Optimal Rates for Regularization of Statistical Inverse Learning Problems
| Autor: | Gilles Blanchard, Nicole Mücke |
|---|---|
| Přispěvatelé: | Institut für Mathematik [Potsdam], Universität Potsdam |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
FOS: Computer and information sciences
Spectral regularization msc:62C20 Inverse Machine Learning (stat.ML) 010103 numerical & computational mathematics msc:62G20 01 natural sciences Regularization (mathematics) [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI] [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG] Statistics - Machine Learning [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] msc:62G05 Applied mathematics ddc:510 0101 mathematics Mathematics Applied Mathematics 010102 general mathematics Institut für Mathematik Minimax convergence rates Inverse problem Minimax Statistical learning Linear map Computational Mathematics Kernel method Computational Theory and Mathematics msc:65J22 Reproducing kernel Hilbert space Exponent Analysis |
| Zdroj: | Foundations of Computational Mathematics Foundations of Computational Mathematics, Springer Verlag, 2018, 18 (4), pp.971-1013. ⟨10.1007/s10208-017-9359-7⟩ |
| ISSN: | 1615-3375 1615-3383 |
| DOI: | 10.1007/s10208-017-9359-7⟩ |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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