On the expected L2-discrepancy of stratified samples from parallel lines

Abstract

We study the expected $\mathcal{L}_2$-discrepancy of stratified samples generated from special equi-volume partitions of the unit square. The partitions are defined via parallel lines that are all orthogonal to the diagonal of the square. It is shown that the expected discrepancy of stratified samples derived from these partitions is a factor 2 smaller than the expected discrepancy of the same number of i.i.d uniformly distributed random points in the unit square. We conjecture that this is best possible among all partitions generated from parallel lines.

Figures (8)

• Figure 1: The partition of the unit square into two convex sets with smallest expected $\mathcal{L}_2$-discrepancy
• Figure 2: Partition of the unit cube into $N=6$ equivolume slices that are orthogonal to the diagonal.
• Figure 3: Intersections of a rectangle with a positive halfspace.
• Figure 4: The intersection of the five positive half-spaces generated by the lines $H_i$ with $1\leq i \leq 5$ without $H_5$ is shaded in gray. The corresponding set $H_I$ for $I=\{2,3\}$ consists of the vertex $C$.
• Figure 5: Illustration of $q_i$ for $N=4$.

Theorems & Definitions (1)

• proof