# Duality induced by an embedding structure of determinantal point process

Hideitsu<EMAIL_ADDRESS>[ Keisuke<EMAIL_ADDRESS>[
The Institute of Statistical Mathematics,
10-3 Midori cho, Tachikawa City, Tokyo, 190-8562, Japan.

###### 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.

and

##  1 Introduction

Determinantal point processes (DPP) on a finite set are a common choice for
modeling probabilities that accommodate repulsive property (Kulesza and
Taskar, 2012). They have found wide application such as document summarization
(Lin and Bilmes, 2012), recommendation systems (Gartrell et al., 2016), and
Bayesian statistics (Kojima and Komaki, 2016; Bardenet and Hardy, 2020;
Ročková and George, 2022), being blessed with efficient algorithms for
inferential tasks (Gillenwater et al., 2014; Affandi et al., 2014; Mariet and
Sra, 2016; Urschel et al., 2017; Dupuy and Bach, 2018; Kawashima and Hino,
2023).

In this paper, we elucidate the information geometric structure of a DPP
model. 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 $\mathrm{e}$-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 $\mathrm{e}$-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 $\mathrm{e}$-embedding curvature (Corollary 1). This partially
$\mathrm{e}$-embedding flat structure endows the Fisher information matrix
with a nice representation using the Hessian of a certain function (Corollary
2). This also presents a duality related to two representations of a DPP
(Lemma 2), a marginal kernel representation and $L$-ensemble kernel
representation.

As a related work, Brunel et al. (2017) investigate the qualitative structure
of log-likelihood of a DPP, which enables us to check the identifiability and
evaluate the null space of the Fisher information matrix. Grigorescu et al.
(2022) investigate the maximization of the log-likelihood from the
computational viewpoint, and Friedman et al. (2024) further analyze the number
of critical points of the log-likelihood using algebraic geometry. Our
analyses complement Brunel et al. (2017) from yet another perspective, the
embedding structure of a DPP. Also, Batmanghelich et al. (2014) utilize the
idea that DPP has an exponential form to variational inference. As
Batmanghelich et al. (2014) mention, DPP does not form an exponential family
in general. Our results measure the extent to which a DPP deviates from an
exponential family and identify the sub structure that can be regarded as an
exponential family.

The rest of the paper is organized in the followings. Section 2 summarizes
DPPs and information geometry. Section 3 presents the main results, and
Section 4 examines the main results through examples. Section 5 gives the
proofs of the results.

##  2 Preliminaries

Before presenting the main results, we summarize preliminaries used in this
paper.

###  2.1 Determinantal point process

Let $\mathcal{Y}:=\\{1,\ldots,m\\}$ be a finite set of size $m$. A
determinantal point process (DPP) on $\mathcal{Y}$ is a random subset $Y$ of
$\mathcal{Y}$ of which distribution is given by

$\displaystyle P(A\subseteq
Y)=\mathrm{det}(K_{A})\quad(A\subseteq\mathcal{Y})$

for some $m\times m$ symmetric real matrix $K$ with all eigenvalues in
$[0,1]$, where $K_{B}$ for a subset $B\subseteq\mathcal{Y}$ denotes the
submatrix of $K$ with indices of columns and rows lying in $B$. This matrix
$K$ is called a marginal kernel. The condition that all eigenvalues of $K$ lie
in $[0,1]$ is necessary to ensure that a DPP forms a probability distribution
(see Section 2 of Kulesza and Taskar (2012) as well as Section 2 of Kawashima
and Hino (2023)). If all eigenvalues of $K$ lie in $(0,1)$, the probability
mass function of a DPP with a marginal kernel $K$ has a useful representation
(Borodin and Rains, 2005)

$\displaystyle P_{L}(Y=A)=\cfrac{\det(L_{A})}{\det(L+I_{m\times
m})}\quad(A\subseteq\mathcal{Y})\quad\text{with}\,\,L:=K(I_{m\times
m}-K)^{-1}.$

Here, $I_{m\times m}$ denotes the identity matrix of size $m$. DPP having a
marginal kernel with all eigenvalues in $(0,1)$ allows this representation and
is called $L$-ensemble. The matrix $L$ is called an $L$-ensemble kernel. We
consider $L$-ensembles and consider the parametric model

$\mathcal{P}_{\mathrm{DPP}}:=\left\\{P_{L}(Y=A)=\frac{\mathrm{det}(L_{A})}{\mathrm{det}(L+I_{m\times
m})}\,:\,\text{$L$ is a positive definite matrix}\right\\}.$

###  2.2 Exponential family and curved exponential family

To introduce our theoretical results, we briefly review exponential families
and curved exponential families (on $\mathcal{Y}$).

An exponential family on $(\mathcal{Y}\,,\,2^{\mathcal{Y}})$ is a family
$\\{P_{\theta}(\cdot)\,:\,\theta=(\theta^{1},\ldots,\theta^{i},\ldots,\theta^{d})^{\top}\in\Theta\subseteq\mathbb{R}^{d}\\}$
of probability functions parameterized by $d$-dimensional parameter $\theta$
that are given by

$P_{\theta}(A)=\exp\left(\sum_{i=1}^{d}\theta^{i}T_{i}(A)-\phi(\theta)\right)\quad(A\subseteq\mathcal{Y}),$

where $T_{i}(A)\,(i=1,\dots,d)$ are a set of measurable functions called
sufficient statistics and $\phi(\theta)$ is a potential function:

$\phi(\theta):=\log\sum_{A\subseteq\mathcal{Y}}\exp\left(\sum_{i=1}^{d}\theta^{i}T_{i}(A)\right).$

The parameterization $\theta$ is called a canonical parameter. In an
exponential family, the Fisher information matrix $(g_{ij}(\theta))_{1\leq
i,j\leq d}$ with respect to $\theta$ is given by

$\displaystyle
g_{ij}(\theta)=\frac{\partial^{2}}{\partial\theta^{i}\partial\theta^{j}}\phi(\theta)=\mathrm{Cov}_{\theta}[T_{i}(A),T_{j}(A)],$
(1)

where $\mathrm{Cov}_{\theta}[\cdot\,,\,\cdot]$ is the covariance with respect
to $P_{\theta}$. From the Legendre duality by the convexity of $\phi(\theta)$,
we can define the dual parameterization $\eta=(\eta_{i})_{i=1,\ldots,d}$ of
$\theta$ called the expectation parameter by
$\eta_{i}=\partial_{i}\phi(\theta)$. This parameterization is actually the
expectation of the sufficient statistics with respect to $P_{\theta}$:

$\displaystyle\eta_{i}=\mathrm{E}_{\theta}[T_{i}(A)],$ (2)

where $\mathrm{E}_{\theta}[\cdot]$ is the expectation with respect to
$P_{\theta}$. The Fisher information matrix $(g^{ij}(\eta))_{1\leq i,j\leq d}$
with respect to $\eta$ is given by

$\displaystyle g^{ij}(\eta)=(g^{-1}(\theta(\eta)))_{ij},$

that is, the $(i,j)$-component of the inverse of the Fisher information matrix
$g(\theta)$ with respect to $\theta$.

