#  Equality cases in the Anantharam–Jog–Nair inequality

Efe Aras, Thomas A. Courtade and Albert Zhang
University of California, Berkeley

( )

###### Abstract

Anantharam, Jog and Nair recently unified the Shannon–Stam inequality and the
entropic form of the Brascamp–Lieb inequalities under a common inequality.
They left open the problems of extremizability and characterization of
extremizers. Both questions are resolved in the present paper.

##  1 Preliminaries

We begin by briefly fixing notation and definitions that will be needed
throughout. A Euclidean space $E$ is a finite-dimensional Hilbert space over
the real field, equipped with Lebesgue measure. For a probability measure
$\mu$ on $E$, absolutely continuous with respect to Lebesgue measure, and a
random vector $X\sim\mu$, we define the Shannon entropy

$h(X)\equiv h(\mu):=-\int_{E}\log\left(\frac{d\mu}{dx}\right)d\mu,$

provided the integral exists. If $\mu$ is not absolutely continuous with
respect to Lebesgue measure, we adopt the convention that $h(\mu):=-\infty$.
We let $\mathcal{P}(E)$ denote the set of probability measures on $E$ having
finite entropies and second moments. When there is no cause for ambiguity, we
adopt the usual notational convention where a random vector $X$ and its law
$\mu$ are used interchangeably. So, for example, writing $X\in\mathcal{P}(E)$
means that $X$ is a random vector taking values in $E$, having finite entropy
and finite second moments.

For $x,y\in E$, we denote the standard (Euclidean) inner product as $x^{T}y$,
and denote the Euclidean metric by $|\cdot|$ (i.e., $|x|:=\sqrt{x^{T}x}$). If
$A:E\to E^{\prime}$ is a linear map between Euclidean spaces $E,E^{\prime}$,
we let $A^{T}:E^{\prime}\to E$ denote its adjoint satisfying

$(Ax)^{T}y=x^{T}(A^{T}y),~{}~{}~{}\forall x\in E,y\in E^{\prime}.$

All of this notation is consistent with the representation of linear maps as
matrices. We let $\mathbf{S}(E)$ denote the set of symmetric linear maps from
$E$ to itself (i.e., $A\in\mathbf{S}(E)$ iff $A=A^{T}$), and
$\mathbf{S}^{+}(E)$ denote the subset of positive definite linear maps (i.e.,
$A\in\mathbf{S}^{+}(E)$ iff $A=A^{T}$ and $x^{T}Ax>0$ for all nonzero $x\in
E$).

For a random vector $X\sim\mu\in\mathcal{P}(E)$, its covariance is defined as
the (positive semidefinite) symmetric linear map

$\operatorname{Cov}(X)=\int_{E}(x-\mathbb{E}[X])(x-\mathbb{E}[X])^{T}d\mu(x)\in\mathbf{S}(E),$

where $\mathbb{E}$ denotes expectation (here, with respect to $\mu$). The
Gaussian distribution on $E$ with mean $m$ and covariance
$\Sigma\in\mathbf{S}^{+}(E)$ is denoted by $N(m,\Sigma)$. A Gaussian random
vector $X$ is said to be isotropic if it has covariance proportional to the
identity map. The standard Gaussian distribution on $E$ is denoted by
$\gamma_{E}$.

Of course, all Euclidean spaces $E,E^{\prime}$ of dimensions $m$ and $n$,
respectively, can always be identified as $\mathbb{R}^{m}$ and
$\mathbb{R}^{n}$, respectively. Moreover, any linear transformation $A:E\to
E^{\prime}$ can be expressed as a real $n\times m$ matrix. Our notation is
chosen to be compatible with this, but for various reasons it is notationally
more convenient to state things abstractly. For example, this avoids ambiguity
that can result from referring to two different Euclidean spaces of the same
dimension.

