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

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