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We consider the problem of identifying, from statistics, a distribution ofdiscrete random variables $X_1,\ldots,X_n$ that is a mixture of $k$ productdistributions. The best previous sample complexity for $n \in O(k)$ was$(1/\zeta)^{O(k^2 \log k)}$ (under a mild separation assumption parameterizedby $\zeta$). The best known lower bound was $\exp(\Omega(k))$. It is known that$n\geq 2k-1$ is necessary and sufficient for identification. We show, for any$n\geq 2k-1$, how to achieve sample complexity and run-time complexity$(1/\zeta)^{O(k)}$. We also extend the known lower bound of $e^{\Omega(k)}$ tomatch our upper bound across a broad range of $\zeta$. Our results are obtainedby combining (a) a classic method for robust tensor decomposition, (b) a novelway of bounding the condition number of key matrices called Hadamardextensions, by studying their action only on flattened rank-1 tensors.

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