Identification of Mixtures of Discrete Product Distributions in Near-Optimal Sample and Time Complexity

We consider the problem of identifying, from statistics, a distribution of discrete random variables X1,ldots,Xn that is a mixture of k product distributions. The best previous sample complexity for n ∈ O(k) was (1/zeta)O(k2 log k) (under a mild separation assumption parameterized by zeta). The best known lower bound was exp(Omega(k)). It is known that n≥ 2k-1 is necessary and sufficient for identification. We show, for any n≥ 2k-1, how to achieve sample complexity and run-time complexity (1/zeta)O(k). We also extend the known lower bound of eOmega(k) to match our upper bound across a broad range of zeta. Our results are obtained by combining (a) a classic method for robust tensor decomposition, (b) a novel way of bounding the condition number of key matrices called Hadamard extensions, by studying their action only on flattened rank-1 tensors.