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A Variational View on Statistical Multiscale Estimation

We present a unifying view on various statistical estimation techniques including penalization, variational and thresholding methods. These estimators will be analyzed in the context of statistical linear inverse problems including nonparametric and change point regression, and high dimensional linear models as examples. Our approach reveals many seemingly unrelated estimation schemes as special instances of a general class of variational multiscale estimators, named MIND (MultIscale Nemirovskii--Dantzig). These estimators result from minimizing certain regularization functionals under convex constraints that can be seen as multiple statistical tests for local hypotheses. For computational purposes, we recast MIND in terms of simpler unconstraint optimization problems via Lagrangian penalization as well as Fenchel duality. Performance of several MINDs is demonstrated on numerical examples.