 , Michael Chertkov 1,2,3 1 Skolkovo Institute of Science and Technology, Moscow, Russia 2 Theoretical Division and Center for Nonlinear Studies, Los Alamos

  1. Inference and Sampling of K 33 -free Ising Models Valerii Likhosherstov 1 , Yury Maximov 1,2 , Michael Chertkov 1,2,3 1 Skolkovo Institute of Science and Technology, Moscow, Russia 2 Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA 3 Graduate Program in Applied Mathematics, University of Arizona, Tucson, AZ, USA June 11, 2019

  2. Definitions and Notations For a graph G = ( V , E ), | V | = N , zero-field Ising model is a distribution over S ∈ {− 1 , +1 } N defined as P ( S = X ) = 1 � Z exp( J e x v x w ) (1) e = { v , w }∈ E where { J e } e ∈ E are pairwise interactions and � � Z ( J ) = exp( J e x v x w ) (2) X ∈{− 1 , +1 } N e = { v , w }∈ E is a partition function . 2 / 7

  3. Problem Overview Question For which graphs G can we compute Z and sample from P ( S )? Fact (Barahona, 1982) Even when G is a two-level square grid, the task of finding Z is NP-hard. Fact (Jerrum & Sinclair, 1993) Even when J > 0, the task of finding Z is #P-complete. 3 / 7

  4. Problem Overview: Planar Zero-field Ising Models Planar zero-field Ising model - a case when G is planar. Theorem Given a planar zero-field Ising model, finding Z and sampling from 3 2 ) time. P ( S ) takes O ( N ◮ Theorem is due to (Kasteleyn, 1963; Wilson, 1997; Schraudolph & Kamenetsky, 2009; Thomas & Middleton, 2009; 2013). ◮ No self-contained description of the algorithm. ◮ Extension to arbitrary genus g with a factor of 4 g (Gallucio & Loebl, 1999). 4 / 7

  5. Algorithm Overview: Graph Decomposition Informal definition A tree of triconnected components T of graph G is a tree decomposition of G into triconnected graphs G t with shared edges. Theorem (Hopcroft & Tarjan, 1973) A tree of triconnected components is unique and can be obtained in O ( N + | E | ) . 5 / 7

  6. Algorithm Overview: Inference of K 33 -free Zero-field Ising Models Lemma (Hall, 1943) Graph G is K 33 -free if and only if its triconnected components are either planar or K 5 . Theorem Given a K 33 -free zero-field Ising model, finding Z and sampling 3 2 ) time. from P ( S ) takes O ( N 6 / 7

  7. Conclusions Main results: 3 ◮ Self-contained description of O ( N 2 ) inference and sampling of planar zero-field Ising models. 3 ◮ O ( N 2 ) inference and sampling of K 33 -free Ising models. ◮ Implementation of the algorithm https://github.com/ValeryTyumen/planar_ising . Poster: “Inference and Sampling of K 33 -free Ising Models”, Valerii Likhosherstov, Yury Maximov, Michael Chertkov. Pacific Ballroom #162 7 / 7

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