Near-optimality of linear recovery from indirect observations
Résumé
We consider the problem of recovering linear image and $Bx$ of a signal and $x$ known to belong to a given convex compact set $X$ from indirect observation $\omega = Ax + \xi$ of $x$ corrupted by random noise $\xi$ with finite covariance matrix. It is shown that under some assumptions on $X$ (satisfied, e.g. when $X$ is the intersection of $K$ concentric ellipsoids/elliptic cylinders, or the unit ball of the spectral norm in the space of matrices) and on the norm $\|\cdot\|$used to measure the recovery error (satisfied, e.g. by $\|\cdot\|_p$-norms, $1\leq p\leq2$, on $\mathbb{R}^m$ and by the nuclear norm on the space of matrices), one can build, in a computationally efficient manner, a "seemingly good“ linear in observations estimate. Further, in the case of zero mean Gaussian observation noise and general mappings $A$ and $B$ have $x\in X$ expected $\|\cdot\|$-loss. These results form an essential extension of classical results [7, 24] and of the recent work [13], where the assumptions on X were more restrictive, and the norm $\|\cdot\|$ was assumed to be the Euclidean one.
Domaines
Statistiques [math.ST]Origine | Fichiers éditeurs autorisés sur une archive ouverte |
---|