These two parameterizations $\theta$ and $\eta$ play important roles in
geometric structure of statistical models (Amari, 1985). For a parametric
model
$\\{P_{\xi}(\cdot):\xi=(\xi^{1},\ldots,\xi^{d})^{\top}\in\Xi\subset\mathbb{R}^{d}\\}$,
a parameterization $\xi$ is called $\mathrm{e}$-affine if it vanishes
$\mathrm{e}$-connection coefficients $\raise
1.5pt\hbox{$\displaystyle\mathop{\Gamma}^{\rm e}$}^{\,}(\xi)$ defined by

$\displaystyle\raise 1.5pt\hbox{$\displaystyle\mathop{\Gamma}^{\rm
e}$}_{ijk}^{\,}(\xi)=\sum_{A\subseteq\mathcal{Y}}P_{\xi}(A)\left[\frac{\partial^{2}}{\partial\xi^{i}\partial\xi^{j}}\log
P_{\xi}(A)\frac{\partial}{\partial\xi^{k}}\log P_{\xi}(A)\right].$

In the same model, a parameterization $\nu=\nu(\xi)$ is called
$\mathrm{m}$-affine if it vanishes $\mathrm{m}$-connection coefficients
$\raise 1.5pt\hbox{$\displaystyle\mathop{\Gamma}^{\rm m}$}^{\,}(\nu)$ defined
by

$\displaystyle\raise 1.5pt\hbox{$\displaystyle\mathop{\Gamma}^{\rm
m}$}_{ijk}^{\,}(\nu)=\sum_{A\subseteq\mathcal{Y}}\frac{1}{P_{\xi(\nu)}(A)}\left[\frac{\partial^{2}}{\partial\nu^{i}\partial\nu^{j}}P_{\xi(\nu)}(A)\frac{\partial}{\partial\nu^{k}}P_{\xi(\nu)}(A)\right].$

In the exponential family, the canonical parameter $\theta$ is
$\mathrm{e}$-affine and the expectation parameter $\eta$ is
$\mathrm{m}$-affine.

A curved exponential family on $(\mathcal{Y},2^{\mathcal{Y}})$ is a family
$\\{P_{u}(\cdot)\,:\,u=(u^{1},\ldots,u^{a},\ldots,u^{d^{\prime}})^{\top}\in\mathcal{U}\subseteq\mathbb{R}^{d^{\prime}}\\}$
of probability functions parameterized by $d^{\prime}$-dimensional parameter
that is smoothly embedded in an exponential family. Here, the embedding of a
model is defined through the embedding of the parameter. Consider a curved
exponential family $\\{P_{u}(\cdot)\,:\,u\in\mathcal{U}\\}$ that is embedded
in a $d$-dimensional exponential family
$\\{P_{\theta}(\cdot)\,:\,\theta\in\Theta\\}$ with a canonical
parameterization
$\theta=(\theta^{1},\cdots,\theta^{i},\cdots,\theta^{d})^{\top}$ and an
expectation parameterization
$\eta=(\eta_{1},\cdots,\eta_{i},\cdots,\eta_{d})^{\top}$. Then, $\theta$ and
$\eta$ are smooth functions of $u$: $\theta=\theta(u)$ and $\eta=\eta(u)$. In
this case, the Jacobian matrices of $\theta$ and $\eta$ with respect to $u$

$\displaystyle B^{i}_{a}(u)=\frac{\partial\theta^{i}}{\partial
u^{a}}\Big{|}_{\theta=\theta(u)}\quad\text{and}\quad
B_{ia}(u)=\frac{\partial\eta_{i}}{\partial u^{a}}\Big{|}_{\eta=\eta(u)}$

are important tools to describe geometric quantities. The Fisher information
matrix $(g_{ab}(u))_{1\leq a,b\leq d^{\prime}}$ with respect to $u$ is
expressed as

$\displaystyle
g_{ab}(u)=\sum_{i=1}^{d}\sum_{j=1}^{d}B^{i}_{a}B^{j}_{b}g_{ij}(\theta(u))=\sum_{i=1}^{d}\sum_{j=1}^{d}B_{ia}B_{jb}g^{ij}(\theta(u))$
(3)

and the $\mathrm{e}/\mathrm{m}$-connection coefficients $\raise
1.5pt\hbox{$\displaystyle\mathop{\Gamma}^{\rm e}$}^{\,}(u)$ and $\raise
1.5pt\hbox{$\displaystyle\mathop{\Gamma}^{\rm m}$}^{\,}(u)$ are given by

$\displaystyle\raise 1.5pt\hbox{$\displaystyle\mathop{\Gamma}^{\rm
e}$}_{abc}^{\,}(u)=\sum_{i=1}^{d}\sum_{j=1}^{d}\frac{\partial
B^{i}_{b}}{\partial u^{a}}B^{j}_{c}g_{ij}(\theta(u))\quad\text{and}\quad\raise
1.5pt\hbox{$\displaystyle\mathop{\Gamma}^{\rm
m}$}_{abc}^{\,}(u)=\sum_{i=1}^{d}\sum_{j=1}^{d}\frac{\partial B_{ib}}{\partial
u^{a}}B_{jc}g^{ij}(\eta(u)).$

The embedding structure also yield the $\mathrm{e}$-embedding curvature tensor
($\mathrm{e}$-Euler–Schouten embedding curvature). To define this, we
introduce an ancillary parameter
$v=(v^{d^{\prime}+1},\ldots,v^{\kappa},\ldots,v^{d})$ to each point
$\theta(u)$ so that the pair $(u,v)$ forms a parameterization of
$\\{P_{\theta}(\cdot):\theta\in\Theta\\}$ with $(u,0)$ implying $\theta(u)$,
and the $(u,v)$-components of the Fisher information matrix $g(u,v)$ with
respect to $(u,v)$ are zero. Using an ancillary parameter, we define the
$\mathrm{e}$-embedding curvature $\raise
1.5pt\hbox{$\displaystyle\mathop{H}^{\rm e}$}_{ab\kappa}^{\,}$ ($1\leq a,b\leq
d^{\prime}$ and $d^{\prime}+1\leq\kappa\leq d$) as

$\displaystyle\raise 1.5pt\hbox{$\displaystyle\mathop{H}^{\rm
e}$}_{ab\kappa}^{\,}(u)=\sum_{A\subseteq\mathcal{Y}}P_{\xi}(A)\left[\frac{\partial^{2}}{\partial
u^{a}\partial u^{b}}\log P_{(u,0)}(A)\frac{\partial}{\partial v^{\kappa}}\log
P_{(u,0)}(A)\right]=\sum_{i=1}^{d}\frac{\partial B^{i}_{b}}{\partial
u^{a}}\frac{\partial\eta_{i}}{\partial v^{\kappa}}\Big{|}_{v=0}.$ (4)

The $d^{\prime}\times d^{\prime}$-matrix

$[\raise 1.5pt\hbox{$\displaystyle\mathop{H}^{\rm
e}$}^{\,}]^{2}_{ab}(u):=\sum_{\kappa=d^{\prime}+1}^{d}\sum_{\lambda=d^{\prime}+1}^{d}\sum_{c=1}^{d^{\prime}}\sum_{d=1}^{d^{\prime}}\raise
1.5pt\hbox{$\displaystyle\mathop{H}^{\rm e}$}_{ac\kappa}^{\,}(u)\raise
1.5pt\hbox{$\displaystyle\mathop{H}^{\rm
e}$}_{bd\lambda}^{\,}(u)g^{\kappa\lambda}(u,0)g^{cd}(u,0)$

represents the square of $\mathrm{e}$-embedding curvature (Efron, 1975), where
$g^{\kappa\lambda}$ and $g^{cd}$ are the inverses of $g_{\kappa\lambda}$ and
$g_{cd}$, respectively. The $\mathrm{e}$-embedding curvature quantifies the
extent to which the model deviates from an exponential family. In fact, the
following holds.

