BM algorithms for noisy data and implicit regression modelling

Abstract

In this paper we consider the problem of finding a set of monomials $\mathcal O$ and a polynomial $f$ whose support is contained in $\mathcal O$, such that (1) $f$ is almost vanishing at a set of points $\mathbb X$ whose coordinates are not known exactly and (2) $\mathcal O$ exhibits structural stability, that is the model/design matrix associated to $\mathcal O$ is full rank for each set of points differing only slightly from $\mathbb X$. We review some numerical versions of the Buchberger-Möller (BM) algorithm for computing the set $\mathcal O$ and the polynomial $f$ and we present a variant, called LDP-LP, which integrates one of these methods with a classical statistical least squares algorithm for implicit regression from [1]. To illustrate the usefulness of these numerical BM algorithms, we review some of their application in the analyses of data sets for which standard techniques did not yield satisfactory results.