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We study the problem of partitioning a small sample of $n$ individuals from amixture of $k$ product distributions over a Boolean cube $\{0, 1\}^K$ accordingto their distributions. Each distribution is described by a vector of allelefrequencies in $\R^K$. Given two distributions, we use $\gamma$ to denote theaverage $\ell_2^2$ distance in frequencies across $K$ dimensions, whichmeasures the statistical divergence between them. We study the case assumingthat bits are independently distributed across $K$ dimensions. This workdemonstrates that, for a balanced input instance for $k = 2$, a certaingraph-based optimization function returns the correct partition with highprobability, where a weighted graph $G$ is formed over $n$ individuals, whosepairwise hamming distances between their corresponding bit vectors define theedge weights, so long as $K = \Omega(\ln n/\gamma)$ and $Kn = \tilde\Omega(\lnn/\gamma^2)$. The function computes a maximum-weight balanced cut of $G$, wherethe weight of a cut is the sum of the weights across all edges in the cut. Thisresult demonstrates a nice property in the high-dimensional feature space: onecan trade off the number of features that are required with the size of thesample to accomplish certain tasks like clustering.

Learning Balanced Mixtures of Discrete Distributions with Small Sample