Throughout, we consider collections of Euclidean spaces $(E_{i})_{i=1}^{k}$,
$(E^{j})_{j=1}^{m}$, and corresponding sets of positive real numbers
$\mathbf{c}=(c_{i})_{i=1}^{k}$ and $\mathbf{d}=(d_{j})_{j=1}^{m}$. A datum is
a triplet $(\mathbf{c},\mathbf{d},\mathbf{B})$ where
$\mathbf{B}=(B_{j})_{j=1}^{m}$ is a collection of linear maps $B_{j}:E_{0}\to
E^{j}$, with common domain $E_{0}:=\oplus_{i=1}^{k}E_{i}$. Given the structure
of $E_{0}$, we let $\pi_{E_{i}}:E_{0}\to E_{i}$ denote the coordinate
projections. A vector $x\in E_{0}$ will frequently be written in its
coordinate representation $x=(x_{1},\dots,x_{k})$, where
$x_{i}=\pi_{E_{i}}(x)$, $1\leq i\leq k$. If $A_{i}:E_{i}\to E_{i}$, $1\leq
i\leq k$, are linear maps, then the direct sum of operators
$A=\oplus_{i=1}^{k}A_{i}$ is a linear map from $E_{0}$ to itself and, without
confusion, can be denoted as the block-diagonal operator

$A=\operatorname{diag}(A_{1},\dots,A_{k}).$

For a set $V$, we let $\operatorname{id}_{V}:V\to V$ denote the identity map
from $V$ to itself. So, as an example of the above, we have
$\operatorname{id}_{E_{0}}=\oplus_{i=1}^{k}\operatorname{id}_{E_{i}}\equiv\operatorname{diag}(\operatorname{id}_{E_{1}},\dots,\operatorname{id}_{E_{k}})$.
Again, this is all compatible with the representation of linear operators as
matrices.

We conclude this section by recording a few associated definitions for
convenience.

######  Definition 1.

A subspace $T\subset E_{0}$ is said to be product-form if it can be written as
$T=\oplus_{i=1}^{k}T_{i}$, where $T_{i}\subset E_{i}$ for $1\leq i\leq k$.

######  Definition 2.

A subspace $T\subset E_{0}$ is said to be critical for
$(\mathbf{c},\mathbf{d},\mathbf{B})$ if it is product-form, and

$\displaystyle\sum_{i=1}^{k}c_{i}\dim(\pi_{E_{i}}T)=\sum_{j=1}^{m}d_{j}\dim(B_{j}T).$

######  Definition 3.

Two data $(\mathbf{c},\mathbf{d},\mathbf{B})$ and
$(\mathbf{c^{\prime}},\mathbf{d^{\prime}},\mathbf{B^{\prime}})$ are said to be
equivalent if $\mathbf{c}=\mathbf{c^{\prime}}$,
$\mathbf{d}=\mathbf{d^{\prime}}$, and there exist invertible linear
transformations $A_{j}:E^{j}\to E^{j}$ and $C_{i}:E_{i}\to E_{i}$ such that

$\displaystyle B^{\prime}_{j}=A_{j}^{-1}B_{j}C^{-1}\hskip 14.22636pt\mbox{for
each $1\leq j\leq m$},$ (1)

where $C:=\operatorname{diag}(C_{1},\dots,C_{k})$.

We remark that, in the special case of $k=1$, the definitions of critical
subspaces and equivalent data coincide with those found in [3]. For general
$k$, all three definitions coincide with those in [8].

##  2 The Anantharam–Jog–Nair inequality

For a datum $(\mathbf{c},\mathbf{d},\mathbf{B})$, Anantharam, Jog and Nair
(AJN) characterized the best (i.e., smallest) constant $C$ such that the
entropy inequality

$\displaystyle\sum_{i=1}^{k}c_{i}h(X_{i})\leq\sum_{j=1}^{m}d_{j}h(B_{j}X)+C$
(2)

holds for any choice of independent random vectors
$X_{i}\in\mathcal{P}(E_{i})$, $1\leq i\leq k$, with $X:=(X_{1},\dots,X_{k})$.
This inequality unifies the Shannon–Stam inequality [21, 22] and the entropic
formulation of the (Euclidean) Brascamp–Lieb inequalities [7, 5] under a
common framework. Extending the Gaussian saturation properties enjoyed by each
(see, e.g., [6] and [15]), Anantharam, Jog and Nair showed that the best
constant can be computed by considering only Gaussian $X_{i}$’s, and gave
necessary and sufficient conditions for finiteness. More precisely, their main
result is the following:

######  Theorem 4 (AJN inequality [1]).

Fix a datum $(\mathbf{c},\mathbf{d},\mathbf{B})$. For any random vectors
$X_{i}\in\mathcal{P}(E_{i})$, $1\leq i\leq k$ and $X=(X_{1},\dots,X_{k})$,

$\displaystyle\sum_{i=1}^{k}c_{i}h(X_{i})-\sum_{j=1}^{m}d_{j}h(B_{j}X)\leq
C_{g}(\mathbf{c},\mathbf{d},\mathbf{B}),$ (3)

where $C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})$ is defined as the supremum of
the LHS over independent Gaussian vectors $(X_{i})_{i=1}^{k}$. Moreover, the
constant $C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})$ is finite if and only if
the following two conditions are satisfied.

