Suppose you are given some dataset 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 is bounded by some a priori specified v between 0 and 1. We propose a method to approach this problem by trying to estimate a function f which 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 preliminary theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabelled data.

}, author = {John C. Platt and Bernhard Sch{\"o}lkopf and John Shawe-Taylor and Alex J. Smola and Robert C. Williamson}, institution = {Microsoft Research}, month = {November}, number = {MSR-TR-99-87}, pages = {30}, title = {Estimating the Support of a High-Dimensional Distribution}, url = {http://research.microsoft.com/apps/pubs/default.aspx?id=69731}, year = {1999}, }