Determinantal Point Process Approximation under Positive and Negative Dependence

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 factorization 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.