On learning mixtures of heavy-tailed distributions

We consider the problem of learning mixtures of arbitrary symmetric distributions. We formulate sufficient separation conditions and present a learning algorithm with provable guarantees for mixtures of distributions that satisfy these separation conditions. Our bounds are independent of the variances of the distributions; to the best of our knowledge, there were no previous algorithms known with provable learning guarantees for distributions having infinite variance and/or expectation. For Gaussians and log-concave distributions, our results match the best known sufficient separation conditions by D. Achlioptas and F. McSherry (2005) and S. Vempala and G. Wang (2004). Our algorithm requires a sample of size O˜(dk), where d is the number of dimensions and k is the number of distributions in the mixture. We also show that for isotropic power-laws, exponential, and Gaussian distributions, our separation condition is optimal up to a constant factor.