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Nearly tight sample complexity bounds for learning mixtures of Gaussians via sample compression schemes
Hassan Ashtiani · Shai Ben-David · Nicholas Harvey · Christopher Liaw · Abbas Mehrabian · Yaniv Plan

Wed Dec 05 07:45 AM -- 09:45 AM (PST) @ Room 210 #100

We prove that ϴ(k d^2 / ε^2) samples are necessary and sufficient for learning a mixture of k Gaussians in R^d, up to error ε in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that O(k d / ε^2) samples suffice, matching a known lower bound.

The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in R^d has an efficient sample compression.

Author Information

Hassan Ashtiani (McMaster University)
Shai Ben-David (University of Waterloo)
Nicholas Harvey (University of British Columbia)
Christopher Liaw (University of British Columbia)
Abbas Mehrabian (Mcgill University)
Yaniv Plan (University of British Columbia)

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