Random Tensors and Planted Cliques

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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 fr\sqrt{n}\log^{Or}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.

Author: S. Charles Brubaker; Santosh Vempala

Source: https://archive.org/