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116
Estimating the Support of a HighDimensional Distribution
, 1999
"... 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 between 0 and 1. We propo ..."
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Cited by 783 (29 self)
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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 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 d...
Smooth Discrimination Analysis
 Ann. Statist
, 1998
"... Discriminant analysis for two data sets in IR d with probability densities f and g can be based on the estimation of the set G = fx : f(x) g(x)g. We consider applications where it is appropriate to assume that the region G has a smooth boundary. In particular, this assumption makes sense if di ..."
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Cited by 146 (3 self)
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Discriminant analysis for two data sets in IR d with probability densities f and g can be based on the estimation of the set G = fx : f(x) g(x)g. We consider applications where it is appropriate to assume that the region G has a smooth boundary. In particular, this assumption makes sense if discriminant analysis is used as a data analytic tool. We discuss optimal rates for estimation of G. 1991 AMS: primary 62G05 , secondary 62G20 Keywords and phrases: discrimination analysis, minimax rates, Bayes risk Short title: Smooth discrimination analysis This research was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 373 "Quantifikation und Simulation okonomischer Prozesse", HumboldtUniversitat zu Berlin 1 Introduction Assume that one observes two independent samples X = (X 1 ; : : : ; X n ) and Y = (Y 1 ; : : : ; Ym ) of IR d valued i.i.d. observations with densities f or g, respectively. The densities f and g are unknown. An additional random variabl...
Theory of classification: A survey of some recent advances
, 2005
"... The last few years have witnessed important new developments in the theory and practice of pattern classification. We intend to survey some of the main new ideas that have led to these recent results. ..."
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Cited by 96 (3 self)
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The last few years have witnessed important new developments in the theory and practice of pattern classification. We intend to survey some of the main new ideas that have led to these recent results.
A classification framework for anomaly detection
 J. Machine Learning Research
, 2005
"... One way to describe anomalies is by saying that anomalies are not concentrated. This leads to the problem of finding level sets for the data generating density. We interpret this learning problem as a binary classification problem and compare the corresponding classification risk with the standard p ..."
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Cited by 71 (6 self)
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One way to describe anomalies is by saying that anomalies are not concentrated. This leads to the problem of finding level sets for the data generating density. We interpret this learning problem as a binary classification problem and compare the corresponding classification risk with the standard performance measure for the density level problem. In particular it turns out that the empirical classification risk can serve as an empirical performance measure for the anomaly detection problem. This allows us to compare different anomaly detection algorithms empirically, i.e. with the help of a test set. Based on the above interpretation we then propose a support vector machine (SVM) for anomaly detection. Finally, we establish universal consistency for this SVM and report some experiments which compare our SVM to other commonly used methods including the standard oneclass SVM. 1
Learning minimum volume sets
 J. Machine Learning Res
, 2006
"... Given a probability measure P and a reference measure µ, one is often interested in the minimum µmeasure set with Pmeasure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence region ..."
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Cited by 39 (7 self)
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Given a probability measure P and a reference measure µ, one is often interested in the minimum µmeasure set with Pmeasure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence regions. This paper addresses the problem of estimating minimum volume sets based on independent samples distributed according to P. Other than these samples, no other information is available regarding P, but the reference measure µ is assumed to be known. We introduce rules for estimating minimum volume sets that parallel the empirical risk minimization and structural risk minimization principles in classification. As in classification, we show that the performances of our estimators are controlled by the rate of uniform convergence of empirical to true probabilities over the class from which the estimator is drawn. Thus we obtain finite sample size performance bounds in terms of VC dimension and related quantities. We also demonstrate strong universal consistency and an oracle inequality. Estimators based on histograms and dyadic partitions illustrate the proposed rules. 1
How to compare different loss functions and their risks
, 2006
"... Many learning problems are described by a risk functional which in turn is defined by a loss function, and a straightforward and widelyknown approach to learn such problems is to minimize a (modified) empirical version of this risk functional. However, in many cases this approach suffers from subst ..."
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Cited by 25 (2 self)
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Many learning problems are described by a risk functional which in turn is defined by a loss function, and a straightforward and widelyknown approach to learn such problems is to minimize a (modified) empirical version of this risk functional. However, in many cases this approach suffers from substantial problems such as computational requirements in classification or robustness concerns in regression. In order to resolve these issues many successful learning algorithms try to minimize a (modified) empirical risk of a surrogate loss function, instead. Of course, such a surrogate loss must be “reasonably related ” to the original loss function since otherwise this approach cannot work well. For classification good surrogate loss functions have been recently identified, and the relationship between the excess classification risk and the excess risk of these surrogate loss functions has been exactly described. However, beyond the classification problem little is known on good surrogate loss functions up to now. In this work we establish a general theory that provides powerful tools for comparing excess risks of different loss functions. We then apply this theory to several learning problems including (costsensitive) classification, regression, density estimation, and density level detection.
Generalization error bounds in semisupervised classification under the cluster assumption
, 2007
"... ..."
Performance guarantee for individualized treatment rules
, 2009
"... S.1 The overfitting problem In this section, we discuss the problem with overfitting due to the potentially large number of pretreatment variables (and/or complex approximation space for Q0) mentioned in Section 4. Consider the setting in which we know that Q0 is linear in the {X, A} variables and ..."
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Cited by 22 (2 self)
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S.1 The overfitting problem In this section, we discuss the problem with overfitting due to the potentially large number of pretreatment variables (and/or complex approximation space for Q0) mentioned in Section 4. Consider the setting in which we know that Q0 is linear in the {X, A} variables and suppose that most coefficients are nonzero (some may be quite small). Then the least squares estimator using the best correct linear model (i.e. the model that contains and only contains variables with truly nonzero coefficients) may result in ITRs with poor Value as compared to the estimator from a more sparse model. Intuitively this occurs when the dimension of {X, A} is too large for the size of the data set. This is similar to the case of stepwise model selection; a solution is to select the model that balances the approximation error with the estimation error instead of keeping all of the correct terms (Massart [3]). Indeed the l1PLS method aims to estimate a parameter possessing small approximation error (i.e. the excess prediction error) and controlled sparsity (which is directly related to the estimation error). As a result, the ITR produced by l1PLS will more reliably have higher Value than the rule produced by the OLS (ordinary least squares) estimator constructed when the correct model is known but is too nonsparse relative to the size of the data set. In the following we use a simple simulation to support this argument. First we generate X = (X1,..., X12), where X1,..., X12 are mutually independent and each Xj is uniformly distributed on [−1, 1]. The treatment A is then generated independently of X from {−1, 1} with probability 1/2 each. The response R is generated from a normal distribution with mean Q0(X, A) = (1, X−12, A, X−12A)ϑ and variance 1, where X−12 = (X1,..., X11) and ϑ ∈ R24 is a vector parameter. We consider