Graph Matrices: Norm Bounds and Applications

TL;DR

This paper establishes nearly tight probabilistic norm bounds for graph matrices, a class of random matrices with dependent entries defined by a fixed graph shape, and demonstrates their applications in combinatorial optimization and proof complexity.

Contribution

The paper introduces a unified approach to bound the norms of graph matrices and applies these bounds to simplify proofs in existing literature and analyze complex systems.

Findings

Derived nearly tight probabilistic norm bounds for graph matrices

Applied bounds to simplify proofs in combinatorial optimization

Extended applications to proof systems and energy bounds

Abstract

In this paper, we derive nearly tight probabilistic norm bounds for a class of random matrices we call graph matrices. While the classical case of symmetric matrices with independent random entries (Wigner's matrices) is a special case, in general, the entries of our matrices will be dependent in a way that can be specified in terms of a fixed-size graph we refer to as the shape. For Wigner's matrices, this shape is $K_2$, the clique on 2 vertices. To prove our norm bounds, we use the trace power method. In a recent series of papers by Potechin and coauthors, graph matrices played a crucial role in proving average-case lower bounds for the Sum-of-Squares (SoS) hierarchy of proof systems, one of the most powerful, but difficult to analyze, techniques in combinatorial optimization. In particular, graph matrices played a crucial role in proving that low-degree SoS cannot refute the existence of a large clique in a random graph and proving that low-degree SoS cannot prove a tight lower bound on the ground state energy of the Sherrington-Kirkpatrick Hamiltonian. In this paper, we give several additional applications of graph matrices. We show that for several technical lemmas in the literature, while the original analyses were quite involved, we can give direct proofs using graph matrices and our norm bounds.