Generating A Maximally Spaced Set Of Bins To Fill For High-Dimensional Space-Filling Latin Hypercube Sampling

In the literature, space-filling Latin hypercube sample designs typically are generated by optimizing some criteria such as maximizing the minimum distance between points or minimizing discrepancy. However, such methods are time consuming and frequently produce designs that are highly regular, which can bias results. A fast way to generate irregular space-filling Latin hypercube sample designs is to randomly distribute the sample points to a pre-selected set of well-spaced bins. Such designs are said to be "binning optimal" and are shown to be irregular. Specifically, Fourier analysis reveals regular patterns in the multi-dimensional spacing of points for the Sobol sequence but not for Binning optimal symmetric Latin hypercube sampling. For M = 2(r) <= 8 dimensions and N = 2(s) >= 2M points, where r and s are non-negative integers, simple patterns can be used to create a list of maximally spaced bins. Good Latin hypercube sample designs for non-power of two dimensions can be generated by discarding excess dimensions. Since the octants/ bins containing the 2 M end points of an "orientation" (a rotated set of orthogonal axes) are maximally spaced, the process of generating the list of octants simplifies to finding a list of maximally spaced orientations. Even with this simplification, the "patterns" for maximally spaced bins in M >= 16 dimensions are not so simple. In this paper, we use group theory to generate 2(M) / (2M) disjoint orientations, and present an algorithm to sort these into maximally spaced order. Conceptually, the procedure works for arbitrarily large numbers of dimensions. However, memory requirements currently preclude even listing the 2(M) / (2M) orientation leaders for M >= 32 dimensions. In anticipation of overcoming this obstacle, we outline a variant of the sorting algorithm with a low memory requirement for use in M >= 32 dimensions.