######  Lemma 1 (Amari and Nagaoka (2000) and Sei (2011)).

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

Although the $\mathrm{m}$-embedding curvature is defined in the same way, we
do not discuss it in this paper and omit it.

##  3 Information-geometric structure of determinantal point processes

This section elucidates the embedding structure of DPP. This connects DPP to
the log-linear model (c.f., Agresti, 1990; Amari, 2001) and endows the DPP
model with interesting properties (Corollaries 1 and 2) and the duality behind
a marginal kernel $K$ and an $L$-ensemble kernel $L$ (Lemma 2).

###  3.1 Embedding structure of determinantal point processes

To present the results, we start with employing the diagonal scaling of a
positive definite kernel $L$ (Marshall and Olkin, 1968), also known as quality
vs. diversity decomposition (Kulesza and Taskar, 2012). Since $L$ is positive
definite, we parameterize $L$ as $L=DRD$ with the diagonal matrix $D$ and the
symmetric matrix $R$, where

$\displaystyle D:=\begin{pmatrix}\sqrt{L_{11}}&0&\cdots&0\\\
0&\sqrt{L_{22}}&\cdots&0\\\ \vdots&\vdots&\vdots&\vdots\\\ 0&0&\cdots&0\\\
0&0&\cdots&\sqrt{L_{mm}}\\\
\end{pmatrix}\,\text{and}\,R:=\begin{pmatrix}1&\rho_{12}&\cdots&\rho_{1m}\\\
\rho_{12}&1&\cdots&\rho_{2m}\\\ \vdots&\vdots&\vdots&\vdots\\\
\rho_{1(m-1)}&\rho_{2(m-1)}&\cdots&\rho_{(m-1)m}\\\
\rho_{1m}&\rho_{2m}&\cdots&1\\\ \end{pmatrix}$

with $\rho_{ij}:=L_{ij}/\sqrt{L_{ii}L_{jj}}$.

The following theorem gives us an embedding structure of DPP as a curved
exponential family. The exponential family in which DPP is embedded is a log-
linear model.

######  Theorem 1.

(a) The DPP model $\mathcal{P}_{\mathrm{DPP}}$ is written as the model
$\mathcal{P}:=\\{P_{u}(A)\,:\,u\in\mathbb{R}^{m}\times(-\infty,0)^{m(m-1)/2}\\}$
parameterized by an $m(m+1)/2$-dimensional parameter

$u=\left(u^{\\{1\\}},\ldots,u^{\\{m\\}},u^{\\{1,2\\}},u^{\\{1,3\\}},\ldots,u^{\\{(m-1),m\\}}\right)$

with the indices sorted in a lexicographical order of which member has the
form

$\displaystyle P_{u}(A)$
$\displaystyle=\exp\left(\sum_{\alpha_{1}\in\mathcal{S}_{1}^{m}}u^{\alpha_{1}}T_{\alpha_{1}}(A)+\sum_{\alpha_{2}\in\mathcal{S}_{2}^{m}}u^{\alpha_{2}}T_{\alpha_{2}}(A)+\sum_{k=3}^{m}\sum_{\mathcal{I}_{k}\in\mathcal{S}_{k}^{m}}\theta^{\mathcal{I}_{k}}(u)T_{\mathcal{I}_{k}}(A)-\psi(u)\right)$
(5)

where

$\displaystyle\mathcal{S}_{k}^{m}$
$\displaystyle:=\left\\{\\{i_{1},\ldots,i_{k}\\}\,:\,1\leq i_{1}<\ldots
i_{k}\leq m\right\\}$ $\displaystyle(1\leq k\leq m),$ $\displaystyle
T_{\mathcal{I}_{k}}(A)$ $\displaystyle:=1_{i_{1}\in A\,,\,\ldots,i_{k}\in
A}\quad\text{with}\quad\mathcal{I}_{k}=\\{i_{1},\ldots,i_{k}\\}$
$\displaystyle(\mathcal{I}_{k}\in\mathcal{S}_{k}^{m}\,\,\text{and}\,\,1\leq
k\leq m),$ $\displaystyle u^{\alpha_{1}}$ $\displaystyle:=\log L_{\alpha_{1}}$
$\displaystyle(\alpha_{1}\in\mathcal{S}_{1}^{m}),$ $\displaystyle
u^{\alpha_{2}}$ $\displaystyle:=\log(1-\rho_{\alpha_{2}}^{2})$
$\displaystyle(\alpha_{2}\in\mathcal{S}_{2}^{m}),$ $\displaystyle\psi(u)$
$\displaystyle:=\sum_{\alpha_{1}\in\mathcal{S}_{1}^{m}}u^{\alpha_{1}}+\log\mathrm{det}\begin{pmatrix}e^{-u^{\\{1\\}}}+1&\cdots&\sqrt{1-e^{u^{\\{1,m\\}}}}\\\
\vdots&\vdots&\vdots\\\
\sqrt{1-e^{u^{\\{1,m\\}}}}&\cdots&e^{-u^{\\{m\\}}}+1\end{pmatrix},$

and $\theta^{\mathcal{I}_{k}}(u)$
($\mathcal{I}_{k}=(i_{1},\ldots,i_{k})\in\mathcal{S}_{k}^{m}$ and $3\leq k\leq
m$) is defined recursively as

$\begin{split}\theta^{\mathcal{I}_{k}}(u)&:=\log\mathrm{det}\left(I_{k\times
k}+\left(1_{a\neq
b}\sqrt{1-e^{u^{\\{a,b\\}}}}\right)_{a,b=i_{1},\ldots,i_{k}}\right)\\\
&\quad\quad-\sum_{l=3}^{k-1}\sum_{\mathcal{J}_{l}\in\mathcal{S}_{l}^{m}\,:\,\mathcal{J}_{l}\subset\mathcal{I}_{k}}\theta^{\mathcal{J}_{l}}(u)-\sum_{\alpha_{2}\in\mathcal{S}_{2}^{m}\,:\,\alpha_{2}\subset\mathcal{I}_{k}}u^{\alpha_{2}}.\end{split}$
(6)

(b) The family $\mathcal{P}$ is the curved exponential family embedded in the
exponential family
$\mathcal{E}:=\\{P_{\theta}(A)\,:\,\theta\in\mathbb{R}^{2^{m}-1}\\}$ given by

$\displaystyle
P_{\theta}(A)=\exp\left(\sum_{k=1}^{m}\sum_{\mathcal{I}_{k}\in\mathcal{S}_{k}^{m}}\theta^{\mathcal{I}_{k}}T_{\mathcal{I}_{k}}(A)-\phi(\theta)\right),$
(7)

where $\phi(\theta)$ is given by

$\displaystyle\phi(\theta):=\log\sum_{A\subseteq\mathcal{Y}}\exp\left(\sum_{k=1}^{m}\sum_{\mathcal{I}_{k}\in\mathcal{S}_{k}^{m}}\theta^{\mathcal{I}_{k}}T_{\mathcal{I}_{k}}(A)\right).$

Note that the DPP model forms an exponential family for $m=1,2$ but do not for
$m\geq 3$. The underlying exponential family of the DPP model is nothing but
the log-linear model (c.f., Agresti, 1990; Amari, 2001); more specifically,
the log-linear model on a partially ordered set (Sugiyama et al., 2016). The
proof of this theorem is given in Section 5.

