Learning Mixtures of Arbitrary Distributions over Large Discrete Domains

Опубликовано 27 июля 2016, 1:36

We give an algorithm for learning a mixture of unstructured distributions. This problem arises in various unsupervised learning scenarios, for example in learning topic models from a corpus of documents spanning several topics. We show how to learn the constituents of a mixture of k arbitrary distributions over a large discrete domain [n]={1,2,�,n\} and the mixture weights, using O(n polylog n) samples. (In the topic-model learning setting, the mixture constituents correspond to the topic distributions.) This task is information-theoretically impossible for k1 under the usual sampling process from a mixture distribution. However, there are situations (such as the above-mentioned topic model case) in which each sample point consists of several observations from the same mixture constituent. This number of observations, which we call the ``sampling aperture'', is a crucial parameter of the problem. We obtain the first bounds for this mixture-learning problem without imposing any assumptions on the mixture constituents. We show that efficient learning is possible exactly at the information-theoretically least-possible aperture of 2k-1. Thus, we achieve near-optimal dependence on n and optimal aperture. While the sample-size required by our algorithm depends exponentially on k, we prove that such a dependence is unavoidable when one considers general mixtures. A sequence of tools contribute to the algorithm, such as concentration results for random matrices, dimension reduction, moment estimations, and sensitivity analysis. Joint work with Leonard Schulman and Chaitanya Swamy.