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Foundations of Computational Mathematics
Erschienen in:

20.06.2017

Optimal Rates for Regularization of Statistical Inverse Learning Problems

verfasst von: Gilles Blanchard, Nicole Mücke

Erschienen in: Foundations of Computational Mathematics | Ausgabe 4/2018

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Abstract

We consider a statistical inverse learning (also called inverse regression) problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points \(X_i\), superposed with an additive noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependency of the constant factor in the variance of the noise and the radius of the source condition set.
Vorheriger Artikel Computing the Homology of Real Projective Sets
Nächster Artikel An Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball

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Fußnoten
1
This can be extended to the case where g is only approximated in \(L^2(\nu )\) by a sequence of functions in \({\mathcal H}_K\). For the sake of the present discussion, only the case where it is assumed \(g \in {\mathcal H}_K\) is of interest.
 
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Metadaten
Titel
Optimal Rates for Regularization of Statistical Inverse Learning Problems
verfasst von
Gilles Blanchard
Nicole Mücke
Publikationsdatum
20.06.2017
Verlag
Springer US
Erschienen in
Foundations of Computational Mathematics / Ausgabe 4/2018
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-017-9359-7

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