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 1 displays the construction of $u$ and $\theta(u)$ using a modification
of the Hasse diagram, where the figure is drawn for $m=4$. We begin with
drawing the Hasse diagram of the power set $2^{\mathcal{Y}}$ of an $m$-element
item $\mathcal{Y}=\\{1,\ldots,m\\}$ ordered by inclusion. A node of the Hasse
diagram corresponds to a subset of the power set, and a directed edge from a
subset $A$ to a subset $B$ is drawn when $A\neq B$, $A\subset B$, and there is
no $C$ distinct from $A$ and $B$ with $A\subset C\subset B$. A directed edge
in the Hasse diagram is called an upward path. We then eliminate all upward
paths starting from singletons $\\{m\\}$ ($1\leq a\leq m$) and the empty set
$\emptyset$. In this modified Hasse diagram, the second and the third layers
from the bottom correspond to the indices for the free parameter $u$. The
layers from the third to the top correspond to the indices for $\theta(u)$.
The recursive construction (6) in Theorem 1 is explained by upward paths in
the diagram. Each element $\theta^{\mathcal{I}_{k}}(u)$
($\mathcal{I}_{k}\in\mathcal{S}^{m}_{k}$, $3\leq k\leq m$) consists of

$\log\mathrm{det}(R_{\mathcal{I}_{k}})=\log\mathrm{det}\left(I_{k\times
k}+\left(1_{a\neq
b}\sqrt{1-e^{u^{\\{a,b\\}}}}\right)_{a,b=i_{1},\ldots,i_{k}}\right)$

subtracted by all $u^{\alpha_{2}}$ with $\alpha_{2}$ in the second layer that
are connected to $\mathcal{I}_{k}$ via upward paths and all
$\theta^{\mathcal{J}_{l}}$ with $\mathcal{J}_{l}$ in the $l$-th layer ($3\leq
l\leq k-1$) that are connected to $\mathcal{I}_{k}$ via upward paths. For
example, in the DPP model with $m=4$, the constrained parameteter
$\theta^{\\{1,2,4\\}}$ is constructed using parameters having upward paths to
$\\{1,2,3\\}$ depicted in the orange edges in Figure 1. Note that parameters
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$ in the second layer,
corresponding to the quality terms in the DPP literature, are not involved in
the construction, which is a key to the following analysis.

######  Remark 1 (Unidentifiability).

The parameterization using $u$ implies the essential unidentifiability issue
of DPPs posed by Brunel et al. (2017). In fact, $u^{\\{a,b\\}}$ only recovers
$\pm\rho_{ab}$ ($1\leq a<b\leq m$), which implies that two DPPs $P_{L}$ and
$P_{L^{\prime}}$ are identical if and only if there exists an $m\times m$
diagonal matrix $\tilde{D}$ with $\pm 1$ diagonal entries such that
$L^{\prime}=\tilde{D}L\tilde{D}$.

We next investigate a structure of the $\mathrm{e}$-embedding curvature.
Although the $\mathrm{e}$-embedding curvature of the DPP model is not
necessarily identically zero, it along with the direction of
$\partial/\partial u^{\alpha_{1}}$ ($\alpha_{1}\in\mathcal{S}_{1}^{m}$) is
shown to vanish, which implies that
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$ can be regarded as the
parameter of an exponential family.

######  Corollary 1.

The $\mathrm{e}$-embedding curvature $\raise
1.5pt\hbox{$\displaystyle\mathop{H}^{\rm e}$}_{\alpha_{1}\alpha\kappa}^{\,}$
vanishes for any $\alpha_{1}\in\mathcal{S}_{1}^{m}$,
$\alpha\in\mathcal{S}_{1}^{m}\cup\mathcal{S}_{2}^{m}$,
$\kappa\in\mathcal{S}_{k}^{m}$ ($k=3,\ldots,m$), and thus the square of the
$\mathrm{e}$-embedding curvature has the following structure:

$\displaystyle\left([\raise 1.5pt\hbox{$\displaystyle\mathop{H}^{\rm
e}$}^{\,}]^{2}_{\alpha\tilde{\alpha}}\right)_{\alpha,\tilde{\alpha}\in\mathcal{S}_{1}^{m}\cup\mathcal{S}_{2}^{m}}=\begin{pmatrix}0_{m\times
m}&0_{m\times m(m-1)/2}\\\ 0_{m(m-1)/2\times m}&*\end{pmatrix},$

where $0_{a\times b}$ ($a,b\in\mathbb{N}$) is the $a\times b$ zero matrix, and
$*$ implies an $m(m-1)/2\times m(m-1)/2$-matrix that is possibly non-zero.

The proof is given in Section 5.

Blessed with the embedding structure indicated by Corollary 1, we obtain the
following expression for the Fisher information matrix with respect to $u$.

######  Corollary 2.

(a) For $1\leq a,b\leq m$, the $(\\{a\\},\\{b\\})$-component of the Fisher
information matrix $\mathcal{G}$ with respect to $u$ is given by

$\displaystyle\mathcal{G}_{\\{a\\}\\{b\\}}(u)=\frac{\partial^{2}}{\partial
u^{\\{a\\}}\partial
u^{\\{b\\}}}\psi(u)=\begin{cases}K_{aa}(1-K_{aa})&(a=b),\\\ -K_{ab}^{2}&(a\neq
b).\end{cases}$

(b) For $\alpha_{1}\in\mathcal{S}_{1}^{m}$ and
$\alpha_{2}\in\mathcal{S}_{2}^{m}$, the $(\alpha_{1},\alpha_{2})$-component of
the Fisher information matrix $\mathcal{G}$ with respect to $u$ is given by

$\displaystyle\mathcal{G}_{\alpha_{1}\alpha_{2}}(u)=\mathrm{det}(K_{\alpha_{1}\cup\alpha_{2}})-\mathrm{det}(K_{\alpha_{1}})\mathrm{det}(K_{\alpha_{2}})+\sum_{\mathcal{B}_{n}\,:\,\alpha_{2}\subsetneq\mathcal{B}_{n}\,,\,2<n\leq
m}\\{\mathrm{det}(K_{\alpha_{2}})\left(1-\mathrm{det}(K_{\mathcal{B}_{n}})\right)\\}\frac{\partial\theta^{\mathcal{B}_{n}}}{\partial
u^{\alpha_{2}}}.$

A key tool for this result is the covariance expression (1) of the Fisher
information matrix for the underlying exponential family of the DPP model
evaluated at a DPP: Let
$\mathcal{I},\mathcal{J}\in\cup_{k=1}^{m}\mathcal{S}_{k}^{m}$. At the
distribution in the log-linear model (7) corresponding to a DPP (that is,
$\theta=\theta(u)$), the $(\mathcal{I},\mathcal{J})$-component of the Fisher
information matrix $\mathcal{G}(\theta)$ with respect to $\theta$ of the log-
linear model (7) is given

$\displaystyle\mathcal{G}_{\mathcal{I}\mathcal{J}}(\theta)\Big{|}_{\theta=\theta(u)}=\mathrm{Cov}_{\theta(u)}[T_{\mathcal{I}},T_{\mathcal{J}}]=\mathrm{det}(K_{\mathcal{I}\cup\mathcal{J}})-\mathrm{det}(K_{\mathcal{I}})\mathrm{det}(K_{\mathcal{J}}).$

The proof of this result is given in Section 5.

