A Thorough View of Exact Inference in Graphs from the Degree-4 Sum-of-Squares Hierarchy

TL;DR

This paper investigates the effectiveness of the degree-4 sum-of-squares hierarchy for exact graph inference, revealing its connection to algebraic connectivity and deriving a new Cheeger-type bound.

Contribution

It provides a graph-theoretic analysis of the degree-4 SoS relaxation, linking dual solutions to Johnson and Kneser graphs and introducing a novel algebraic connectivity bound.

Findings

Degree-4 SoS relaxation improves exact recoverability.

Dual solutions relate to Johnson and Kneser graph edge weights.

New Cheeger-type bound for algebraic connectivity of signed graphs.

Abstract

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.