{"doi":"10.3389/fams.2020.00031","title":"A Witness Function Based Construction of Discriminative Models Using Hermite Polynomials","abstract":"In machine learning, we are given a dataset of the form {(x j , y j )} M j=1 , drawn as i.i.d. samples from an unknown probability distribution ; the marginal distribution for the x j 's being * , and the marginals of the k th class * k (x) possibly overlapping. We address the problem of detecting, with a high degree of certainty, for which x we have * k (x) > * 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.","journal":"Frontiers in Applied Mathematics and Statistics","year":2020,"id":107460,"datarank":0.0,"base_score":0.0,"endowment":0.0,"self_citation_contribution":0.0,"citation_network_contribution":0.0,"self_endowment_contribution":0.0,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":6,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9592,"is_data_producer":false,"deposit_databanks":null,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2020-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":416446,"name":"Xiuyuan Cheng","orcid":"0000-0002-1034-6019","position":1,"is_corresponding":false},{"id":516489,"name":"Alexander Cloninger","orcid":"0000-0002-1423-9624","position":2,"is_corresponding":false},{"id":516488,"name":"H. N. Mhaskar","orcid":"0000-0001-8793-4321","position":0,"is_corresponding":true}],"reference_count":56,"raw_metadata":null,"created_at":"2026-07-18T23:12:34.898963Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}