A Witness Function Based Construction of Discriminative Models Using Hermite Polynomials - Mhaskar, Hrushikesh N.; Cheng, Xiuyuan; Cloninger, Alexander (Frontiers in Applied Mathematics and Statistics, 2020)
In machine learning, we are given a dataset of the form $\{(x_j,y_j)\}_{j=1}^M$, drawn as i.i.d. samples from an unknown probability distribution $μ$; the marginal distribution for the $x_j$'s being $μ^*$, and the marginals of the $k^{th}$ class $μ_k^*(x)$ possibly overlapping. We address the problem of detecting, with a high degree of certainty, for which $x$ we have $μ_k^*(x)>μ_i^*(x)$ for all $i≠ k$. We propose that rather than using a positive kernel such as the Gaussian for estimation of these measures, using a non-positive kernel that preserves a large number of moments of these measures yields an optimal approximation. We use multi-variate Hermite polynomials for this purpose, and prove optimal and local approximation results in a supremum norm in a probabilistic sense. Together with a permutation test developed with the same kernel, we prove that the kernel estimator serves as a ``witness function'' in classification problems. Thus, if the value of this estimator at a point $x$ exceeds a certain threshold, then the point is reliably in a certain class. This approach can be used to modify pretrained algorithms, such as neural networks or nonlinear dimension reduction techniques, to identify in-class vs out-of-class regions for the purposes of generative models, classification uncertainty, or finding robust centroids. This fact is demonstrated in a number of real world data sets including MNIST, CIFAR10, Science News documents, and LaLonde data sets.