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(Neural Computation. 2001;13:1443-1471.)
© 2001 The MIT Press

Estimating the Support of a High-Dimensional Distribution

Bernhard Schölkopf

Microsoft Research Ltd, Cambridge CB2 3NH, U.K.

John C. Platt

Microsoft Research, Redmond, WA 98052, U.S.A.

John Shawe-Taylor

Royal Holloway, University of London, Egham, Surrey TW20 OEX, U.K.

Alex J. Smola

Department of Engineering, Australian National University, Canberra 0200, Australia

Robert C. Williamson

Department of Engineering, Australian National University, Canberra 0200, Australia

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.




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