Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
Optimal outlier removal in high-dimensional
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Learning Halfspaces with Malicious Noise
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Learning Halfspaces with Malicious Noise
The Journal of Machine Learning Research
Optimal consensus set for annulus fitting
DGCI'11 Proceedings of the 16th IAPR international conference on Discrete geometry for computer imagery
O(n 3logn) time complexity for the optimal consensus set computation for 4-Connected Digital Circles
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
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We study the problem of finding an outlier-free subset of a set of points (or a probability distribution) in n-dimensional Euclidean space. As in [BFKV 99], a point x is defined to be a β-outlier if there exists some direction w in which its squared distance from the mean along w is greater than β times the average squared distance from the mean along w. Our main theorem is that for any ε 0, there exists a (1 - ε) fraction of the original distribution that has no O(n/ε(b + logn/ε))-outliers, improving on the previous bound of O(n7b/ε). This is asymptotically the best possible, as shown by a matching lower bound. The theorem is constructive, and results in a 1/1-ε approximation to the following optimization problem: given a distribution µ (i.e. the ability to sample from it), and a parameter ε 0, find the minimum β for which there exists a subset of probability at least (1 - ε) with no β-outliers.