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Title: Robustly Learning a Gaussian: Getting Optimal Error, Efficiently
We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise — where an ε-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error O(ε) in the total variation distance, which is optimal up to a universal constant that is independent of the dimension. In the case where just the mean is unknown, our robustness guarantee is optimal up to a factor of and the running time is polynomial in d and 1/ε. When both the mean and covariance are unknown, the running time is polynomial in d and quasipolynomial in 1/ε. Moreover all of our algorithms require only a polynomial number of samples. Our work shows that the same sorts of error guarantees that were established over fifty years ago in the one-dimensional setting can also be achieved by efficient algorithms in high-dimensional settings.  more » « less
Award ID(s):
1741137
NSF-PAR ID:
10079743
Author(s) / Creator(s):
; ; ; ; ;
Date Published:
Journal Name:
Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms
ISSN:
1071-9040
Page Range / eLocation ID:
2683-2702
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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