Duality induced by an embedding structure of determinantal point process

Specifically, we clarify the embedding structure of a DPP model in the exponential family of log-linear models (c.f., Agresti, 1990; Amari, 2001) in Theorem 1. Models embedded in exponential families are called curved exponential families. Information geometry (Amari, 1985) provides a measure, the e-embedding curvature tensor (Efron, 1975; Reeds, 1975; Amari, 1982; Sei, 2011), to quantify the extent to which a curved exponential family deviates from an exponential family. To check the e-embedding curvature as well as the Fisher information matrix, we apply the diagonal scaling (Marshall and Olkin, 1968), also known as the quality vs. diversity decomposition in the DPP literature (Kulesza and Taskar, 2012), to an L-ensemble kernel of a DPP model and then evaluate them, which clarifies that the subset of parameters related to the item-wise effects (quality terms) has zero e-embedding curvature (Corollary 1).