Learning mixtures of arbitrary distributions over large discrete domains

Yuval Rabani, Leonard J. Schulman, Chaitanya Swamy

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

17 Scopus citations

Abstract

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(npolylog n) samples. (In the topic-model learning setting, the mixture constituents correspond to the topic distributions.) This task is information-theoretically impossible for k > 1 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.

Original languageAmerican English
Title of host publicationITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science
PublisherAssociation for Computing Machinery
Pages207-223
Number of pages17
ISBN (Print)9781450322430
DOIs
StatePublished - 2014
Event2014 5th Conference on Innovations in Theoretical Computer Science, ITCS 2014 - Princeton, NJ, United States
Duration: 12 Jan 201414 Jan 2014

Publication series

NameITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science

Conference

Conference2014 5th Conference on Innovations in Theoretical Computer Science, ITCS 2014
Country/TerritoryUnited States
CityPrinceton, NJ
Period12/01/1414/01/14

Keywords

  • Convex geometry
  • Linear programming
  • Mixture learning
  • Moment methods
  • Randomized algorithms
  • Spectral techniques
  • Topic models

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