Statistics Seminar :: De-noising signals of unknown local structure

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We focus on recovering a non-parametric signal observed in Gaussian noise along an equidistant $d$-dimensional grid. Typical nonparametric regression estimates are aimed at recovering smooth signals $f$ and utilize the fact that such an $f$ is locally well approximated by an algebraic polynomial; thus, we can recover $f$ at a point from nearby observations as if $f$ were a polynomial, which is an easy-to-estimate entity. The research to be outlined in the talk is motivated by the simple observation that polynomials are not the only ``easy to estimate'' entities; the latter property is shared, e.g., by products of polynomials and harmonic oscillations. However, the traditional nonparametric techniques fail to recover signals of the latter type, except for the case when the frequencies of oscillations are known in advance or are low; the reason is that a good estimate of a product of algebraic polynomial and a harmonic oscillation requires a priori knowledge of the frequencies. In the talk, we present a kind of universal nonparametric estimate, based on Linear Programming, with the following property: whenever the signal to be estimated admits an ``easy to estimate'' local approximation $g$ (one which can be recovered well by an {sl unknown in advance} convolution filter), as it is the case for smooth signals, products of smooth signals and harmonic oscillations, etc., our estimate recovers the signal nearly as well as a hypothetical estimate utilizing this ``existing in the nature'' (and unknown in reality) filter. We demonstrate also that the family of ``easy to estimate'' signals is pretty rich, specifically, contains a number of important generic families of signals and is closed with respect to basic operations like modulation $g(x)mapsto g(x)cos(omega^Tx+phi)$ and summation.


  • Workflow Status: Published
  • Created By: Barbara Christopher
  • Created: 10/08/2010
  • Modified By: Fletcher Moore
  • Modified: 10/07/2016


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