Generalizing Boundary Points
The complexity of numerical domain partitioning depends on the number of potential cut points. In multiway partitioning this dependency is often quadratic, even exponential. Therefore, reducing the number of candidate cut points is important. For a large family of attribute evaluation functions only boundary points need to be considered as candidates. We prove that an even more general property holds for many commonly-used functions. Their optima are located on the borders of example segments in which the relative class frequency distribution is static. These borders are a subset of boundary points. Thus, even less cut points need to be examined for these functions. The results shed a new light on the splitting properties of common attribute evaluation functions and they have practical value as well. The functions that are examined also include non-convex ones. Hence, the property introduced is not just another consequence of the convexity of a function.