Learning Mixtures of Arbitrary Distributions over Large Discrete Domains
Abstract
We give an algorithm for learning a mixture of {\em unstructured} distributions. This problem arises in various unsupervised learning scenarios, for example in learning {\em 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 $[n]=\{1,2,\dots,n\}$ 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 $k>1$ 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 {\em "sampling aperture"}, is a crucial parameter of the problem. We obtain the {\em first} bounds for this mixture-learning problem {\em 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 {\em 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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2012
- DOI:
- 10.48550/arXiv.1212.1527
- arXiv:
- arXiv:1212.1527
- Bibcode:
- 2012arXiv1212.1527R
- Keywords:
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- Computer Science - Machine Learning;
- Computer Science - Data Structures and Algorithms;
- F.2.2;
- G.2;
- G.3
- E-Print:
- Update of previous version with improved aperture and sample-size lower bounds