Variational Multiscale Nonparametric Regression: Algorithms and Implementation

TL;DR

This paper introduces the MIND estimator for nonparametric regression, combining statistical optimality with computational algorithms like Chambolle-Pock, ADMM, and semismooth Newton, demonstrated through simulations and image tests.

Contribution

It develops a multiscale estimator (MIND) that integrates convex smoothness functionals with various dictionaries and provides explicit algorithms for efficient computation.

Findings

Chambolle-Pock algorithm performs best for speed and convergence.

MIND estimator achieves statistical optimality in image denoising.

Numerical experiments validate the effectiveness of proposed algorithms.

Abstract

Many modern statistically efficient methods come with tremendous computational challenges, often leading to large-scale optimisation problems. In this work, we examine such computational issues for recently developed estimation methods in nonparametric regression with a specific view on image denoising. We consider in particular certain variational multiscale estimators which are statistically optimal in minimax sense, yet computationally intensive. Such an estimator is computed as the minimiser of a smoothness functional (e.g., TV norm) over the class of all estimators such that none of its coefficients with respect to a given multiscale dictionary is statistically significant. The so obtained multiscale Nemirowski-Dantzig estimator (MIND) can incorporate any convex smoothness functional and combine it with a proper dictionary including wavelets, curvelets and shearlets. The computation of MIND in general requires to solve a high-dimensional constrained convex optimisation problem with a specific structure of the constraints induced by the statistical multiscale testing criterion. To solve this explicitly, we discuss three different algorithmic approaches: the Chambolle-Pock, ADMM and semismooth Newton algorithms. Algorithmic details and an explicit implementation is presented and the solutions are then compared numerically in a simulation study and on various test images. We thereby recommend the Chambolle-Pock algorithm in most cases for its fast convergence. We stress that our analysis can also be transferred to signal recovery and other denoising problems to recover more general objects whenever it is possible to borrow statistical strength from data patches of similar object structure.