Probabilistic Kernel Support Vector Machines
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We propose a probabilistic enhancement of standard kernel Support Vector Machines for binary classification, in order to address the case when, along with given data sets, a description of uncertainty (e.g., error bounds) may be available on each datum. In the present paper, we specifically consider Gaussian distributions to model uncertainty. Thereby, our data consist of pairs (x_i,Σ_i), i∈{1,...,N}, along with an indicator y_i∈{-1,1} to declare membership in one of two categories for each pair. These pairs may be viewed to represent the mean and covariance, respectively, of random vectors ξ_i taking values in a suitable linear space (typically R^n). Thus, our setting may also be viewed as a modification of Support Vector Machines to classify distributions, albeit, at present, only Gaussian ones. We outline the formalism that allows computing suitable classifiers via a natural modification of the standard "kernel trick." The main contribution of this work is to point out a suitable kernel function for applying Support Vector techniques to the setting of uncertain data for which a detailed uncertainty description is also available (herein, "Gaussian points").
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