Direct and inverse problems in machine learning

We analyze an inverse noisy regression model under random design with the aim of estimating the unknown target function based on a given set of data, drawn according to some unknown probability distribution. Our estimators are all constructed by kernel methods, which depend on a Reproducing Kernel Hilbert Space structure using spectral regularization methods. A first main result establishes upper and lower bounds for the rate of convergence under a given source condition assumption, restricting the class of admissible distributions. But since kernel methods scale poorly when massive datasets are involved, we study one example for saving computation time and memory requirements in more detail. We show that Parallelizing spectral algorithms also leads to minimax optimal rates of convergence provided the number of machines is chosen appropriately. We emphasize that so far all estimators depend on the assumed a-priori smoothness of the target function and on the eigenvalue decay of the kernel covariance operator, which are in general unknown. To obtain good purely data driven estimators constitutes the problem of adaptivity which we handle for the single machine problem via a version of the Lepskii principle.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Potsdam

Year

2017

Author dc:creator

• Mücke, Nicole

Contributors dc:contributor

• Blanchard, Gilles

• inverse problems
• kernel methods
• minimax optimality
• inverse Probleme
• Kern Methoden
• Minimax Optimalität

Statement dc:rights

• Keine öffentliche Lizenz: Unter Urheberrechtsschutz

Repository record source_url

https://publishup.uni-potsdam.de/frontdoor/index/index/docId/40347

OAI identifier oai:identifier

oai:kobv.de-opus4-uni-potsdam:40347