Distance-based optimal sampling in a hypercube: Energy potentials for high-dimensional and low-saturation designs

In this paper, the family of phi(p) optimization criteria for space-filling designs is critically reviewed, with a focus on its behavior in moderate to large dimensions, especially for small sample sizes (low saturations of the design domain). Problems that arise during the standard use of the phi(p) criteria for the optimization of point sets in standard hypercubic design domains are identified and adequate remedies are proposed. It is shown how the distance exponent in the distance-based criteria should be dependent on the domain dimension. In cases of small sample sizes, we propose utilizing multiple repetitions of a periodic hyper-toroidal domain. We show that the naive use of the phi(p) criterion for the construction of optimized designs can produce undesired orthogonal grid patterns (either complete or incomplete). We show how this behavior is related to the directional non-uniformity of hypercubical volume considered in the objective function, and we propose a simple remedy that involves limiting the interaction to a rotationally symmetrical neighborhood. Use of the recently proposed minimum image convention may provide too crude an approximation of the full periodic extension of the design space. We propose that a finite but sufficiently large interaction radius be considered for the evaluation of the pairwise potential. The upper bound on the interaction radius can be set to contain a sufficient number of points within the periodically repeated domain. These enhancements are embodied in the proposed psi(p) criterion for space-filling designs. We show that the new criterion favors designs with better space-filling property, better projection properties and also with lower discrepancy. Euclidean distances among points within high-dimensional objects tend to concentrate and the resolution between distances decreases. We show that despite the decreasing contrast of distances, the desired resolution ability of the refined criterion is retained even when this isotropic metric is used.