######  Remark 2 (Conditional potential function).

The first part of Corollary 2 conveys an implication on the role of $\psi(u)$:
that is, the Fisher information matrix for
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$ is characterized by the
sub-matrix of Hessian of $\psi(u)$. Because of this characterization, we call
$\psi(u)$ a conditional potential function. This property is not obvious since
the DPP model is not an exponential family. In fact, this characterization
using the Hessian matrix is not applicable for the whole Fisher information
matrix: $\mathcal{G}(u)\neq\nabla^{2}_{u}\psi(u)$ in general. Blessed with the
$\mathrm{e}$-embedding flat structure of
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$, the part of the Fisher
information matrix related to
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$ has this property.

######  Remark 3 (Non-orthogonality).

From Corollary 2 (b), item-wise effects (quality terms)
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$ and pair-wise effects
(diversity terms) $(u^{\alpha_{2}})_{\alpha_{2}\in\mathcal{S}_{2}^{m}}$ are
not necessarily orthogonal in the Fisher information matrix: for
$\alpha_{1}\in\mathcal{S}_{1}^{m}$ and $\alpha_{2}\in\mathcal{S}_{2}^{m}$,
$\mathcal{G}_{\alpha_{1}\alpha_{2}}(u)\neq 0$ in general. A disappointing
message from this is that estimation of
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$ and that of
$(u^{\alpha_{2}})_{\alpha_{2}\in\mathcal{S}_{2}^{m}}$ cannot be separated to
attain efficient estimation.

###  3.2 Duality in determinantal point processes

Theorem 1 clarifies the role of an $L$-ensemble kernel in describing the DPP
model as a curved exponential family. What is a role of a marginal kernel $K$
in describing a DPP as a statistical model? The following lemma answers this
question and implies that the expectation parameter $\eta=\eta(u)$ of the
underlying exponential family (7) restricted to the DPP model is given by a
marginal kernel $K$. This relationship between the $K$ and $L$ matrices
exhibits a duality within the DPP model, revealing how both matrices play
pivotal roles in the model’s geometric structure.

######  Lemma 2.

The expectation parameter of (7) evaluated at a DPP is written as

$\eta_{\mathcal{I}_{k}}=\mathrm{det}(K_{\mathcal{I}_{k}})\quad(\mathcal{I}_{k}\in\mathcal{S}_{k}^{m},\quad
k=1,\ldots,m).$

Further, the expectation parameter $\eta_{\alpha_{1}}$ for
$\alpha_{1}\in\mathcal{S}_{1}^{m}$ evaluated at a DPP is written as

$\displaystyle\eta_{\alpha_{1}}=(\partial/\partial u^{\alpha_{1}})\psi(u).$
(8)

###### Proof.

Since $\eta_{\mathcal{I}_{k}}$ is equal to
$\mathrm{E}_{\theta}[T_{\mathcal{I}_{k}}]$, and
$\mathrm{E}_{\theta}[T_{\mathcal{I}_{k}}]$ at a DPP is equal to
$\mathrm{det}(K_{\mathcal{I}_{k}})$, we conclude the first assertion. The
score vector of the DPP model with respect to
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$ is given by

$(\partial/\partial u^{\alpha_{1}})\log
P_{u}(A)=T_{\alpha_{1}}(A)-(\partial/\partial
u^{\alpha_{1}})\psi(u)\quad(\alpha_{1}\in\mathcal{S}_{1}^{m}).$

Then taking the expectation of both sides yields

$0=\eta_{\alpha_{1}}-(\partial/\partial
u^{\alpha_{1}})\psi(u),\quad(\alpha_{1}\in\mathcal{S}_{1}^{m})$

which concludes the second assertion. ∎

######  Remark 4 (Relation to the Laplace expansion).

The latter claim is also verified via the Laplace expansion. We take
$\eta_{\\{1\\}}=K_{11}$ as an example. By working with the diagonal scaling
$L=DRD$, $K_{11}$ has the form

$K_{11}=\left(L(L+I_{m\times
m})^{-1}\right)_{11}=\left(DR(R+D^{-2})^{-1}D^{-1}\right)_{11}=\left(R(R+D^{-2})^{-1}\right)_{11}.$

Let $\Delta_{ij}$ ($1\leq i,j\leq m$) denote the $(i,j)$-cofactor of
$(R+D^{-2})$. Using the cofactors of $(R+D^{-2})$, we get the following
expression of $K_{11}$:

$K_{11}=\sum_{j=1}^{m}R_{1j}\left((R+D^{-2})^{-1}\right)_{j1}=\sum_{j=1}^{m}R_{1j}\frac{\Delta_{j1}}{\mathrm{det}(R+D^{-2})}=\frac{\Delta_{11}+\sum_{j=2}^{m}\rho_{1j}\Delta_{j1}}{\mathrm{det}(R+D^{-2})}.$

Note that the Laplace expansion yields

$\mathrm{det}(R+D^{-2})=\sum_{j=1}^{m}(R+D^{-2})_{1j}\Delta_{j1}=(1+e^{-u^{\\{1\\}}})\Delta_{11}+\sum_{j=2}^{m}\rho_{1j}\Delta_{j1},$

which gives

$\displaystyle
K_{11}=\frac{\left\\{(1+e^{-u^{\\{1\\}}})\Delta_{11}+\sum_{j=2}^{m}\rho_{1j}\Delta_{j1}\right\\}-e^{-u^{\\{1\\}}}\Delta_{11}}{\mathrm{det}(R+D^{-2})}=1-e^{-u^{\\{1\\}}}\frac{\Delta_{11}}{\mathrm{det}(R+D^{-2})}.$
(9)

At the same time, the partial derivative $(\partial/\partial
u^{\\{1\\}})\psi(u)$ is written as

$\displaystyle(\partial/\partial u^{\\{1\\}})\psi(u)$
$\displaystyle=(\partial/\partial
u^{\\{1\\}})\\{\log\mathrm{det}D^{2}+\log\mathrm{det}(R+D^{-2})\\}$
$\displaystyle=1+\frac{\partial(R+D^{-2})_{11}}{\partial
u^{\\{1\\}}}\frac{\Delta_{11}}{\mathrm{det}(R+D^{-2})}$
$\displaystyle=1-e^{-u^{\\{2\\}}}\frac{\Delta_{11}}{\mathrm{det}(R+D^{-2})}.$
(10)

Thus, combining (9) with (10) also verifies the second claim of Lemma 2.

Duality revealed by Lemma 2 suggests us yet another parameterization of the
DPP model:

$\omega:=(\eta_{\\{1\\}},\ldots,\eta_{\\{m\\}}\,;\,u^{\\{1,2\\}},\ldots,u^{\\{(m-1),m\\}}).$

This type of parameterization has been often used in the analyses of the log-
linear model (Amari, 2001; Sugiyama et al., 2016) and more generally models
with the dual foliation structure such as the minimum information dependence
model (Sei and Yano, 2024). It is called a mixed parameterization. It may not
be obvious that the mixed parameterization is well-defined since the DPP model
is a curved exponential family. By employing a property of the conditional
potential function $\psi(u)$, we ensure the well-definedness of the mixed
parameterization $\omega$ in the DPP model as follows.

######  Lemma 3.

There is a one-to-one correspondence between $\omega$ and $u$.

