Random Tensors and Planted Cliques

TL;DR

This paper extends understanding of the spectral properties of r-parity tensors of random graphs, showing their 2-norm is bounded by a function of n and r, and links this to the planted clique problem.

Contribution

It proves a bound on the 2-norm of r-parity tensors for random graphs, generalizing previous results and connecting tensor spectral norms to the planted clique problem.

Findings

2-norm of r-parity tensor is at most f(r)√n log^{O(r)} n

Established a connection between the planted clique problem and tensor norm approximation

Extended known bounds from r=3 to general r

Abstract

The r-parity tensor of a graph is a generalization of the adjacency matrix, where the tensor's entries denote the parity of the number of edges in subgraphs induced by r distinct vertices. For r=2, it is the adjacency matrix with 1's for edges and -1's for nonedges. It is well-known that the 2-norm of the adjacency matrix of a random graph is O(\sqrt{n}). Here we show that the 2-norm of the r-parity tensor is at most f(r)\sqrt{n}\log^{O(r)}n, answering a question of Frieze and Kannan who proved this for r=3. As a consequence, we get a tight connection between the planted clique problem and the problem of finding a vector that approximates the 2-norm of the r-parity tensor of a random graph. Our proof method is based on an inductive application of concentration of measure.