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In machine learning, we are given a dataset of the form$\{(\mathbf{x}_j,y_j)\}_{j=1}^M$, drawn as i.i.d. samples from an unknownprobability distribution $\mu$; the marginal distribution for the$\mathbf{x}_j$'s being $\mu^*$. We propose that rather than using a positivekernel such as the Gaussian for estimation of these measures, using anon-positive kernel that preserves a large number of moments of these measuresyields an optimal approximation. We use multi-variate Hermite polynomials forthis purpose, and prove optimal and local approximation results in a supremumnorm in a probabilistic sense. Together with a permutation test developed withthe same kernel, we prove that the kernel estimator serves as a `witnessfunction' in classification problems. Thus, if the value of this estimator at apoint $\mathbf{x}$ exceeds a certain threshold, then the point is reliably in acertain class. This approach can be used to modify pretrained algorithms, suchas neural networks or nonlinear dimension reduction techniques, to identifyin-class vs out-of-class regions for the purposes of generative models,classification uncertainty, or finding robust centroids. This fact isdemonstrated in a number of real world data sets including MNIST, CIFAR10,Science News documents, and LaLonde data sets.

A witness function based construction of discriminative models using Hermite polynomials