Adjusted least squares fitting of algebraic hypersurfaces

Konstantin Usevich (1) , Ivan Markovsky (2)

1 GIPSA-CICS - GIPSA - Communication Information and Complex Systems
2 ELEC - Department of Fundamental Electricity and Instrumentation

We consider the problem of fitting a set of points in Euclidean space by an algebraic hypersurface. We assume that points on a true hypersurface, described by a polynomial equation, are corrupted by zero mean independent Gaussian noise, and we estimate the coefficients of the true polynomial equation. The adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator. The adjusted least squares estimator is based on constructing a quasi-Hankel matrix, which is a bias-corrected matrix of moments. For the case of unknown noise variance, the estimator is defined as a solution of a polynomial eigenvalue problem. In this paper, we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation.

Mots clés: Quasi-Hankel matrix; errors-in-variables; hypersurface fitting; statistical estimation; curve fitting; Hermite polynomials; affine invariance; subspace clustering

Domaines: Traitement du signal et de l'image [eess.SP]; Analyse numérique [math.NA]; Géométrie algorithmique [cs.CG]; Calcul [stat.CO]

Konstantin Usevich, Ivan Markovsky. Adjusted least squares fitting of algebraic hypersurfaces. Linear Algebra and its Applications, 2015, 502, pp.243-274. ⟨10.1016/j.laa.2015.07.023⟩. ⟨hal-01093214⟩