@article{Rastogi2020,
  author    = {Rastogi, Abhishake},
  title     = {Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems},
  series = {Communications on Pure and Applied Analysis},
  volume    = {19},
  journal   = {Communications on Pure and Applied Analysis},
  number    = {8},
  publisher = {American Institute of Mathematical Sciences},
  address   = {Springfield},
  issn      = {1534-0392},
  doi       = {10.3934/cpaa.2020183},
  pages     = {4111 -- 4126},
  year      = {2020},
  abstract  = {In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.},
  language  = {en}
}
@unpublished{BlanchardKraemer2016,
  author    = {Blanchard, Gilles and Kr{\"a}mer, Nicole},
  title     = {Convergence rates of kernel conjugate gradient for random design regression},
  volume    = {5},
  number    = {8},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-94195},
  pages     = {31},
  year      = {2016},
  abstract  = {We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.},
  language  = {en}
}
@article{BlanchardKraemer2016,
  author    = {Blanchard, Gilles and Kraemer, Nicole},
  title     = {Convergence rates of Kernel Conjugate Gradient for random design regression},
  series = {Analysis and applications},
  volume    = {14},
  journal   = {Analysis and applications},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0219-5305},
  doi       = {10.1142/S0219530516400017},
  pages     = {763 -- 794},
  year      = {2016},
  abstract  = {We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient (CG) algorithm, where regularization against over-fitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L-2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.},
  language  = {en}
}