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While several papers have investigated computationally and statisticallyefficient methods for learning Gaussian mixtures, precise minimax bounds fortheir statistical performance as well as fundamental limits in high-dimensionalsettings are not well-understood. In this paper, we provide precise informationtheoretic bounds on the clustering accuracy and sample complexity of learning amixture of two isotropic Gaussians in high dimensions under small meanseparation. If there is a sparse subset of relevant dimensions that determinethe mean separation, then the sample complexity only depends on the number ofrelevant dimensions and mean separation, and can be achieved by a simplecomputationally efficient procedure. Our results provide the first step of atheoretical basis for recent methods that combine feature selection andclustering.

Minimax Theory for High-dimensional Gaussian Mixtures with Sparse Mean Separation