###### Proof.

Given $(u^{\alpha_{2}})_{\alpha_{2}\in\mathcal{S}_{2}^{m}}$, $\psi(u)$ is
strictly convex by Corollary 2. This, together with Lemma 2 implies that the
function
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}\mapsto(\partial\phi(u)/\partial
u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$ is injective, which
completes the proof. ∎

######  Remark 5 (The Kullback–Leiber divergence).

Using the mixed parameterization, we obtain an interesting formula for the
Kullback–Leibler divergence between DPPs. Let $u$ and $\tilde{u}$ be two
points in $\mathbb{R}^{m}\times(-\infty,0)^{m(m-1)/2}$ with
$u^{\alpha_{2}}=\tilde{u}^{\alpha_{2}}$ ($\alpha_{2}\in\mathcal{S}_{2}^{m}$).
Then we have

$\displaystyle
D[\,P_{u}\,:\,P_{\tilde{u}}\,]:=\sum_{A\subseteq\mathcal{Y}}P_{u}(A)\log\frac{P_{u}(A)}{P_{\tilde{u}}(A)}=\psi^{*}(\omega)+\psi(\tilde{u})-\sum_{\alpha_{1}\in\mathcal{S}_{1}^{m}}\tilde{u}^{\alpha_{1}}\eta_{\alpha_{1}},$

where $\omega$ denotes the mixed parameterization of $u$, and
$\psi^{*}(\omega)$ is the Legendre transform of $\psi(u)$ given
$(u^{\alpha_{2}})_{\alpha_{2}\in\mathcal{S}_{2}^{m}}$:

$\psi^{*}(\omega)=\max_{u^{\alpha_{1}}\in\mathbb{R}^{m}}\left\\{u^{\alpha_{1}}\eta_{\alpha_{1}}-\psi(u)\right\\}.$

This type of expression appears in the Kullback–Leibler divergence for an
exponential family. Endowed with the $\mathrm{e}$-embedding flat structure
(Corollary 1) of $(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{{m}}}$, the
Kullback–Leibler divergence for the DPP model partially enjoys this
expression.

##  4 Examples

This section analytically checks the results by using DPP with $m=2$. It also
analyzes the structure of DPP with $m=3$ by using the numerical computation.

###  4.1 Determinantal point process with $m=2$

We begin with DPP with $m=2$. Observe that with the diagonal scaling $L=DRD$,
all probabilities are given by

$\displaystyle P_{L}(\emptyset)$ $\displaystyle=e^{-c(L)},\quad
P_{L}(\\{1\\})=e^{\log L_{11}-c(L)},\quad P_{L}(\\{2\\})=e^{\log
L_{22}-c(L)},$ $\displaystyle P_{L}(\\{1,2\\})$ $\displaystyle=e^{\log
L_{11}+\log L_{22}+\log|1-\rho_{12}^{2}|-c(L)},$

where $c(L):=\log\mathrm{det}(L+I_{2\times 2})$. This implies that the DPP
model with $m=2$ forms an exponential family

$\displaystyle
P_{L}(A)=P_{u}(A)=e^{u^{\\{1\\}}T_{\\{1\\}}(A)+u^{\\{2\\}}T_{\\{2\\}}(A)+u^{\\{1,2\\}}T_{\\{1,2\\}}(A)-\psi(u)},$

where

$\displaystyle T_{\\{1\\}}(A)$ $\displaystyle=1_{1\in A},\quad
T_{\\{2\\}}(A)=1_{2\in A},\quad T_{\\{1,2\\}}(A)=1_{1,2\in A}\quad\text{and}$
$\displaystyle u^{\\{1\\}}$ $\displaystyle=\log L_{11},\quad u^{\\{2\\}}=\log
L_{22},\quad u^{\\{1,2\\}}=\log(1-\rho_{12}^{2})$ $\displaystyle\psi(u)$
$\displaystyle=u^{\\{1\\}}+u^{\\{2\\}}+\log\left\\{e^{u^{\\{1,2\\}}}+e^{-u^{\\{1\\}}}+e^{-u^{\\{2\\}}}+e^{-u^{\\{1\\}}-u^{\\{2\\}}}\right\\}.$

The conditional potential function $\psi(u)$ becomes an actual potential
function. Figure 2 (a) displays the contour plot of $\psi(u)$ with respect to
$u^{\\{1\\}}$ and $u^{\\{2\\}}$ at $u^{\\{1,2\\}}=-0.1$, from which we see the
convex shape of $\psi(u)$ with respect to $u^{\\{1\\}}$ and $u^{\\{2\\}}$. The
$\mathrm{e}$-embedding curvature vanishes because this DPP model is an
exponential family.

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.

We check the duality in a marginal kernel $K$ and an $L$-ensemble kernel $L$.
Working with the identity $K=L(L+I_{2\times 2})^{-1}=R(R+D^{-2})^{-1}$, we
obtain the identities

$\displaystyle K_{11}$
$\displaystyle=\frac{e^{-u^{\\{2\\}}}+e^{u^{\\{1,2\\}}}}{e^{-u^{\\{1\\}}}+e^{-u^{\\{2\\}}}+e^{-u^{\\{1\\}}-u^{\\{2\\}}}+e^{u^{\\{1,2\\}}}},\quad
K_{22}=\frac{e^{-u^{\\{1\\}}}+e^{u^{\\{1,2\\}}}}{e^{-u^{\\{1\\}}}+e^{-u^{\\{2\\}}}+e^{-u^{\\{1\\}}-u^{\\{2\\}}}+e^{u^{\\{1,2\\}}}},$
$\displaystyle\mathrm{det}(K)$
$\displaystyle=\frac{e^{u^{\\{1,2\\}}}}{e^{-u^{\\{1\\}}}+e^{-u^{\\{2\\}}}+e^{-u^{\\{1\\}}-u^{\\{2\\}}}+e^{u^{\\{1,2\\}}}}.$

Together with the differentiation of the (conditional) potential function
$\psi(u)$, we confirm the duality formulae:

$\displaystyle K_{11}(=:\eta_{\\{1\\}})=\frac{\partial\psi(u)}{\partial
u^{\\{1\\}}},\quad K_{22}(=:\eta_{\\{2\\}})=\frac{\partial\psi(u)}{\partial
u^{\\{2\\}}},\quad\text{and}\quad\mathrm{det}(K)(=:\eta_{\\{1,2\\}})=\frac{\partial\psi(u)}{\partial
u^{\\{1,2\\}}}.$

Consider the Fisher information matrix for the mixed parameterization
$\omega=(\eta_{\\{1\\}},\eta_{\\{2\\}}\,;\,\theta^{\\{1,2\\}})$. Observe that
the DPP also has the form

$P_{\theta}(A)=(\eta_{\\{1\\}}-\eta_{\\{1,2\\}})1_{A=\\{1\\}}+(\eta_{\\{2\\}}-\eta_{\\{1,2\\}})1_{A=\\{2\\}}+\eta_{\\{1,2\\}}1_{A=\\{1,2\\}}+(1-\eta_{\\{1\\}}-\eta_{\\{2\\}}+\eta_{\\{1,2\\}})1_{A=\emptyset}.$

This, together with $(\partial/\partial\theta^{\\{1,2\\}})\log
P_{\theta}(A)=T_{\\{1,2\\}}(A)-\mathrm{E}_{\theta}[T_{\\{1,2\\}}(A)]$, gives

