Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors

Lepski, Oleg; Mammen, Enno; Spokoiny, Vladimir

doi:10.1214/aos/1069362731

Cited by 223 publications

(138 citation statements)

References 17 publications

Supporting

Mentioning

137

Contrasting

Order By: Relevance

“…From a more practical point of view, minimax adaptivity guarantees good accuracy of the estimator for a wide choice of functions. Thus, the estimator automatically 'adapts' to the unknown smoothness of the underlying function (see, for example [44]).…”

Section: Definition 4 An Estimator θ Is Called Minimax Adaptive On Tmentioning

confidence: 99%

Nonparametric statistical inverse problems

2008

View full text Add to dashboard Cite

We explain some basic theoretical issues regarding nonparametric statistics applied to inverse problems. Simple examples are used to present classical concepts such as the white noise model, risk estimation, minimax risk, model selection and optimal rates of convergence, as well as more recent concepts such as adaptive estimation, oracle inequalities, modern model selection methods, Stein's unbiased risk estimation and the very recent risk hull method.

show abstract

Section: Definition 4 An Estimator θ Is Called Minimax Adaptive On Tmentioning

confidence: 99%

Nonparametric statistical inverse problems

2008

View full text Add to dashboard Cite

show abstract

“…We were unable to obtain oracle inequalities in the spirit of the Goldenshluger-Lepski method, see Goldenshluger and Lepski [28,29,30], due to the non-Euclidean character of the support of f P : our route goes along the more classical approach of Lepski et al [46]. Obtaining oracle inequalities in this framework remain an open problem.…”

Section: Theorem 31 (Upper Bound)mentioning

confidence: 99%

“…Section 3.3 focuses on the case of one-dimensional submanifolds M when d = 1, where we explicitly construct a kernel estimator that achieves the minimax rate of convergence, revisiting the estimator (2) of Pelletier [54] and relying on the Isomap algorithm. In Section 3.4, we implement Lepski's algorithm on the bandwidth of our kernel estimators, following Lepski et al [46]; this achieves smoothness adaptation w.r.t. α ∧ β.…”

Section: Organisation Of the Papermentioning

confidence: 99%

Density estimation on an unknown submanifold

Berenfeld

Hoffmann

2021

Electron. J. Statist.

View full text Add to dashboard Cite

We investigate the estimation of a density f from a n-sample on an Euclidean space R D , when the data are supported by an unknown submanifold M of possibly unknown dimension d < D, under a reach condition. We investigate several nonparametric kernel methods, with datadriven bandwidths that incorporate some learning of the geometry via a local dimension estimator. When f has Hölder smoothness β and M has regularity α, our estimator achieves the rate n −α∧β/(2α∧β+d) for a pointwise loss. The rate does not depend on the ambient dimension D and we establish that our procedure is asymptotically minimax for α ≥ β. Following Lepski's principle, a bandwidth selection rule is shown to achieve smoothness adaptation. We also investigate the case α ≤ β: by estimating in some sense the underlying geometry of M , we establish in dimension d = 1 that the minimax rate is n −β/(2β+1) proving in particular that it does not depend on the regularity of M . Finally, a numerical implementation is conducted on some case studies in order to confirm the practical feasibility of our estimators.

show abstract

“…Before passing on to the CT reconstruction, we mention that a similar algorithm was devised by Lepski, Mammen, and Spokoiny in [13]. Both works implement a general scheme of Lepski [14] and seem to share the same approach.…”

Section: A Lepski-goldenshluger-nemirovsky Estimatormentioning

confidence: 99%

Spatially-Adaptive Reconstruction in Computed Tomography Using Neural Networks

Boublil

Elad

Shtok

et al. 2015

IEEE Trans. Med. Imaging

View full text Add to dashboard Cite

Abstract-We propose a direct reconstruction algorithm for Computed Tomography, based on a local fusion of a few preliminary image estimates by means of a non-linear fusion rule. One such rule is based on a signal denoising technique which is spatially adaptive to the unknown local smoothness. Another, more powerful fusion rule, is based on a neural network trained off-line with a high-quality training set of images. Two types of linear reconstruction algorithms for the preliminary images are employed for two different reconstruction tasks. For an entire image reconstruction from full projection data, the proposed scheme uses a sequence of Filtered Back-Projection algorithms with a gradually growing cut-off frequency. To recover a Region Of Interest only from local projections, statistically-trained linear reconstruction algorithms are employed. Numerical experiments display the improvement in reconstruction quality when compared to linear reconstruction algorithms.

show abstract

Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors

Cited by 223 publications

References 17 publications

Nonparametric statistical inverse problems

Nonparametric statistical inverse problems

Density estimation on an unknown submanifold

Spatially-Adaptive Reconstruction in Computed Tomography Using Neural Networks

Contact Info

Product

Resources

About