Adapting to unknown noise level in sparse deconvolution

TL;DR

This paper introduces the Concomitant Beurling Lasso (CBLasso), a method for sparse spike deconvolution that simultaneously estimates the target measure and unknown noise level with theoretical guarantees.

Contribution

The paper proposes CBLasso, a semi-definite program that adapts to unknown noise levels, extending super-resolution theory with practical noise estimation.

Findings

Consistent noise level estimation under unknown noise conditions

Theoretical guarantees match state-of-the-art super-resolution results

New tail estimate of non-Gaussian process used in proofs

Abstract

In this paper, we study sparse spike deconvolution over the space of complex-valued measures when the input measure is a finite sum of Dirac masses. We introduce a modified version of the Beurling Lasso (BLasso), a semi-definite program that we refer to as the Concomitant Beurling Lasso (CBLasso). This new procedure estimates the target measure and the unknown noise level simultaneously. Contrary to previous estimators in the literature, theory holds for a tuning parameter that depends only on the sample size, so that it can be used for unknown noise level problems. Consistent noise level estimation is standardly proved. As for Radon measure estimation, theoretical guarantees match the previous state-of-the-art results in Super-Resolution regarding minimax prediction and localization. The proofs are based on a bound on the noise level given by a new tail estimate of the supremum of a stationary non-Gaussian process through the Rice method.