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Article Dans Une Revue Mathematical Statistics and Learning Année : 2018

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, 1p2, 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 xX 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.
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Dates et versions

hal-02047192 , version 1 (07-01-2025)

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Anatoli B. Juditsky, Arkadi Nemirovski. Near-optimality of linear recovery from indirect observations. Mathematical Statistics and Learning, 2018, 1 (2), pp.171-225. ⟨10.4171/MSL/1-2-2⟩. ⟨hal-02047192⟩
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