  1. (i)

Scaling condition: It holds that

$\displaystyle\sum_{i=1}^{k}c_{i}\dim(E_{i})=\sum_{j=1}^{m}d_{j}\dim(E^{j}).$
(4)

  2. (ii)

Dimension condition: For all product-form subspaces $T\subset E_{0}$,

$\displaystyle\sum_{i=1}^{k}c_{i}\dim(\pi_{E_{i}}T)\leq\sum_{j=1}^{m}d_{j}\dim(B_{j}T).$
(5)

Anantharam, Jog and Nair left open the question of extremizability. That is,
when do there exist random vectors $(X_{i})_{i=1}^{k}$ such that (3) is met
with equality, and what form do any such extremizers take? The goal of this
paper is to answer both questions completely. The first question is addressed
in Section 3, and the second in Section 4.

The precise characterization of extremizers is somewhat complicated, but the
general idea is easily understood in the context of a toy example. For
$\lambda\in(0,1)$, the following holds: If $(X,Y)$ is independent of $Z$, and
$Y$ and $Z$ are of the same dimension, then

$\displaystyle\lambda h(X,Y)+(1-\lambda)h(Z)\leq\lambda
h(X)+h(\lambda^{1/2}Y+(1-\lambda)^{1/2}Z).$ (6)

This inequality is obtained by a concatenation of subadditivity of entropy and
the Shannon–Stam inequality. Restricting attention to cases where all
entropies are finite, we can use known equality cases for both to assert that
$(X,Y)$ and $Z$ are extremizers in (6) if and only if (i) $X$ and $Y$ are
independent; and (ii) $Y$ and $Z$ are Gaussian with identical covariances.

Roughly speaking, all extremizers of the AJN inequality (3) resemble the above
example. That is, extremizers are characterized by a rigid factorization into
independent components, where some components can have any distribution, and
the remaining are necessarily Gaussian with covariances that are typically
linked in some way.

Our approach leverages an assemblage of techniques developed by various
researchers. In particular, the question of extremizability is addressed by
identifying a suitable notion of “AJN-geometricity”, and showing that all
extremizable data are equivalent to AJN-geometric data. This parallels the
approach developed by Bennett, Carbery, Christ and Tao [3] for the functional
form of the Brascamp–Lieb inequalities, which by duality [7] can be realized
as an instance of (3). The Gaussian saturation property of AJN-geometric data
is established by a stochastic argument involving the Föllmer drift (see
Appendix A for definitions and properties), inspired by Lehec’s stochastic
proof of the Shannon–Stam inequality [14]. This stochastic proof lends itself
to identifying the structure of extremizers (when they exist), by combining
key ideas from Valdimarsson’s characterization of optimizers in the functional
Brascamp–Lieb inequalities [23] together with tools from Eldan and
Mikulincer’s work on the stability of the Shannon–Stam inequality [11].

##  3 Extremizability and Geometricity

We first address the question of when (3) is extremizable. To make things
precise, we say that a datum $(\mathbf{c},\mathbf{d},\mathbf{B})$ is
extremizable if $C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})$ is finite and there
exist independent $X_{i}\in\mathcal{P}(E_{i})$, $1\leq i\leq k$ such that (3)
is met with equality. We say that $(\mathbf{c},\mathbf{d},\mathbf{B})$ is
Gaussian-extremizable if $C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})$ is finite
and there exist independent Gaussian $(X_{i})_{i=1}^{k}$ meeting (3) with
equality.

In analogy to definitions made in the context of Brascamp–Lieb inequalities,
we define the class of AJN-geometric data below. Their significance to (3) is
the same as that of geometric data to inequalities of Brascamp–Lieb-type. In
particular, we will see that all (Gaussian-)extremizable instances of (3) are
equivalent to AJN-geometric data.

