Estimating the Support of a High-Dimensional Distribution-英文文献.pdf

Estimating the Support of a High-Dimensional Distribution z Bernhard Sch¨olkopf , John C. Platt , John Shawe-Taylory x , Alex J. Smola , Robert C. Williamsonx Microsoft Research Ltd, 1 Guildhall Street, Cambridge CB2 3NH, UK z Microsoft Research, 1 Microsoft Way, Redmond, WA, USA y Royal Holloway, University of London, Egham, UK x Department of Engineering, Australian National University, Canberra 0200, Australia bsc@, jplatt@, john@dcs.rhbnc.ac.uk Alex.Smola@.au, Bob.Williamson@.au 27 November 1999 Technical Report MSR-TR-99-87 Microsoft Research Microsoft Corporation One Microsoft Way Redmond, WA 98052 Abstract Suppose you are given some dataset drawn from an underlying proba- bility 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 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