arXiv subjects

Maxime Ferreira Da Costa

Publications and source records attributed to Maxime Ferreira Da Costa.

21 records · Page 2Linked to original sources

arXiv2018

A Tight Converse to the Spectral Resolution Limit via Convex Programming

It is now well understood that convex programming can be used to estimate the frequency components of a spectrally sparse signal from $2m+1$ uniform temporal measurements. It is conjectured that a phase transition on the success of the total-variation regularization occurs when the distance between the spectral components of the signal to estimate crosses $1/m$. We prove the necessity part of this conjecture by demonstrating that this regularization can fail whenever the spectral distance of the signal of interest is asymptotically equal to $1/m$.

cs.IT↗

arXiv2017

Exact Dimensionality Reduction for Partial Line Spectra Estimation Problems

Line spectral estimation theory aims to estimate the off-the-grid spectral components of a time signal with optimal precision. Recent results have shown that it is possible to recover signals having sparse line spectra from few temporal observations via the use of convex programming. However, the computational cost of such approaches remains the major flaw to their application to practical systems. This work investigates the recovery of spectrally sparse signal from low-dimensional partial measurements. It is shown in the first part of this paper that, under a light assumption on the sub-sampling matrix, the partial line spectral estimation problems can be relaxed into a low-dimensional semidefinite program. The proof technique relies on a novel extension of the Gram parametrization to subspaces of trigonometric polynomials. The second part of this work focuses on the analysis of two particular sub-sampling patterns: multirate sampling and random selection sampling. It is shown that those sampling patterns guarantee perfect recovery of the line spectra, and that the reconstruction can be achieved in a poly-logarithmic time with respect to the full observation case. Moreover, the sub-Nyquist recovery capabilities of such sampling patterns are highlighted. The atomic soft thresholding method is adapted in the presented framework to estimate sparse spectra in noisy environments, and a scalable algorithm for its resolution is proposed.

cs.IT↗

arXiv2017

Low Dimensional Atomic Norm Representations in Line Spectral Estimation

The line spectral estimation problem consists in recovering the frequencies of a complex valued time signal that is assumed to be sparse in the spectral domain from its discrete observations. Unlike the gridding required by the classical compressed sensing framework, line spectral estimation reconstructs signals whose spectral supports lie continuously in the Fourier domain. If recent advances have shown that atomic norm relaxation produces highly robust estimates in this context, the computational cost of this approach remains, however, the major flaw for its application to practical systems. In this work, we aim to bridge the complexity issue by studying the atomic norm minimization problem from low dimensional projection of the signal samples. We derive conditions on the sub-sampling matrix under which the partial atomic norm can be expressed by a low-dimensional semidefinite program. Moreover, we illustrate the tightness of this relaxation by showing that it is possible to recover the original signal in poly-logarithmic time for two specific sub-sampling patterns.

cs.IT↗