######  Definition 5 (AJN-Geometric datum).

A datum $(\mathbf{c},\mathbf{d},\mathbf{B})$ is said to be AJN-geometric if

  1. (i)

$B_{j}B_{j}^{T}=\operatorname{id}_{E^{j}}$ for each $1\leq j\leq m$; and

  2. (ii)

we have the operator identity

$\displaystyle\sum_{j=1}^{m}d_{j}\pi_{E_{i}}B^{T}_{j}B_{j}\pi^{T}_{E_{i}}=c_{i}\operatorname{id}_{E_{i}},\hskip
14.22636pt\mbox{for each~{}}1\leq i\leq k.$ (7)

######  Remark 6.

Conditions (i)-(ii) together imply the scaling condition (4). This can be seen
by taking traces in (7), summing from $i=1,\dots,k$, and using the cyclic and
linearity properties of trace together with (ii).

AJN-geometric data have the convenient property that
$C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})=0$, and they are extremizable by
standard Gaussians. We summarize as a formal proposition.

######  Proposition 7.

If $(\mathbf{c},\mathbf{d},\mathbf{B})$ is AJN-geometric, then
$C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})=0$ and $X\sim
N(0,\operatorname{id}_{E_{0}})$ achieves equality in (3).

###### Proof.

We’ll use the properties of the Föllmer drift summarized in Appendix A. Begin
by fixing centered $\mu_{i}\in\mathcal{P}(E_{i})$, $1\leq i\leq k$, and let
$(W_{t})_{t\geq 0}$ be a Brownian motion on $E_{0}$ with
$\operatorname{Cov}(W_{1})=\operatorname{id}_{E_{0}}$. By Theorem 28 and (54),
there is a drift $U_{t}=\int_{0}^{t}u_{s}ds$ such that $\mathbb{E}[u_{t}]=0$
and $(\pi_{E_{i}}(u_{t}))_{i=1}^{k}$ are independent for all $0\leq t\leq 1$,

$\displaystyle(W_{1}+U_{1})\sim\mu_{1}\otimes\cdots\otimes\mu_{k},$ (8)

and
$D(\mu_{i}\|\gamma_{E_{i}})=\frac{1}{2}\int_{0}^{1}\mathbb{E}|\pi_{E_{i}}(u_{s})|^{2}ds$
for each $1\leq i\leq k$. Therefore,

$\displaystyle\sum_{i=1}^{k}c_{i}D(\mu_{i}\|\gamma_{E_{i}})$
$\displaystyle=\frac{1}{2}\mathbb{E}\int_{0}^{1}\sum_{i=1}^{k}c_{i}|\pi_{E_{i}}(u_{s})|^{2}ds$
$\displaystyle=\frac{1}{2}\mathbb{E}\int_{0}^{1}\sum_{j=1}^{m}d_{j}|B_{j}{u}_{s}|^{2}ds$
(9)
$\displaystyle\geq\sum_{j=1}^{m}d_{j}D(B_{j}\sharp(\mu_{1}\otimes\cdots\otimes\mu_{k})\|\gamma_{E^{j}}),$
(10)

where (9) follows from (7) and the properties of $u_{t}$, and (10) follows
from (8) and Proposition 26 (with construction (52)) because
$B_{j}W_{1}\sim\gamma_{E^{j}}$, due to
$B_{j}B_{j}^{T}=\operatorname{id}_{E^{j}}$ by assumption. Now, expanding the
relative entropies in terms of Shannon entropies and second moments, the
second-moment terms cancel due to independence and (7), giving

$\displaystyle\sum_{i=1}^{k}c_{i}h(X_{i})\leq\sum_{j=1}^{m}d_{j}h(B_{j}X)$
(11)

for any $X_{i}\sim\mu_{i}\in\mathcal{P}(E_{i})$ and
$X\sim\otimes_{i=1}^{k}\mu_{i}$, where the centering assumption can be removed
due to translation invariance of Shannon entropy. The fact that
$X\sim\gamma_{E_{0}}$ is an extremizer follows immediately from the scaling
condition (4) (see Remark 6) and the observation that
$B_{j}X\sim\gamma_{E^{j}}$ (since $B_{j}B_{j}^{T}=\operatorname{id}_{E^{j}}$).
∎

######  Remark 8.

In the case where the datum is such that (3) coincides with the Shannon–Stam
inequality, the above proof reduces to that of Lehec [14]. The new idea is
identifying and incorporating the “correct” definition of AJN-geometricity.
When $k=1$, the AJN inequality (3) coincides with the entropic form of the
Brascamp–Lieb inequalities, and the definition of AJN-geometricity reduces to
the the definition of geometricity for Brascamp–Lieb data found in [3].

