OPTIMAL PARTITIONS OF DATA IN HIGHER DIMENSIONS

Consider piece-wise constant approximations to a function of several parameters, and the problem of finding the best such approximation from measurements at a set of points in the parameter space. We find good approximate solutions to this problem in two steps: (1) partition the parameter space into cells, one for each of the N data points, and (2) collect these cells into blocks, such that within each block the function is constant to within measurement uncertainty.

We describe a branch-and-bound algorithm for finding the optimal partition into connected blocks, as well as an O(N2) dynamic programming algorithm that finds the exact global optimum over this exponentially large search space, in a data space of any dimension. This second solution relaxes the connectivity constraint, and requires additivity and convexity conditions on the block fitness function, but in practice none of these items cause problems.