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, ..., X_n that is a mixture of k product distributions. The best previous sample complexity for n ∈ O(k) was (1/ζ)^{O(k2 logk)} (under a mild separation assumption parameterized by ζ). The best known lower bound was exp(Ω(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/ζ)^O(k). We also extend the known lower bound of e^Ω(k) to match our upper bound across a broad range of ζ. 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.