AJN-geometric data have a relatively straightforward geometric interpretation.
In particular, first note that each $E_{i}$ has a natural isometric embedding
into $E_{0}$ via the inclusion $\pi^{T}_{E_{i}}:E_{i}\to E_{0}$. If
$(\mathbf{c},\mathbf{d},\mathbf{B})$ is AJN-geometric then
$B_{j}B_{j}^{T}=\operatorname{id}_{E^{j}}$, which means that each $E^{j}$ can
be isometrically embedded into $E_{0}$ by the map $B_{j}^{T}:E^{j}\to E_{0}$.
In this way, we can consider $(E_{i})_{i=1}^{k}$ and $(E^{j})_{j=1}^{m}$ to be
subspaces of $E_{0}$, and $\Pi_{E_{i}}:=\pi^{T}_{E_{i}}\pi_{E_{i}}$ and
$\Pi_{E^{j}}:=B_{j}^{T}B_{j}$ define the orthogonal projections of $E_{0}$
onto $E_{i}$ and $E^{j}$, respectively. Thus, the geometric instances of the
AJN inequality (3) can be restated in a way that dispenses with the specific
linear maps $\mathbf{B}$ as follows.

######  Corollary 9.

Let $E^{1},\dots,E^{m}$ be subspaces of $E_{0}=\oplus_{i=1}^{k}E_{i}$. If
$\mathbf{c}$ and $\mathbf{d}$ satisfy

$\displaystyle\sum_{j=1}^{m}d_{j}\Pi_{E_{i}}\Pi_{E^{j}}\Pi_{E_{i}}=c_{i}\Pi_{E_{i}},\hskip
14.22636pt\mbox{for each~{}}1\leq i\leq k,$ (12)

then for any independent $X_{i}\in\mathcal{P}(E_{i})$, $1\leq i\leq k$, and
$X=(X_{1},\dots,X_{m})$,

$\displaystyle\sum_{i=1}^{k}c_{i}h(\Pi_{E_{i}}X)\leq\sum_{j=1}^{m}d_{j}h(\Pi_{E^{j}}X).$
(13)

Equality is achieved for $X\sim N(0,\operatorname{id}_{E_{0}})$.

######  Remark 10.

Entropies in (13) are computed with respect to Lebesgue measure on the
subspace being projected upon. In particular, we have
$h(\Pi_{E_{i}}X)=h(X_{i})$, but have chosen to write (13) in a way to
emphasize the symmetry of the inequality.

With the above definitions in hand, the following completely characterizes the
(Gaussian-)extremizable instances of Theorem 4. It is the main result of this
section, and specializes to the extremizability results in [3] for the
Brascamp–Lieb functional inequalities when $k=1$.

######  Theorem 11.

The following are equivalent:

  1. (i)

$(\mathbf{c},\mathbf{d},\mathbf{B})$ is extremizable.

  2. (ii)

$(\mathbf{c},\mathbf{d},\mathbf{B})$ is Gaussian-extremizable.

  3. (iii)

There are $K_{i}\in\mathbf{S}^{+}(E_{i})$, $1\leq i\leq k$, satisfying

$\displaystyle\sum_{j=1}^{m}d_{j}\pi_{E_{i}}B^{T}_{j}(B_{j}KB_{j}^{T})^{-1}B_{j}\pi^{T}_{E_{i}}=c_{i}K_{i}^{-1},\hskip
14.22636pt1\leq i\leq k,$ (14)

where $K:=\operatorname{diag}(K_{1},\dots,K_{k})$.

  4. (iv)

$(\mathbf{c},\mathbf{d},\mathbf{B})$ is equivalent to an AJN-geometric datum.

######  Remark 12.

For $(K_{i})_{i=1}^{k}$ satisfying (14), the Gaussians $X_{i}\sim N(0,K_{i})$,
$1\leq i\leq k$ are extremal in (3). In fact, the proof of Theorem 11 will
show that if $X_{i}\in\mathcal{P}(E_{i})$, $1\leq i\leq k$ are extremal in
(3), then the covariances $K_{i}=\operatorname{Cov}(X_{i})$ necessarily
satisfy (14).

