A Simple Algorithm for Clustering Discrete Distributions

We propose a simple, projection-based algorithm for clustering mixtures of discrete (Bernoulli) distributions. Unlike previous approaches that rely on coordinate-specific ``combinatorial projections,'' our algorithm is rotationally invariant and works by projecting samples onto approximate centers obtained via a $k$-means computation on the best rank-$k$ approximation of the data matrix. This resolves a conjecture of McSherry on the existence of such geometric algorithms for discrete distributions. The same algorithm also applies to continuous distributions such as high-dimensional Gaussians, providing a unified approach across distribution types. We prove that the algorithm succeeds under a natural separation condition on the cluster centers.