TY  - JOUR
A1  - Blanchard, Gilles
A1  - Hoffmann, Marc
A1  - Reiss, Markus
T1  - Optimal adaptation for early stopping in statistical inverse problems
JF  - SIAM/ASA Journal on Uncertainty Quantification
N2  - For linear inverse problems Y = A mu + zeta, it is classical to recover the unknown signal mu by iterative regularization methods ((mu) over cap,(m) = 0,1, . . .) and halt at a data-dependent iteration tau using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error parallel to A((mu) over cap (()(tau)) - mu)parallel to(2) is controlled. In the context of statistical estimation with stochastic noise zeta, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared- error E[parallel to((mu) over cap (()(tau)) - mu)parallel to(2)]. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularization methods. The proofs use bias and variance transfer techniques from weak prediction error to strong L-2-error, as well as convexity arguments and concentration bounds for the stochastic part. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically.
KW  - linear inverse problems
KW  - early stopping
KW  - discrepancy principle
KW  - adaptive estimation
KW  - oracle inequality
KW  - Landweber iteration
Y1  - 2018
U6  - https://doi.org/10.1137/17M1154096
SN  - 2166-2525
VL  - 6
IS  - 3
SP  - 1043
EP  - 1075
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  -