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Estimating the support of a high-dimensional distribution.


Schölkopf,  B
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Schölkopf, B., Platt JC, Shawe-Taylor J, Smola, A., & Williamson, R. (2001). Estimating the support of a high-dimensional distribution. Neural Computation, 13(7), 1443-1471. doi:10.1162/089976601750264965.

Suppose you are given some data set drawn from an underlying probability distribution P and you want to estimate a “simple” subset S of input space such that the probability that a test point drawn from P lies outside of S equals some a priori specified value between 0 and 1. We propose a method to approach this problem by trying to estimate a function f that is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. The expansion coefficients are found by solving a quadratic programming problem, which we do by carrying out sequential optimization over pairs of input patterns. We also provide a theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabeled data.