$\displaystyle\mathcal{G}_{\\{1\\}\\{1,2\\}}(\omega)=\sum_{A\subseteq
2^{\\{1,2\\}}}P_{\theta}(A)\frac{\partial}{\partial\eta_{\\{1\\}}}\log
P_{\theta}(A)\frac{\partial}{\partial\theta^{\\{1,2\\}}}\log
P_{\theta}(A)=\frac{\partial\psi(u)}{\partial\eta_{\\{1\\}}\partial
u^{\\{1,2\\}}}.$

###  4.2 Determinantal point process with $m=3$

We then work with DPP of $m=3$. In this case, we get the following form of the
DPP model as the curved exponential family (5):

$\displaystyle P_{u}(A)=e^{\sum_{a=1}^{3}u^{\\{a\\}}T_{\\{a\\}}(A)+\sum_{1\leq
a<b\leq
3}u^{\\{a,b\\}}T_{\\{a,b\\}}(A)+\theta^{\\{1,2,3\\}}T_{\\{1,2,3\\}}(A)-\psi(u)},$

where

$\displaystyle T_{\\{a\\}}(A)$ $\displaystyle=1_{a\in A},\quad
u^{\\{a\\}}=\log L_{aa}\quad(a=1,2,3)$ $\displaystyle T_{\\{a,b\\}}(A)$
$\displaystyle=1_{a\in,b\in A},\quad
u^{\\{a,b\\}}=\log(1-\rho_{ab}^{2})\quad(1\leq a<b\leq 3),$

and

$\displaystyle\begin{split}\theta^{\\{1,2,3\\}}&=\log\left(e^{u^{\\{1,2\\}}}+e^{u^{\\{2,3\\}}}+e^{u^{\\{1,3\\}}}+2\sqrt{(1-e^{u^{\\{1,2\\}}})(1-e^{u^{\\{2,3\\}}})(1-e^{u^{\\{1,3\\}}})}-2\right)\\\
&\quad-u^{\\{1,2\\}}-u^{\\{2,3\\}}-u^{\\{1,3\\}}.\end{split}$

The conditional potential function $\psi(u)$ is given by

$\displaystyle\psi(u)$ $\displaystyle=u^{\\{1\\}}+u^{\\{2\\}}+u^{\\{3\\}}$
$\displaystyle\quad+\log\left((1+e^{-u^{\\{1\\}}})(1+e^{-u^{\\{2\\}}})(1+e^{-u^{\\{3\\}}})+2\sqrt{(1-e^{u^{\\{1,2\\}}})(1-e^{u^{\\{1,3\\}}})(1-e^{u^{\\{2,3\\}}})}\right.$
$\displaystyle\left.\quad\quad\quad\quad-(1-e^{u^{\\{1,2\\}}})(1+e^{-u^{\\{3\\}}})-(1-e^{u^{\\{1,3\\}}})(1+e^{-u^{\\{2\\}}})-(1-e^{u^{\\{2,3\\}}})(1+e^{-u^{\\{1\\}}})\right).$

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 $u^{\\{1,3\\}}=-0.8$

Here we see shapes of the conditional potential function $\psi(u)$ and the
embedding $\theta^{\\{1,2,3\\}}(u)$. Figure 2 (b) exhibits the shape of the
conditional potential function $\psi(u)$. Although it has a relatively complex
shape, the contour plot reveals the convex shape of $\psi(u)$ with respect to
$u^{\\{1\\}}$ and $u^{\\{2\\}}$. Figure 3 shows the embedding structure of the
DPP model via
$\theta^{\\{1,2,3\\}}(u^{\\{1,2\\}},u^{\\{2,3\\}},u^{\\{1,3\\}})$. The
embedding $\theta^{\\{1,2,3\\}}(u^{\\{1,2\\}},u^{\\{2,3\\}},u^{\\{1,3\\}})$ is
symmetric as described by the explicit form and depicted by Figure 3. The
gradient of $\theta^{\\{1,2,3\\}}(u^{\\{1,2\\}},u^{\\{2,3\\}},u^{\\{1,3\\}})$
with respect to $(u^{\\{1,2\\}},u^{\\{2,3\\}})$ heavily depends on
$u^{\\{1,3\\}}$ as in Figure 3.

##  5 Proofs

This section presents the proofs for Theorem 1, and Corollaries 1 and 2.

###### Proof of Theorem 1.

Before the main proof, note that by using the diagonal scaling $L=DRD$,
$\log\mathrm{det}(L+I_{m\times m})$ is written as $\psi(u)$ in the following
manner:

$\displaystyle\log\mathrm{det}(L+I_{m\times m})$
$\displaystyle=\log\mathrm{det}D^{2}+\log\mathrm{det}(R+D^{-2})$
$\displaystyle=\sum_{i=1}^{m}\log L_{ii}+\log\mathrm{det}(R+D^{-2})$
$\displaystyle=\psi(u).$

For the main proof, we consider $P_{u}(A)$ in four cases: $|A|=1$, $|A|=2$,
$|A|=3$, $|A|\geq 4$.

Case 1 ($A=\\{i\\}$, $1\leq i\leq m$): For $A=\\{i\\}$, the probability
$P_{u}(A)$ is written as

$\displaystyle P_{u}(A)$ $\displaystyle=\exp(\log
L_{ii}-\psi(u))=\exp(u^{\\{i\\}}-\psi(u))=\exp\left(\sum_{\alpha_{1}\in\mathcal{S}_{1}^{m}}u^{\alpha_{1}}T_{\alpha_{1}}(A)-\psi(u)\right),$

which completes Case 1.

Case 2 ($A=\\{i,j\\}$, $1\leq i<j\leq m$): Consider $A=\\{i,j\\}$. By the
diagonal scaling of $L$, the probability $P_{u}(A)$ is written as

$\displaystyle P_{u}(A)$
$\displaystyle=\exp(\log\mathrm{det}L_{\\{i,j\\}}-\psi(u))=\exp(\log\mathrm{det}D_{\\{i,j\\}}R_{\\{i,j\\}}D_{\\{i,j\\}}-\psi(u)).$

This, together with $\mathrm{det}R_{\\{i,j\\}}=1-\rho_{ij}^{2}$, yields

$\displaystyle P_{u}(A)$
$\displaystyle=\exp(u^{\\{i\\}}+u^{\\{j\\}}+u^{\\{i,j\\}}-\psi(u))=\exp\left(\sum_{\alpha_{1}\in\mathcal{S}_{1}^{m}}u^{\alpha_{1}}T_{\alpha_{1}}(A)+\sum_{\alpha_{2}\in\mathcal{S}_{2}^{m}}u^{\alpha_{2}}T_{\alpha_{2}}(A)-\psi(u)\right),$

which completes Case 2.

