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.
翻译:在图形中进行推断是几个机器学习问题的共同任务,例如图像分割、社区探测等。对于一个指定的非定向连接图形,我们处理从对每个边缘的单一腐败观察中完全恢复一个未知的结点的地面真相二进制标签的统计问题。这样的问题可以作为布林超立方的二次组合优化问题来表述,在这些问题之前,人们能够(极有可能和多式时间)完全恢复具有等离子测量数的图表的地面真相标签。对于一个特定非定向连接图形,我们处理从每个边缘的单一腐败观察中完全恢复一个未知的结点的地面真相二进制标签的统计问题。在这项工作中,我们用一个被称为“平方之和方之和”等级的放松等级来描述调和问题。根据关于精确恢复性改进的经验证据,我们把注意力集中在4级 SoS放松,并开始从图表的偏重角度来理解这种改进的起源,在节点(如完整的图表、完整的图表的完整图表、定期扩张)中,我们用硬质的直径的直度来显示我们图表的直径直径直径直径直的直径直径直径的直径的直径的直径的直度,我们的方法。我们通过直径直径直到直径直径直径直到直的直的直的直的直的直的直的直的直径直径直径直度,我们显示,我们展示的分辨率的分辨率的分辨率的解的分辨率的分辨率的分辨率的分辨率的分辨率的分辨率的分辨率的解的解的分辨率的分辨率的分辨率的分辨率的分辨率的分辨率的解的分辨率的解到直径直径直径直径直。