A Variational View on Statistical Multiscale Estimation

Abstract

We present a unifying view on various statistical estimation techniques including penalization, variational, and thresholding methods. These estimators are 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, called 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.

Data-Proximal Complementary ℓ1-TV Reconstruction for Limited Data Computed Tomography

“…However, the total variation's monoscale nature does not achieve optimal reconstruction in the presence of noise [15,16], while wavelet-or curvelet-based priors [17,18] struggle to eliminate typical limited angle artifacts. This problem stems from the observation that extending the missing sinogram with zero values results in reconstructions with a smaller ℓ 1 norm compared to non-zero extensions.…”

“…However, since this is a monoscale approach, there is a trade-off between noise reduction and preservation of features at certain scales. Natural images have features across multiple scales which become either over or under smoothed depending on the particular choice of the regularization parameter [15,16]. This already has negative impact for fully sampled tomographic systems or simple denoising.…”