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Duality induced by an embedding structure of determinantal point process

Hideitsu Hino, Keisuke Yano

TL;DR

This work analyzes the information-geometric structure of determinantal point processes on finite sets and proves that a DPP is embedded as a curved exponential family under a diagonal scaling of the $L$-ensemble kernel. The authors show that a subset of parameters (the singleton quality terms) are partially $e$-embedding flat, enabling a Hessian representation of the Fisher information w.r.t. those parameters via the conditional potential $\psi(u)$, and they establish a duality between marginal kernel $K$ and the $L$-ensemble kernel $L$. They provide a mixed parameterization that yields a KL-divergence expression analogous to exponential families and demonstrate the results with concrete $m=2$ and $m=3$ examples, clarifying when DPPs are truly exponential (e.g., $m\le2$) and when they are curved. The findings offer a geometric lens for identifiability and inference in DPPs and connect the $K$ and $L$ representations through information-geometric duality.

Abstract

This paper investigates the information geometrical structure of a determinantal point process (DPP). It demonstrates that a DPP is embedded in the exponential family of log-linear models. The extent of deviation from an exponential family is analyzed using the $\mathrm{e}$-embedding curvature tensor, which identifies partially flat parameters of a DPP. On the basis of this embedding structure, the duality related to a marginal kernel and an $L$-ensemble kernel is discovered.

Duality induced by an embedding structure of determinantal point process

TL;DR

This work analyzes the information-geometric structure of determinantal point processes on finite sets and proves that a DPP is embedded as a curved exponential family under a diagonal scaling of the -ensemble kernel. The authors show that a subset of parameters (the singleton quality terms) are partially -embedding flat, enabling a Hessian representation of the Fisher information w.r.t. those parameters via the conditional potential , and they establish a duality between marginal kernel and the -ensemble kernel . They provide a mixed parameterization that yields a KL-divergence expression analogous to exponential families and demonstrate the results with concrete and examples, clarifying when DPPs are truly exponential (e.g., ) and when they are curved. The findings offer a geometric lens for identifiability and inference in DPPs and connect the and representations through information-geometric duality.

Abstract

This paper investigates the information geometrical structure of a determinantal point process (DPP). It demonstrates that a DPP is embedded in the exponential family of log-linear models. The extent of deviation from an exponential family is analyzed using the -embedding curvature tensor, which identifies partially flat parameters of a DPP. On the basis of this embedding structure, the duality related to a marginal kernel and an -ensemble kernel is discovered.
Paper Structure (12 sections, 6 theorems, 67 equations, 3 figures)

This paper contains 12 sections, 6 theorems, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Determinantal point process
  4. Exponential family and curved exponential family
  5. Information-geometric structure of determinantal point processes
  6. Embedding structure of determinantal point processes
  7. Duality in determinantal point processes
  8. Examples
  9. Determinantal point process with $m=2$
  10. Determinantal point process with $m=3$
  11. Proofs
  12. Acknowledgement

Key Result

Lemma 1

Under suitable conditions, the square of $\mathrm{e}$-embedding curvature vanishes if and only if the model is an exponential family.

Figures (3)

  • Figure 1: Construction of parameters $(u^{\alpha})_{\alpha\in\mathcal{S}_{1}^{m}\cup\mathcal{S}_{2}^{m}}$ and $(\theta^{\mathcal{I}})_{\mathcal{I}\in\mathcal{S}_{3}^{m}\cup\cdots\cup \mathcal{S}_{m}^{m}}$ in the DPP model with $m=4$. After drawing the Hasse diagram for the power set, we eliminate upward paths starting from singletons and the emptyset. In the resulting diagram, nodes correspond to the indices for parameters, and directed edges represent the dependence of parameters. Nodes in the same layer have the same cardinality. For example, $\theta^{\{1,2,4\}}$ is constructed using parameters having directed edges to $\{1,2,4\}$ as indicated in orange edges.
  • Figure 2: Contour plots of conditional potential functions $\psi(u)$ for DPPs with $m=2$ and $m=3$. (a) Conditional potential function for the DPP model with $m=2$. The evaluation fixes the value of $u^{\{1,2\}}$ to $-0.1$; (b) Conditional potential function for the DPP model with $m=3$. The evaluation fixes the values of $u^{\{3\}},u^{\{1,2\}},u^{\{2,3\}},u^{\{3,1\}}$ to $-0.1, \log(1-{0.5}^2)$, respectively.
  • Figure 3: Plots of $\theta^{\{1,2,3\}}(u^{\{1,2\}}, u^{\{2,3\}}, u^{\{1,3\}})$. (a) the plot at $u^{\{1,3\}}=-0.1$; (b) the plot at $u^{\{1,3\}}=-0.5$; (c) the plot at $u^{\{1,3\}}=-0.8$

Theorems & Definitions (16)

  • Lemma 1: Amari_Nagaoka and Sei_2011
  • Theorem 1
  • Remark 1: Unidentifiability
  • Corollary 1
  • Corollary 2
  • Remark 2: Conditional potential function
  • Remark 3: Non-orthogonality
  • Lemma 2
  • proof
  • Remark 4: Relation to the Laplace expansion
  • ...and 6 more