# A witness function based construction of discriminative models using Hermite polynomials

arXiv: https://arxiv.org/abs/1901.02975 · HTML: https://mlsanity.com/paper/1901.02975/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2019.

## Built on by (most cited first, 4 of 4)

- [Local approximation of operators](https://mlsanity.com/paper/2202.06392/) (2022)
- [Kernel distance measures for time series, random fields and other structured data](https://mlsanity.com/paper/2109.14752/) (2021)
- [Approximation by non-symmetric networks for cross-domain learning](https://mlsanity.com/paper/2305.03890/) (2023)
- [Active Learning Classification from a Signal Separation Perspective](https://mlsanity.com/paper/2502.16425/) (2025)

## Abstract (arXiv)

In machine learning, we are given a dataset of the form (xj,yj)j=1M, drawn as i.i.d. samples from an unknown probability distribution μ; the marginal distribution for the xj's being μ^*. 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.