Exact computation of canonical GKD in dimensions four and higher

Establish whether canonical Geometric Kernel Diversity based on the Student-t2 radial kernel admits efficient exact algorithms in dimensions d ≥ 4.

Background

The paper gives exact polynomial-time algorithms for canonical Geometric Kernel Diversity in dimensions one, two, and three, using Voronoi decompositions and divergence-based boundary integration. In arbitrary dimensions, it instead provides a randomized relative-error approximation algorithm.

The exact higher-dimensional problem remains unresolved because Voronoi diagrams can have rapidly increasing complexity and the resulting higher-dimensional face integrals lack comparably simple exact evaluations.

References

For GKD itself, it remains open whether canonical GKD admits efficient exact algorithms in dimensions d ≥ 4, and whether other kernels retaining the axiomatic guarantees lead to simpler or faster computation.

— Geometric Kernel Diversity: An Axiomatic Diversity Measure in Euclidean Space  (2610.19570 - Ban et al., 14 Oct 2026) in Section 7, Conclusion