Multiscale interpretation of taut string estimation and its connection to Unbalanced Haar wavelets

TL;DR

This paper establishes a unified multiscale framework connecting the taut string and Unbalanced Haar wavelet methods for nonparametric function estimation, revealing their similarities and potential for mutual improvement.

Contribution

It provides a novel multiscale interpretation of the taut string method and links it to the multiscale Unbalanced Haar wavelet approach, enhancing understanding of both techniques.

Findings

Unified multiscale representation for both methods

Insight into the relationship between taut string and Haar wavelets

Potential for methods to learn from each other

Abstract

We compare two state-of-the-art non-linear techniques for nonparametric function estimation via piecewise constant approximation: the taut string and the Unbalanced Haar methods. While it is well-known that the latter is multiscale, it is not obvious that the former can also be interpreted as multiscale. We provide a unified multiscale representation for both methods, which offers an insight into the relationship between them as well as suggesting lessons both methods can learn from each other.