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

TL;DR

This paper extends the theory of continuous sparse regularisation with Beurling-LASSO, proving the LPC condition for the sinc-4 kernel, introducing kernel switch analysis, and improving error bounds adaptively for mixture model estimation.

Contribution

It proves the LPC condition for the sinc-4 kernel, introduces kernel switch analysis for error bounds, and enhances BLASSO guarantees to be adaptive to noise levels.

Findings

The sinc-4 kernel satisfies the LPC assumption.

Kernel switch analysis enables error bounds for various kernels.

BLASSO guarantees can be adapted to noise levels.

Abstract

This paper advances the general theory of continuous sparse regularisation on measures with the Beurling-LASSO (BLASSO). This TV-regularised convex program on the space of measures allows to recover a sparse measure using a noisy observation from a 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. In this paper, we prove that the ``sinc-4'' kernel, used for signal recovery and mixture problems, does satisfy the LPC assumption. Furthermore, we introduce the kernel switch analysis, which allows to leverage on a known LPC-kernel as a pivot kernel to prove error bounds. Together, these results provide easy-to-check conditions to get error bounds for a large family of translation-invariant model kernels. Besides, we also show that known BLASSO guarantees can be made adaptive to the noise level. 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 mixture model estimation, using band-limiting smoothing and sketching techniques to reduce the computational burden of BLASSO.