Effective regions and kernels in continuous sparse regularisation, with application to sketched mixtures

Abstract: This paper advances the general theory of continuous sparse regularisation on measures with the Beurling-LASSO (BLASSO). This TV-regularized convex program on the space of measures allows to recover a sparse measure using a noisy observation from an appropriate measurement operator. While previous works have uncovered the central role played by this operator and its associated kernel in order to get estimation error bounds, the latter requires a technical local positive curvature (LPC) assumption to be verified on a case-by-case basis. In practice, this yields only few LPC-kernels for which this condition is proved. At the heart of our contribution lies the kernel switch, which uncouples the model kernel from the LPC assumption: it enables to leverage any known LPC-kernel as a pivot kernel to prove error bounds, provided embedding conditions are verified between the model and pivot RKHS. We increment the list of LPC-kernels, proving that the "sinc-4" kernel, used for signal recovery and mixture problems, does satisfy the LPC assumption. Furthermore, we also show that the BLASSO localisation error around the true support decreases with the noise level, leading to effective near regions. This improves on known results where this error is fixed with some parameters depending on the model kernel. We illustrate the interest of our results in the case of translation-invariant mixture model estimation, using bandlimiting smoothing and sketching techniques to reduce the computational burden of BLASSO.