2026/07/21 by So Anzai, Hideitsu Hino
#math.ST #stat.TH
Determinantal point processes (DPPs) are widely used as probabilistic models for diverse random subsets, but their approximation error under model misspecification has not been fully characterized. We study the population-level approximation of a strictly positive target distribution p* by DPPs under the forward Kullback-Leibler divergence. Using information-geometric analysis and the standard quality-diversity decomposition of an L-ensemble kernel, in which the diagonal quality component Q encodes item-specific weights and the diversity component D controls repulsive interactions among items, we show that the quality component can be chosen uniquely to match all first-order inclusion probabilities of p*. The DPP approximation problem therefore reduces to the optimization of the diversity component. This reduction yields a global optimality result for attractively dependent targets: under conditions including weak positive association, the independent product distribution with the same first-order marginals, corresponding to D=I, is an optimal DPP approximation. For more general target distributions, positively correlated pairs yield lower bounds on the approximation error. On the repulsive side, a matching of disjoint negatively correlated pairs yields an upper bound on the approximation error, or equivalently a guaranteed improvement over the independent approximation. We further study local optimality around D=I and, more generally, around block-diagonal diversity matrices by analyzing perturbations between their blocks.