As a preliminary observation, we note that the extremizers in (3) are closed
under convolutions. This fact can be extracted from the doubling argument in
[1]; we state and prove it here for completeness.

######  Proposition 13.

Fix a datum $(\mathbf{c},\mathbf{d},\mathbf{B})$ that is extremizable for the
AJN inequality (3). Let $X=(X_{1},\dots,X_{k})$ and $Y=(Y_{1},\dots,Y_{k})$
each satisfy (3) with equality. If $X,Y$ are independent, then
$X+Y=(X_{1}+Y_{1},\dots,X_{k}+Y_{k})$ also satisfies (3) with equality.

###### Proof.

Define $Z^{+}=(Z_{1}^{+},\dots,Z_{k}^{+})$ and
$Z^{-}=(Z_{1}^{-},\dots,Z_{k}^{-})$, where

$Z_{i}^{+}:=\frac{1}{\sqrt{2}}(X_{i}+Y_{i}),\hskip
14.22636ptZ_{i}^{-}:=\frac{1}{\sqrt{2}}(X_{i}-Y_{i}),\hskip 14.22636pt1\leq
i\leq k.$

Observe that

$\displaystyle\sum_{i=1}^{k}c_{i}(h(X_{i})+h(Y_{i}))$
$\displaystyle=\sum_{i=1}^{k}c_{i}h(X_{i},Y_{i})$ (15)
$\displaystyle=\sum_{i=1}^{k}c_{i}\left(h(Z_{i}^{+})+h(Z_{i}^{-}|Z_{i}^{+})\right)$
(16)
$\displaystyle\leq\sum_{j=1}^{m}d_{j}\left(h(B_{j}Z^{+})+h(B_{j}Z^{-}|Z^{+})\right)+2C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})$
(17)
$\displaystyle\leq\sum_{j=1}^{m}d_{j}\left(h(B_{j}Z^{+})+h(B_{j}Z^{-}|B_{j}Z^{+})\right)+2C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})$
(18)
$\displaystyle=\sum_{j=1}^{m}d_{j}\left(h(B_{j}X,B_{j}Y)\right)+2C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})$
(19)
$\displaystyle=\sum_{j=1}^{m}d_{j}(h(B_{j}X)+h(B_{j}Y))+2C_{g}(\mathbf{c},\mathbf{d},\mathbf{B}).$
(20)

In the above, (15) is due to independence; (16) follows due to orthogonality
of the transformation $(X_{i},Y_{i})\to(Z_{i}^{+},Z_{i}^{-})$ and the chain
rule; (17) is two applications of (3); (18) follows because conditioning
reduces entropy; (19) is due to the chain rule and orthogonality of the
transformation $(B_{j}Z^{+},B_{j}Z^{-})\to(B_{j}X,B_{j}Y)$; (20) is again due
to independence.

Since $X$ and $Y$ are extremal by assumption, we have equality throughout.
This implies $Z^{+}$ is also extremal, and hence we conclude $X+Y$ is extremal
by the scaling condition (4). ∎

###### Proof of Theorem 11.

$(i)\Rightarrow(ii)$: Let $X$ be an extremizer in (3), and put
$Z_{n}:=n^{-1/2}\sum_{\ell=1}^{n}X^{(i)}$, where $X^{(1)},X^{(2)},\dots$ are
i.i.d. copies of $X$, which we assume to be zero-mean without loss of
generality. By an application of Proposition 13 and the scaling condition (4)
(which holds by finiteness of $C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})$), we
have that $Z_{n}$ is an extremizer in (3) for all $n\geq 1$. By an application
of the entropic central limit theorem [2, 6], it follows that $Z\sim
N(0,\operatorname{Cov}(X))$ is also an extremizer in (3).

$(ii)\Rightarrow(i)$: This follows immediately from Theorem 4.