Case 3 ($A=\\{i,j,k\\}$, $1\leq i<j<k\leq m$): Consider $A=\\{i,j\\}$. By the
same procedure as before, we get

$\displaystyle
P_{u}(A)=\exp(\log\mathrm{det}L_{\\{i,j,k\\}}-\psi(u))=\exp\left(u^{\\{i\\}}+u^{\\{j\\}}+u^{\\{k\\}}+\log\mathrm{det}R_{\\{i,j,k\\}}-\psi(u)\right).$

Observe $A=\\{i,j\\}\cup\\{j,k\\}\cup\\{k,i\\}$. Then

$u^{\\{i,j\\}}T_{\\{i,j\\}}(A)+u^{\\{j,k\\}}T_{\\{j,k\\}}(A)+u^{\\{i,k\\}}T_{\\{i,k\\}}(A)=\sum_{\alpha_{2}\subset\\{i,j,k\\}}u^{\alpha_{2}}T_{\alpha_{2}}(A)=u^{\\{i,j\\}}+u^{\\{j,k\\}}+u^{\\{i,k\\}}$

and thus we get

$\displaystyle
P_{u}(A)=\exp\left(\sum_{\alpha_{1}\in\mathcal{S}_{1}^{m}}u^{\alpha_{1}}T_{\alpha_{1}}(A)+\sum_{\alpha_{2}\in\mathcal{S}_{2}^{m}}u^{\alpha_{2}}T_{\alpha_{2}}(A)+\log\mathrm{det}R_{\\{i,j,k\\}}-\sum_{\alpha_{2}\subset(i,j,k)}u^{\alpha_{2}}T_{\alpha_{2}}(A)\right).$

Together with the formula

$\log\mathrm{det}R_{\\{i,j,k\\}}=\log\mathrm{det}\left(I_{3\times
3}+(\sqrt{1-e^{u^{\\{a,b\\}}}})_{\\{a,b\\}\subset\\{i,j,k\\}}\right),$

we complete Case 3.

Case 4 ($|A|=k\geq 4$): For $A=\mathcal{I}_{k}$ and $l\leq k$, we decompose
$A$ as

$A=\mathcal{I}_{k}=\cup_{\mathcal{J}_{l}\subset\mathcal{I}_{k}}\mathcal{J}_{l}$

and we get

$\sum_{\mathcal{J}_{l}\subset\mathcal{I}_{k}}\theta^{\mathcal{J}_{l}}(u)T_{\mathcal{J}_{l}}(A)=\sum_{\mathcal{J}_{l}\in\mathcal{S}_{l}^{m}}\theta^{\mathcal{J}_{l}}(u)T_{\mathcal{J}_{l}}(A)=\sum_{\mathcal{J}_{l}\subset\mathcal{I}_{k}}\theta^{\mathcal{J}_{l}},$

which, together with the same procedure as in Case 3, completes the proof of
(a). The proof of (b) is obvious. ∎

###### Proof of Corollary 1.

Observe that we have

$\displaystyle B=\frac{\partial\theta}{\partial u}=\begin{pmatrix}I_{m\times
m}&0_{m\times m(m-1)/2}\\\ 0_{m(m-1)/2\times m}&I_{m(m-1)/2\times m(m-1)/2}\\\
0_{(2^{m}-1-m(m+1)/2)\times m}&C\end{pmatrix}$

where $C$ is the $((2^{m}-1-m(m+1)/2)\times m(m-1)/2)$-matrix that corresponds
to $\partial\theta^{\mathcal{I}_{k}}/\partial u^{\alpha_{2}}$
($\mathcal{I}_{k}\in\mathcal{S}_{k}^{m}$, $3\leq k\leq m$,
$\alpha_{2}\in\mathcal{S}_{2}^{k}$). So, the Jacobian matrix
$\partial\theta/\partial u$ does not depend on
$(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$. Together with equation
(4), this completes the proof. ∎

###### Proof of Corollary 2.

Consider (a). Using the expression of the score vector of the DPP model with
respect to $(u^{\alpha_{1}})_{\alpha_{1}\in\mathcal{S}_{1}^{m}}$

$(\partial/\partial u^{\alpha_{1}})\log
P_{u}(A)=T_{\alpha_{1}}(A)-(\partial/\partial
u^{\alpha_{1}})\psi(u)\quad(\alpha_{1}\in\mathcal{S}_{1}^{m}),$

and the expression of the Fisher information matrix
$\mathcal{G}(u)=\mathrm{E}_{\theta(u)}[-\nabla_{u}^{2}\log P_{u}(A)]$, we get

$\mathcal{G}_{\alpha_{1}\tilde{\alpha}_{1}}(u)=\frac{\partial^{2}\psi(u)}{\partial
u^{\alpha_{1}}\partial u^{\tilde{\alpha}_{1}}}.$

Next, observe that the Jacobian matrix has the form

$\displaystyle B=\frac{\partial\theta}{\partial u}=\begin{pmatrix}I_{m\times
m}&0_{m\times m(m-1)/2}\\\ 0_{m(m-1)/2\times m}&I_{m(m-1)/2\times m(m-1)/2}\\\
0_{(2^{m}-1-m(m+1)/2)\times m}&C\end{pmatrix}$ (11)

where $C$ is the $((2^{m}-1-m(m+1)/2)\times m(m-1)/2)$-matrix that corresponds
to $\partial\theta^{\mathcal{I}_{k}}/\partial u^{\alpha_{2}}$
($\mathcal{I}_{k}\in\mathcal{S}_{k}^{m}$, $3\leq k\leq m$,
$\alpha_{2}\in\mathcal{S}_{2}^{k}$). Denote the Fisher information matrix with
respect to $\theta$ of the log-linear model evaluated at $\theta=\theta(u)$ by
$M$ and write it in the block-matrix notion:

$\displaystyle M=\begin{pmatrix}M_{ss}&M_{sp}&M_{sh}\\\
M_{ps}&M_{pp}&M_{ph}\\\ M_{hs}&M_{hp}&M_{hh}\end{pmatrix},$

where $M_{ss}$ is an $m\times m$ sub-matrix, $M_{sp}$ is an $m\times m(m-1)/2$
sub-matrix, $M_{sh}$ is an $m\times(2^{m}-1-m(m+1)/2)$ sub-matrix, and the
remaining sub-matrix are defined in similar ways. Then the expression (3)
gives

$\displaystyle\mathcal{G}(u)=B^{\top}MB=\begin{pmatrix}M_{ss}&M_{sp}+M_{sh}C\\\
M_{ps}+C^{\top}M_{hs}&*\end{pmatrix},$

where $*$ is a suitable matrix. By using equation (1), for
$\alpha_{1},\tilde{\alpha}_{1}\in\mathcal{S}_{1}^{m}$, the
$(\alpha_{1},\tilde{\alpha}_{1})$-component of $M_{ss}$ is expressed as

$\displaystyle(M_{ss})_{\alpha_{1}\tilde{\alpha}_{1}}=\mathrm{Cov}_{\theta(u)}[T_{\alpha_{1}},T_{\tilde{\alpha}_{1}}]=\mathrm{det}(K_{\alpha_{1}\cup\tilde{\alpha}_{1}})-\mathrm{det}(K_{\alpha_{1}})\mathrm{det}(K_{\tilde{\alpha}_{1}}),$

which concludes (a). Noting that $\partial\theta^{\mathcal{I}_{k}}/\partial
u^{\alpha_{2}}$ ($\mathcal{I}_{k}\in\mathcal{S}_{k}^{m}$, $3\leq k\leq m$,
$\alpha_{2}\in\mathcal{S}_{2}^{m}$) is zero if
$\alpha_{2}\subset\mathcal{I}_{k}$ (see also Figure 1), we obtain (b). ∎

##  6 Acknowledgement

The authors thank Takahiro Kawashima for helpful comments to the early version
of this work. This work is supported by JSPS KAKENHI (21H05205, 21K12067,
23K11024, JP22H03653, 23H04483), MEXT (JPJ010217).

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