In this paper, the family of ϕ_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 ϕ_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 naïve use of the ϕ_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 ψ_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.

在本文中，在家庭φ _p为空间填充设计的优化标准是严格审查，在中度至较大的尺寸重点是其行为，尤其是对于小的样品量（设计域的低饱和度）。确定了在标准超立方设计域中优化点集的标准使用ϕ _p标准期间出现的问题，并提出了适当的补救措施。

它显示了基于距离的标准中的距离指数应如何取决于域维。在小样本量的情况下，我们建议利用周期性超环形域的多次重复。我们证明了ϕ _p的天真使用优化设计的构造标准可能会产生不希望的正交网格图案（完整或不完整）。我们展示了此行为与目标函数中考虑的超立方体体积的方向不均匀性之间的关系，并提出了一种简单的方法，该方法包括将相互作用限制在旋转对称的邻域内。使用最近提出的最小图像约定可能会使设计空间的整个周期性扩展过于粗略。我们建议考虑有限但足够大的相互作用半径来评估成对电位。可以将交互作用半径的上限设置为在周期性重复的域内包含足够数量的点。这些增强体现在建议中ψ _p判据空间填充设计。我们表明，新标准青睐具有更好的空间填充特性，更好的投影特性以及更低的差异的设计。

高维对象内各点之间的欧几里得距离趋于集中，并且距离之间的分辨率降低。我们显示，尽管距离的对比度降低，但即使使用此各向同性度量标准，也可以保留精简标准的所需分辨率。