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Optimal adaptation for early stopping in statistical inverse problems

Blanchard, Gilles; Hoffmann, Marc; Reiß, Markus (2018), Optimal adaptation for early stopping in statistical inverse problems, SIAM/ASA Journal on Uncertainty Quantification, 6, 3, p. 1043-1075. 10.1137/17M1154096

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-01426253
Date
2018
Journal name
SIAM/ASA Journal on Uncertainty Quantification
Volume
6
Number
3
Publisher
SIAM
Pages
1043-1075
Publication identifier
10.1137/17M1154096
Metadata
Show full item record
Author(s)
Blanchard, Gilles cc
Institute of Mathematics
Hoffmann, Marc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Reiß, Markus
Institute for Mathematics, Humboldt university
Abstract (EN)
For linear inverse problems Y = Aµ + ξ, it is classical to recover the unknown signal µ by iterative regularisation methods (µ (m) , m = 0, 1,. . .) so that the weak (or prediction) error A(µ (τ) − µ) 2 is controlled for some early stopping rule τ based on a discrepancy principle. In the context of statistical estimation with stochastic noise ξ, we study oracle adaptation in strong squared-error E µ (τ) − µ 2. We give precise lower bounds for estimation by early stopping. For a stopping rule based on the residual process oracle adaptation bounds are established for general linear iterative 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. For Sobolev balls the adaptation bounds are shown to match the lower bounds. Adaptive early stopping for the Landweber and spectral cutoff methods are studied in further detail.
Subjects / Keywords
Linear inverse problems; Early stopping; Discrepancy principle; Adaptive estimation; Oracle inequalities

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