Geometric Kernel Diversity: An Axiomatic Diversity Measure in Euclidean Space

Quantifying the diversity of a set of objects is a fundamental problem in machine learning and statistics. Mironov and Prokhorenkova (2025) recently showed that three axioms-monotonicity, uniqueness, and continuity-are crucial for a reliable diversity measure and that none of the standard metrics in the literature satisfy all three. They also constructed two measures that do satisfy the axioms, but both are NP-hard to compute, leaving open the question of whether efficient axiomatic diversity measures can exist. We study a Euclidean-restricted version of this question. We introduce Geometric Kernel Diversity (GKD), which associates a radial kernel solid with each point and measures the volume of their union. We show that canonical GKD satisfies all three Mironov-Prokhorenkova axioms on Euclidean-realizable inputs. For canonical GKD, we give exact polynomial-time algorithms in dimensions d 3. In arbitrary dimensions, we give a randomized algorithm that returns a (1 plus or minus epsilon) -approximation with probability at least 1-δ in time O (nd over epsilon squared δ) for n points. The axiomatic guarantee applies to exact canonical GKD; the randomized algorithm approximates this deterministic quantity. We further show that the two diversity measures of Mironov and Prokhorenkova (2025) remain NP-hard on Euclidean-realizable inputs and, unless NP = RP, admit no FPRAS even on this restricted domain.