The Hidden Hubs Problem

We introduce the following hidden hubs model H(n, k, σ0, σ1): the input is an n × n random matrix A with a subset S of k special rows (hubs); entries in rows outside S are generated from the Gaussian distribution p0 = N(0, σ 0), while for each row in S, an unknown subset of k of its entries are generated from p1 = N(0, σ 1), σ1 > σ0, and the rest of the entries from p0. The special rows with higher variance entries can be viewed as hidden higher-degree hubs. The problem we address is to identify the hubs efficiently. The planted Gaussian Submatrix Model is the special case where the higher variance entries must all lie in a k × k submatrix. If k ≥ c √ n lnn, just the row sums are sufficient to find S in the general model. For the Gaussian submatrix problem (and the related planted clique problem), this can be improved by a √ lnn factor to k ≥ c √ n by spectral or combinatorial methods. We give a polynomial-time algorithm to identify all the hidden hubs with high probability for k ≥ n0.5−δ for some δ > 0, when σ 1 > 2σ 0 . The algorithm extends to the setting where planted entries might have different variances, each at least σ 1 . We also show a nearly matching lower bound: for σ 1 ≤ 2σ 0 , there is no polynomial-time Statistical Query algorithm for distinguishing between a matrix whose entries are all from N(0, σ 0) and a matrix with k = n 0.5−δ hidden hubs for any δ > 0. The lower bound as well as the algorithm are related to whether the chi-squared distance of the two distributions diverges. At the critical value σ 1 = 2σ 2 0 , we show that the hidden hubs problem can be solved for k ≥ c √ n(lnn), improving on the naive row sum-based method.