Approximation complexity for piecewise monotone functions and real data

Abstract

This paper investigates the relationship between approximation error and complexity. A variety of complexity measures are used, including: the number of alternating strictly monotone segments; computation time; and for piecewise linear approximations, the number of linear segments. The results apply to piecewise monotone functions and to finite maps from reals to reals, i.e., real data. We provide a theoretical framework expressing the exact relationship between approximation error and the number of alternating strictly monotone segments. We provide a linear-time algorithm taking an error bound as input and returning a minimal segmentation of the approximated function's domain such that there exists an approximation, alternatingly strictly monotone on the segments, with error less than the given bound. For real data, we provide a suboptimal tradeoff between approximation error and number of linear segments in piecewise linear approximation. The results are obtained by extending the theory of best piecewise monotone approximation to piecewise monotone functions, and by application of a new concept, scale-dependent monotonicity.