Adapting to unknown noise level in sparse deconvolution

In this paper, we study sparse spike deconvolution over the space of complex-valued mea-sures when the input measure is a finite sum of Dirac masses. We introduce a modified versionof the Beurling Lasso (BLasso), a semi-definite program that we refer to as the ConcomitantBeurling Lasso (CBLasso). This new procedure estimates the target measure and the un-known noise level simultaneously. Contrary to previous estimators in the literature, theoryholds for a tuning parameter that depends only on the sample size, so that it can be usedfor unknown noise level problems. Consistent noise level estimation is standardly proved. Asfor Radon measure estimation, theoretical guarantees match the previous state-of-the-art re-sults in Super-Resolution regarding minimax prediction and localization. The proofs are basedon a bound on the noise level given by a new tail estimate of the supremum of a stationarynon-Gaussian process through the Rice method.Key-words: Deconvolution; Convex regularization; Inverse problems; Model selection; Concomi-tant Beurling Lasso; Square-root Lasso; Scaled-Lasso, Sparsity; Rice method;