Spike detection from inaccurate samplings

Abstract

ABSTRACT. This article investigates the support detection problem using the LASSO estimator in the space of measures. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments...) using an ℓ 1 -regularization procedure. In particular, we provide an explicit quantitative localization of the spikes.

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

“…In addition to these exact recovery results, errors in spike detection and noise robustness are of great interest as well. Fernandez-Granda analyzed error bounds of constrained L 1 minimization in [18], while the unconstrained version was addressed in [34] under a Gaussian noise model as well as in [2] for any sampling scheme. The robustness of spike detection was discussed in [15].…”

Compressed sensing applied to weather radar

“…There are extensive works in compressed sensing literature that discuss recovering sparse signals using secondary signal support structures, such as structured sparsity ) (tree-sparsity (Baraniuk et al 2010), block sparsity (Stojnic et al 2009), and Ising models (Cevher et al 2008)), spike trains Azais et al 2013), nonuniform sparsity (Khajehnejad et al 2009;Vaswani and Lu 2010), and multiple measurement vectors (MMVs) (Duarte and Eldar 2011). However, these approaches assume discrete-valued signal parameters while, in the spectrum estimation problem, frequencies are continuous-valued.…”

Approximate support recovery of atomic line spectral estimation: A tale of resolution and precision

“…In [5], the authors derived suboptimal bounds for the Gaussian noise model. In [36], the authors extended this line of research to general measurement schemes beyond Fourier samples using the Beurling-LASSO (B-LASSO) program. The B-LASSO programs, which minimizes a least-squares term plus the measure total variation norm, is mathematically equivalent to the atomic norm formulation.…”

“…The B-LASSO programs, which minimizes a least-squares term plus the measure total variation norm, is mathematically equivalent to the atomic norm formulation. All these works [36,5,35] cannot guarantee the recovery of exactly one frequency in each neighborhood of the true frequencies. In this regard, the work by Duval and Peyré [37] obtained a stronger result by showing that as long as the SNR is large enough and the sources are well-separated, then total variation norm regularization can recover the correct number of the Diracs with both the coefficient error and the frequency error scale as the 2 norm of the noise.…”