Title: Multiscale Change-point Segmentation: Beyond Step Functions,
Abstract: Modern multiscale type segmentation methods are known to detect multiple
change-points with high statistical accuracy, while allowing for fast
computation. Underpinning theory has been developed mainly for models that
assume the signal as a piecewise constant function. In this paper this will be
extended to certain function classes beyond such step functions in a
nonparametric regression setting, revealing certain multiscale segmentation
methods as robust to deviation from such piecewise constant functions. Our main
finding is the adaptation over such function classes for a universal
thresholding, which includes bounded variation functions, and (piecewise)
Hölder functions of smoothness order $ 0 < \alpha \le1$ as special cases.
From this we derive statistical guarantees on feature detection in terms of
jumps and modes. Another key finding is that these multiscale segmentation
methods perform nearly (up to a log-factor) as well as the oracle piecewise
constant segmentation estimator (with known jump locations), and the best
piecewise constant approximants of the (unknown) true signal. Theoretical
findings are examined by various numerical simulations.