Adaptive Hausdorff Estimation of Density Level Sets

Consider the problem of estimating the -level set G = {x : f(x) } of an unknown d-dimensional density function f based on n independent observations X1, . . . , Xn from the density. This problem has been addressed under global error criteria related to the symmetric set difference. However, in certain applications such as anomaly detection and clustering, a more uniform mode of convergence is desirable to ensure that the estimated set is close to the target set everywhere. The Hausdorff error criterion provides this degree of uniformity and hence is more appropriate in such situations. It is known that the minimax optimal rate of convergence for the Hausdorff error is (n/ log n)-1/(d+2) for level sets with Lipschitz boundaries, where the parameter characterizes the regularity of the density around the level of interest. However, the estimators proposed in previous work achieve this rate for very restricted classes of sets (e.g. the boundary fragment and star-shaped sets) that effect...

• Type: Conference
• Year: 2008
• Where: COLT
• Authors: Aarti Singh, Robert Nowak, Clayton Scott
• Keywords: COLT 2008, Hausdorff Error, Machine Learning, Minimax Optimal Rates, Multiple Connected Components