Chi-squared Amplification: Identifying Hidden Hubs

We consider the following general hidden hubs model: 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) probability distribution p0 ∼ N(0, σ 0); for each row in S, some 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 them efficiently. This model includes and significantly generalizes the planted Gaussian Submatrix Model, where the special entries are all in a k × k submatrix. There are two well-known barriers: if k ≥ c √ n lnn, just the row sums are sufficient to find S in the general model. For the submatrix problem, this can be improved by a √ lnn factor to k ≥ c√n by spectral methods or combinatorial methods. In the variant with p0 = ±1 (with probability 1/2 each) and p1 ≡ 1, neither barrier has been broken (in spite of much effort, particularly for the submatrix version, which is called the Planted Clique problem.) Here, we break both these barriers for the general model with Gaussian entries. We give a polynomial-time algorithm to identify all the hidden hubs with high probability for k ≥ n for some δ > 0, when σ 1 > 2σ 2 0 . The algorithm extends easily to the setting where planted entries might have different variances each at least as large as 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 general hidden hubs problem can be solved for k ≥ c√n(lnn)1/4, improving on the naive row sum-based method.