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Early stopping for statistical inverse problems via truncated SVD estimation

Blanchard, Gilles; Hoffman, Marc; Reiß, Markus (2018), Early stopping for statistical inverse problems via truncated SVD estimation, Electronic journal of statistics, 12, 2, p. 3204-3231. 10.1214/18-ejs1482

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Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-01966326
Date
2018
Journal name
Electronic journal of statistics
Volume
12
Number
2
Pages
3204-3231
Publication identifier
10.1214/18-ejs1482
Metadata
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Author(s)
Blanchard, Gilles cc
Institut für Mathematik [Potsdam]
Hoffman, Marc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Reiß, Markus
Institut für Mathematik [Berlin]
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.
Subjects / Keywords
or-acle inequalities; Linear inverse problems; truncated SVD; spec-tral cut-off; early stopping; discrepancy principle; adaptive estimation

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