Stability and super-resolution of generalized spike recovery
For researchers in signal processing and inverse problems, this work extends spike recovery theory to include derivatives and offers practical guidance for super-resolution.
The paper analyzes the numerical conditioning of recovering generalized spikes (Diracs and derivatives) from noisy Fourier samples, providing stability bounds for well-separated spikes and a regularization scheme for super-resolution that is shown to be near-optimal in practice.
We consider the problem of recovering a linear combination of Dirac delta functions and derivatives from a finite number of Fourier samples corrupted by noise. This is a generalized version of the well-known spike recovery problem, which is receiving much attention recently. We analyze the numerical conditioning of this problem in two different settings depending on the order of magnitude of the quantity $Nη$, where $N$ is the number of Fourier samples and $η$ is the minimal distance between the generalized spikes. In the "well-conditioned" regime $Nη\gg1$, we provide upper bounds for first-order perturbation of the solution to the corresponding least-squares problem. In the near-colliding, or "super-resolution" regime $Nη\to0$ with a single cluster, we propose a natural regularization scheme based on decimating the samples \textendash{} essentially increasing the separation $η$ \textendash{} and demonstrate the effectiveness and near-optimality of this scheme in practice.