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The Small Ball Method Applied to Regression

Mullink, J.M. (2023) The Small Ball Method Applied to Regression.

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Abstract:Many problems in applied mathematics amount to estimating an object (i.e. some parameter, function or measure) from measurements. Examples encompass regression and missing value imputation in statistics and computational tomography (ct) and medical resonance imaging (mri) in imaging. So the object one needs to recover is unknown and one only has access to a set of (noisy) measurements. The goal of the small ball method is to formulate sufficient conditions under which it is possible to (approximately) recover the unknown object. In order to recover the unknown object one wants to construct an estimator using only the measurements. Of course one wants this estimator to be close to the unknown object in a pre-specified sense. A large family of estimators can be written as the minimizer of an empirical risk functional. The empirical risk functional only depends on the measurements. The problem is that in general these estimators depend in a very complicated way on the measurements. Only in a limited number of examples it is possible to write down a closed form expression for the estimator. The small ball method can be used to formulate recovery guarantees for empirical risk minimizers. In this report we first of all we describe the small ball method. The main difference between the approach taken here and previous work is that we introduce a delocalized small ball assumption (DSBA). This is a weaker variant of the small ball assumption. In some situations the DSBA holds, but the classical small ball assumption fails to hold. Examples are spaces of Sobolev and Hölder continuous functions. Also uniformly bounded function spaces satisfy the DSBA. We also look at some applications where we partially extend the small ball method beyond the regression setup. 3
Item Type:Essay (Master)
Faculty:EEMCS: Electrical Engineering, Mathematics and Computer Science
Subject:31 mathematics
Programme:Applied Mathematics MSc (60348)
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