De-Noising Signals of Unknown Local Structure

• Activity Number: 39
• Type: Invited
• Date/Time: Sunday, August 3, 2003 : 4:00 PM to 5:50 PM
• Sponsor: IMS
• Author(s): Arkadi Nemirovskii*+
• Companies: William Davidson Faculty of Industrial Engineering and Management
• Keywords: nonparametric regression ; adaptive asymptotically minmax optimal regression estimates

Abstract

We focus on recovering a nonparametric multivariate signal observed in the standard Gaussian noise along an equidistant grid. The standard nonparametric regression estimators are aimed at recovering smooth signals and heavily exploit the fact that locally such a signal can be well-approximated by a polynomial, which is "easy to estimate"--a polynomial of given degree can be recovered at a parametric rate by a known in advance linear estimator. We define a wider family of "easy to estimate signals"--those which can be recovered at a parametric rate by convolving the observations with appropriate kernels and demonstrate that these signals can be recovered at a nearly parametric rate even in the case when the corresponding "good kernels" are not known in advance. Based on this result, we develop asymptotically nearly optimal, in the minmax sense, recovering routines for multivariate "modulated" signals (sums of a fixed number of products of harmonic oscillations of unknown frequencies and signals from a Sobolev class with unknown smoothness parameters), for univariate signals satisfying differential inequalities with unknown differential operators of a given order and for some other nontraditional classes of regression functions.