hypercubesampling

Hypercube sampling is a space-filling, stratified sampling design used for generating points in high-dimensional spaces. It treats the input domain as a unit hypercube [0,1]^d and partitions it into equal-volume hypercubes (cells) by dividing each axis into a fixed number of intervals. Each resulting cell represents a subregion of the domain.

In the simplest randomized variant, one point is drawn from within each hypercube, typically by adding a random offset inside the cell. A deterministic variant may use the cell centers as sample locations. The design aims to achieve broad coverage of the space, reducing clustering that can occur with pure random sampling and improving uniformity across multiple dimensions.

Hypercube sampling is related to other space-filling designs such as Latin hypercube sampling (LHS) and low-discrepancy sequences. While LHS enforces an even distribution along each axis, hypercube sampling emphasizes joint coverage across the multi-dimensional grid by using the grid of cells. In practice, a total number of samples is obtained by selecting a subset of cells or by using a fixed grid size that yields the desired total.

Applications include numerical integration, surrogate modeling, computer experiments, and design of experiments, where uniform exploration of a high-dimensional input space is advantageous. Variants may favor purely deterministic placement or incorporate randomness to preserve statistical properties.

Advantages of hypercube sampling include simple construction and improved space-filling properties over naive Monte Carlo in moderate dimensions. Limitations arise from the exponential growth of cells with dimension, which can limit scalability in very high-dimensional problems, and from potential sensitivity to how cells are selected or randomized.