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Early stopping for statistical inverse problems via truncated SVD estimation

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euclid.ejs.1538121641.pdf (531.1Kb)
Date
2018
Link to item file
https://hal.archives-ouvertes.fr/hal-01966326
Dewey
Probabilités et mathématiques appliquées
Sujet
or-acle inequalities; Linear inverse problems; truncated SVD; spec-tral cut-off; early stopping; discrepancy principle; adaptive estimation
Journal issue
Electronic journal of statistics
Volume
12
Number
2
Publication date
2018
Article pages
3204-3231
DOI
http://dx.doi.org/10.1214/18-ejs1482
URI
https://basepub.dauphine.fr/handle/123456789/18660
Collections
  • CEREMADE : Publications
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Author
Blanchard, Gilles
27961 Institut für Mathematik [Potsdam]
Hoffman, Marc
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Reiß, Markus
4560 Institut für Mathematik [Berlin]
Type
Article accepté pour publication ou publié
Abstract (EN)
We consider truncated SVD (or spectral cutoff , projection) es-timators for a prototypical statistical inverse problem in dimension D. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level m ∈ {1,. .. , D} only based on the knowledge of the first m singular values and vectors. We analyse in detail whether sequential early stopping rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity.

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