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 ω=Ax+ξ of x corrupted by random noise ξ 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 ‖⋅‖used to measure the recovery error (satisfied, e.g. by ‖⋅‖p-norms, 1≤p≤2, on Rm 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∈X expected ‖⋅‖-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 ‖⋅‖ was assumed to be the Euclidean one.
Domaines
Statistiques [math.ST]Origine | Fichiers éditeurs autorisés sur une archive ouverte |
---|