Robust Estimators in High Dimensions without the Computational Intractability
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1604.06443.pdf
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Author(s)
Diakonikolas, Ilias
Kane, Daniel M.
Stewart, Alistair
Kamath, Gautam Chetan
Li, Jerry Zheng
Moitra, Ankur
Date Issued
December 2016
Journal
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Diakonikolas, Ilias, et al. "Robust Estimators in High Dimensions without the Computational Intractability." 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), 9-11 October, New Brunswick, New Jersey, 2016, IEEE, 2016, pp. 655–64.
Version
Original manuscript
Abstract
We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an epsilon fraction of the samples. Such questions have a rich history spanning statistics, machine learning and theoretical computer science. Even in the most basic settings, the only known approaches are either computationally inefficient or lose dimension dependent factors in their error guarantees. This raises the following question: Is high-dimensional agnostic distribution learning even possible, algorithmically? In this work, we obtain the first computationally efficient algorithms for agnostically learning several fundamental classes of high-dimensional distributions: (1) a single Gaussian, (2) a product distribution on the hypercube, (3) mixtures of two product distributions (under a natural balancedness condition), and (4) mixtures of k Gaussians with identical spherical covariances. All our algorithms achieve error that is independent of the dimension, and in many cases depends nearly-linearly on the fraction of adversarially corrupted samples. Moreover, we develop a general recipe for detecting and correcting corruptions in high-dimensions, that may be applicable to many other problems.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1109/FOCS.2016.85
