Variantes sur un th\'eor\`eme de Cand\`es, Romberg et Tao

TL;DR

This paper explores variations of a theorem related to signal reconstruction from sparse frequency data, emphasizing classical Fourier analysis over random matrices, and discusses conditions for nearly certain reconstruction.

Contribution

It introduces modifications to the CRT theorem using classical Fourier analysis, expanding understanding of signal reconstruction without relying on random matrices.

Findings

Reconstruction probability approaches 1 with large C

Classical Fourier analysis can replace random matrix methods

Conditions for reconstructing all signals with s points are discussed

Abstract

Variations on a theorem of Cand\`es, Romberg and Tao The CRT theorem reconstructs a signal from a sparse set of frequencies, a paradigm of Compressed sensing. The signal is assumed to be carried by a small number of points, s, in a large cyclic set, of order N; the frequencies consist of C s log N points chosen randomly in Z/N Z; the reconstruction is based on a minimal extrapolation in the Wiener algebra of Z/N Z of the restriction of the Fourier transform of the signal to the chosen set of frequencies. The probability of reconstructing the signal is nearly 1 when C is large. The statement should be modified when we want all signals carried by s points to be reconstructed in that way. The CRT approach is based on random matrices, here the approach is classical Fourier analysis.