On Learning Mixtures of Heavy-Tailed Distributions

ABSTRACT

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 knownwith 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 [1, 15]. 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.Wealso show that for isotropic power-laws, exponential, and Gaussian distributions, our separation condition is optimal up to a constant factor.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Distribution functions

Additional Classification:
  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Probabilistic algorithms (including Monte Carlo)

General Terms:
Algorithms, Theory