Multiscale Change-point Segmentation: Beyond Step Functions

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 which assume the signal as an unknown piecewise constant function. In this paper this will be extended to certain function classes beyond step functions in a nonparametric regression setting, revealing certain multiscale segmentation methods as robust to deviation from such piecewise constant functions. Although these methods are designed for step functions, our main finding is its adaptation over such function classes for a universal thresholding. On the one hand, this includes nearly optimal convergence rates for step functions with increasing number of jumps. On the other hand, for models which are characterized by certain approximation spaces, we obtain nearly optimal rates as well. This includes bounded variation functions, and (piecewise) H\"{o}lder functions of smoothness order $0 < \alpha \le1$. All results are formulated in terms of $L^p$-loss ($0 < p < \infty$) both almost surely and in expectation. Theoretical findings are examined by various numerical simulations.