Braunstein Alfredo, Catania Giovanni, Dall'Asta Luca
Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129, Torino, Italy.
Italian Institute for Genomic Medicine, Via Nizza 52, 10126, Torino, Italy.
Phys Rev Lett. 2019 Jul 12;123(2):020604. doi: 10.1103/PhysRevLett.123.020604.
Computing marginal distributions of discrete or semidiscrete Markov random fields (MRFs) is a fundamental, generally intractable problem with a vast number of applications in virtually all fields of science. We present a new family of computational schemes to approximately calculate the marginals of discrete MRFs. This method shares some desirable properties with belief propagation, in particular, providing exact marginals on acyclic graphs, but it differs with the latter in that it includes some loop corrections; i.e., it takes into account correlations coming from all cycles in the factor graph. It is also similar to the adaptive Thouless-Anderson-Palmer method, but it differs with the latter in that the consistency is not on the first two moments of the distribution but rather on the value of its density on a subset of values. The results on finite-dimensional Isinglike models show a significant improvement with respect to the Bethe-Peierls (tree) approximation in all cases and with respect to the plaquette cluster variational method approximation in many cases. In particular, for the critical inverse temperature β_{c} of the homogeneous hypercubic lattice, the expansion of (dβ_{c})^{-1} around d=∞ of the proposed scheme is exact up to d^{-4} order, whereas the latter two are exact only up to d^{-2} order.
计算离散或半离散马尔可夫随机场（MRF）的边际分布是一个基本问题，通常难以处理，在几乎所有科学领域都有大量应用。我们提出了一种新的计算方案族，用于近似计算离散MRF的边际分布。该方法与信念传播有一些理想的特性，特别是在无环图上能提供精确的边际分布，但它与信念传播的不同之处在于它包含一些循环修正；即它考虑了因子图中所有循环产生的相关性。它也类似于自适应的 Thouless-Anderson-Palmer 方法，但不同之处在于一致性不是关于分布的前两个矩，而是关于其在一组值的子集上的密度值。有限维类伊辛模型的结果表明，在所有情况下，相对于贝塞 - 皮尔斯（树）近似有显著改进，在许多情况下相对于面元团簇变分法近似也有显著改进。特别是，对于均匀超立方晶格的临界逆温度βₑ，所提出方案在d = ∞附近对(dβₑ)⁻¹的展开式在d⁻⁴阶之前是精确的，而后两者仅在d⁻²阶之前是精确的。
