Structure Learning of H-colorings

We study the structure learning problem for graph homomorphisms, commonly referred to as Hcolorings, including the weighted case which corresponds to spin systems with hard constraints. The learning problem is as follows: for a fixed (and known) constraint graphH with q colors and an unknown graph G = (V,E) with n vertices, given uniformly random H-colorings of G, how many samples are required to learn the edges of the unknown graph G? We give a characterization of H for which the problem is identifiable for everyG, i.e., we can learn G with an infinite number of samples. We focus particular attention on the case of proper vertex q-colorings of graphs of maximum degree d where intriguing connections to statistical physics phase transitions appear. We prove that when q > d the problem is identifiable and we can learn G in poly(d, q) × O(n logn) time. In contrast for soft-constraint systems, such as the Ising model, the best possible running time is exponential in d. When q ≤ d we prove that the problem is not identifiable, and we cannot hope to learn G. When q < d − √ d + Θ(1) we prove that even learning an equivalent graph (any graph with the same set of H-colorings) is computationally hard—sample complexity is exponential in n in the worst-case. For the q-colorings problem, the threshold for efficient learning seems to be connected to the uniqueness/nonuniqueness phase transition at q = d. We explore this connection for generalH-colorings and prove that under a well-known condition in statistical physics, known as Dobrushin uniqueness condition, we can learn G in poly(d, q) ×O(n2 logn) time. Georgia Institute of Technology, USA. Email: {ablanca3,chenzongchen,vigoda}@gatech.edu. Research supported in part by NSF grants CCF-1617306 and CCF-1563838. University of Rochester, USA. Email: [email protected]. Research supported in part by NSF grant CCF-1318374.