从四次平方和层次透视图中的精确推理
A Thorough View of Exact Inference in Graphs from the Degree-4 Sum-of-Squares Hierarchy
摘要：
在图中执行推理是若干机器学习问题中的常见任务，例如，图像分割、社区发现等。对于一个给定的无向连通图，我们解决的统计问题，准确地恢复一个未知的地面真理二进制标记的节点从一个单一的损坏的观察每个边缘。这样的问题可以被公式化为布尔超立方体上的二次组合优化问题，其中之前已经示出了可以（以高概率并且在多项式时间内）精确地恢复具有相对于节点的数量增长的等周数的图的地面真实标记（例如，完全图，正则扩展器）。在这项工作中，我们应用一个强大的层次结构的松弛，被称为平方和（SoS）的层次结构，组合问题。出于经验证据的改进，在确切的可恢复性，我们集中我们的注意力在度-4 SoS放松，并着手了解这种改进的起源从图论的角度来看。我们表明，放松问题的对偶的解决方案是有关的约翰逊和Kneser图，其中的权重满足SoS约束，直观地允许输入图，以增加其代数连通性的边缘权重。最后，作为我们的分析的副产品，我们得到了一个新的Cheeger型下界的代数连通图的符号边权重。
英文摘要：
Performing inference in graphs is a common task within several machine learning problems, e.g., image segmentation, community detection, among others. For a given undirected connected graph, we tackle the statistical problem of exactly recovering an unknown ground-truth binary labeling of the nodes from a single corrupted observation of each edge. Such problem can be formulated as a quadratic combinatorial optimization problem over the boolean hypercube, where it has been shown before that one can (with high probability and in polynomial time) exactly recover the ground-truth labeling of graphs that have an isoperimetric number that grows with respect to the number of nodes (e.g., complete graphs, regular expanders). In this work, we apply a powerful hierarchy of relaxations, known as the sum-of-squares (SoS) hierarchy, to the combinatorial problem. Motivated by empirical evidence on the improvement in exact recoverability, we center our attention on the degree-4 SoS relaxation and set out to understand the origin of such improvement from a graph theoretical perspective. We show that the solution of the dual of the relaxed problem is related to finding edge weights of the Johnson and Kneser graphs, where the weights fulfill the SoS constraints and intuitively allow the input graph to increase its algebraic connectivity. Finally, as byproduct of our analysis, we derive a novel Cheeger-type lower bound for the algebraic connectivity of graphs with signed edge weights.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
