Electronic Journal of Statistics

Early stopping for statistical inverse problems via truncated SVD estimation

Gilles Blanchard, Marc Hoffmann, and Markus Reiß

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We consider truncated SVD (or spectral cut-off, projection) estimators 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 $\widehat{m}\in \{1,\ldots ,D\}$ only based on the knowledge of the first $\widehat{m}$ 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.

Article information

Electron. J. Statist., Volume 12, Number 2 (2018), 3204-3231.

Received: June 2018
First available in Project Euclid: 28 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65J20: Improperly posed problems; regularization 62G07: Density estimation

Linear inverse problems truncated SVD spectral cut-off early stopping discrepancy principle adaptive estimation oracle inequalities

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Blanchard, Gilles; Hoffmann, Marc; Reiß, Markus. Early stopping for statistical inverse problems via truncated SVD estimation. Electron. J. Statist. 12 (2018), no. 2, 3204--3231. doi:10.1214/18-EJS1482. https://projecteuclid.org/euclid.ejs/1538121641

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