We study the following structure learning problem for H -colorings. For a fixed (and known) constraint graph H with q colors, given access to uniformly random H -colorings of an unknown graph G=(V,E) , how many samples are required to learn the edges of G ? We give a characterization of the constraint graphs H for which the problem is identifiable for every G and show that there are identifiable constraint graphs for which one cannot hope to learn every graph G efficiently. We provide refined results for the case of proper vertex q -colorings of graphs of maximum degree d . In particular, we prove that in the tree uniqueness region (i.e., when q≤ d ), the problem is identifiable and we can learn G in poly( d,q )× O(n 2 log n ) time. In the tree non-uniqueness region (i.e., when q≤ d), we show that the problem is not identifiable and thus G cannot be learned. Moreover, when q ≤ d - √d + Θ (1), we establish 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. We further explore the connection between the efficiency/hardness of the structure learning problem and the uniqueness/non-uniqueness phase transition for general H -colorings and prove that under a well-known uniqueness condition in statistical physics, we can learn G in poly( d,q )× O(n 2 log n ) time.

中文翻译：

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H色的结构学习

我们研究以下结构学习问题H-着色。对于固定（已知）的约束图H和q颜色，可以访问均匀随机H-未知图形的着色G=(V,E)，需要多少样本来学习边缘G? 我们给出了约束图的特征H对于每个人来说，问题都是可识别的G并表明存在可识别的约束图，人们不可能希望学习每个图G有效率的。我们为正确顶点的情况提供了精确的结果q- 最大程度的图形着色d. 特别地，我们证明了在树的唯一性区域（即，当q≤d)，问题是可识别的，我们可以学习G在聚（d,q)× O(n2日志n） 时间。在树的非唯一性区域（即当 q≤d 时），我们表明问题不可识别，因此G无法学习。此外，当q≤d- √d + Θ (1)，我们确定即使学习一个等价图（任何具有相同集合的图H-colorings) 计算困难——样本复杂度是指数级的n在最坏的情况下。我们进一步探讨了结构学习问题的效率/难度与一般的唯一性/非唯一性相变之间的联系H-着色并证明在统计物理学中众所周知的唯一性条件下，我们可以学习G在聚（d,q)× O(n2日志n） 时间。