Mismatched sparse denoiser requires overestimating the support length

A well-known result [1, Lemma 3.4] states that, without noise, it is better to overestimate the support of a sparse signal, since, if the estimated support includes the true support, the reconstruction is perfect. In this paper, we investigate whether this result holds also in the presence of noise. First, we derive the covariance matrix of the signal estimate when the observation matrix is Gaussian, generalizing existing results. Then, we show that, even in the noisy case, overestimating the support length is the preferred solution, as the error incurred by missing some signal components dominates the overall error variance. Finally, an upper bound of the estimated support length is provided to avoid excessive noise amplification.