正负相依下的行列式点过程逼近
Determinantal Point Process Approximation under Positive and Negative Dependence
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中文总结 AI 辅助
研究DPPs在正向Kullback-Leibler散度下对严格正目标分布$p^*$的逼近，利用信息几何分析和质量-多样性分解，将问题简化为多样性分量优化，得出吸引相依目标的全局最优结果及一般目标分布逼近误差的界，并研究了局部最优性。
中文摘要 AI 辅助
行列式点过程（DPPs）被广泛用作各种随机子集的概率模型，但其在模型误设下的逼近误差尚未完全表征。我们研究了在正向Kullback-Leibler散度下，DPPs对严格正目标分布$p^*$的总体水平逼近。利用信息几何分析和$L$-系综核的标准质量-多样性分解，其中对角质量分量$Q$编码特定项目权重，多样性分量$D$控制项目间的排斥相互作用，我们表明质量分量可唯一选择以匹配$p^*$的所有一阶包含概率。DPP逼近问题因此简化为多样性分量的优化。对于吸引相依目标，这一简化产生了全局最优性结果：在包括弱正关联等条件下，具有相同一阶边际的独立乘积分布（对应于$D = I$）是最优DPP逼近。对于更一般的目标分布，正相关对给出了逼近误差的下界。在排斥方面，不相交负相关对的匹配给出了逼近误差的上界，或等效地保证了相对于独立逼近的改进。我们还通过分析它们块之间的扰动，进一步研究了围绕$D = I$以及更一般地围绕块对角多样性矩阵的局部最优性。
英文摘要
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.
