Non-uniform spline recovery from small degree polynomial approximation

TL;DR

This paper addresses the problem of recovering sparse measures and non-uniform spline knots from polynomial approximations, providing theoretical bounds and a grid-free recovery method via semidefinite programming.

Contribution

It introduces a novel framework for measure and spline knot recovery from polynomial data, including quantitative bounds and a semidefinite programming approach.

Findings

Support recovery bounds under minimal separation

Support and amplitude error estimates

Grid-free knot localization via semidefinite programming

Abstract

We investigate the sparse spikes deconvolution problem onto spaces of algebraic polynomials. Our framework encompasses the measure reconstruction problem from a combination of noiseless and noisy moment measurements. We study a TV-norm regularization procedure to localize the support and estimate the weights of a target discrete measure in this frame. Furthermore, we derive quantitative bounds on the support recovery and the amplitudes errors under a Chebyshev-type minimal separation condition on its support. Incidentally, we study the localization of the knots of non-uniform splines when a Gaussian perturbation of their inner-products with a known polynomial basis is observed (i.e. a small degree polynomial approximation is known) and the boundary conditions are known. We prove that the knots can be recovered in a grid-free manner using semidefinite programming.