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Local nearest neighbour classification with applications to semi-supervised learning

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In this talk I will present a new asymptotic expansion for the global excess risk of a local k-nearest neighbour classifier, where the choice of k may depend upon the test point. This expansion elucidates conditions under which the dominant contribution to the excess risk comes from the locus of points at which each class label is equally likely to occur. Moreover, I will present results which show that, provided the d-dimensional marginal distribution of the features has a finite ρth moment for some ρ>4 (as well as other regularity conditions), a local choice of k can yield a rate of convergence of the excess risk of O(n^(-4/(d+4))), where n is the sample size, whereas for the standard k-nearest neighbour classifier, our theory would require d≥5 and ρ>4d/(d−4) finite moments to achieve this rate. Motivated by these results, I will introduce a new k-nearest neighbour classifier for semi-supervised learning problems, where the unlabelled data are used to obtain an estimate of the marginal feature density, and fewer neighbours are used for classification when this density estimate is small.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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