@article{BlanchardHoffmannReiss2018,
  author    = {Blanchard, Gilles and Hoffmann, Marc and Reiss, Markus},
  title     = {Optimal adaptation for early stopping in statistical inverse problems},
  series = {SIAM/ASA Journal on Uncertainty Quantification},
  volume    = {6},
  journal   = {SIAM/ASA Journal on Uncertainty Quantification},
  number    = {3},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {2166-2525},
  doi       = {10.1137/17M1154096},
  pages     = {1043 -- 1075},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}