$(ii)\Rightarrow(iii)$: If $(\mathbf{c},\mathbf{d},\mathbf{B})$ is Gaussian-
extremizable, then there exist $K^{*}_{i}\in\mathbf{S}^{+}(E_{i})$, $1\leq
i\leq k$ which maximize

$(K_{i})_{i=1}^{k}\mapsto\sum_{i=1}^{k}c_{i}\log\det(K_{i})-\sum_{j=1}^{m}d_{j}\log\det(B_{j}KB_{j}^{T}),$

where $K:=\operatorname{diag}(K_{1},\dots,K_{k})$ (note this implies
$B_{j}K^{*}B_{j}^{T}$ is invertible for each $1\leq j\leq m$). This means, for
any index $i$ and any $A_{i}\in\mathbf{S}(E_{i})$, we can consider the
perturbation $K_{i}=K_{i}^{*}+\epsilon A_{i}$ for $\epsilon$ sufficiently
small, and the function value cannot increase. By first-order Taylor
expansion, this implies

$\displaystyle c_{i}\langle A_{i},(K^{*}_{i})^{-1}\rangle$
$\displaystyle=\sum_{j=1}^{m}d_{j}\langle
B_{j}\pi_{E_{i}}^{T}A_{i}\pi_{E_{i}}B_{j}^{T},(B_{j}K^{*}B_{j}^{T})^{-1}\rangle$
$\displaystyle=\Big{\langle}A_{i},\sum_{j=1}^{m}d_{j}\pi_{E_{i}}B^{T}_{j}(B_{j}K^{*}B_{j}^{T})^{-1}B_{j}\pi^{T}_{E_{i}}\Big{\rangle},$

where $\langle\cdot,\cdot\rangle$ is the Hilbert–Schmidt (trace) inner
product. By arbitrariness of $A_{i}$, we conclude (14).

$(iii)\Rightarrow(iv)$: Let $K$ be as in (14). The equivalent datum
$(\mathbf{c},\mathbf{d},\mathbf{B^{\prime}})$ defined by

$B_{j}^{\prime}=(B_{j}KB_{j}^{T})^{-1/2}B_{j}K^{1/2},~{}~{}1\leq j\leq m$

is AJN-geometric. Indeed, $B_{j}^{\prime}B_{j}^{\prime
T}=\operatorname{id}_{E^{j}}$ and (14) gives

$\displaystyle\sum_{j=1}^{m}d_{j}\pi_{E_{i}}B_{j}^{\prime
T}B_{j}^{\prime}\pi^{T}_{E_{i}}=\sum_{j=1}^{m}d_{j}K_{i}^{1/2}\pi_{E_{i}}B_{j}(B_{j}KB_{j}^{T})^{-1}B_{j}\pi^{T}_{E_{i}}K_{i}^{1/2}=c_{i}\operatorname{id}_{E_{i}}.$

$(iv)\Rightarrow(ii)$: Let $(\mathbf{c},\mathbf{d},\mathbf{B^{\prime}})$ be
the geometric datum equivalent to $(\mathbf{c},\mathbf{d},\mathbf{B})$. In the
notation of (1), for any $X_{i}\in\mathcal{P}(E_{i})$, $1\leq i\leq k$ and
$X=(X_{1},\dots,X_{k})$, we have by a change of variables

$\displaystyle\sum_{i=1}^{k}c_{i}h(X_{i})-\sum_{j=1}^{m}d_{j}h(B_{j}X)$
$\displaystyle=\sum_{i=1}^{k}c_{i}h(C_{i}X_{i})-\sum_{i=1}^{k}c_{i}\log\det(C_{i})-\sum_{j=1}^{m}d_{j}h(B_{j}^{\prime}CX)-\sum_{j=1}^{m}d_{j}\log\det(A_{j})$
$\displaystyle=\sum_{i=1}^{k}c_{i}h(Y_{i})-\sum_{j=1}^{m}d_{j}h(B_{j}^{\prime}Y)-\sum_{i=1}^{k}c_{i}\log\det(C_{i})-\sum_{j=1}^{m}d_{j}\log\det(A_{j}),$

where we have defined $Y_{i}:=C_{i}X_{i}$, and $Y=(Y_{1},\dots,Y_{k})$. Since
each $C_{i}$ is invertible, it is clear that $X$ is a (Gaussan-)extremizer for
$(\mathbf{c},\mathbf{d},\mathbf{B})$ if and only if $Y$ is a
(Gaussan-)extremizer for $(\mathbf{c},\mathbf{d},\mathbf{B^{\prime}})$. The
latter is Gaussian-extremizable by the assumption of geometricity and
Proposition 7, so the claim follows. ∎

######  Remark 14.

We remark that Theorem 4 can be derived as a limiting case of the forward-
reverse Brascamp–Lieb inequalities [16]; details can be found in [8, Section
4]. There is a counterpart notion of geometricity for the forward-reverse
Brascamp–Lieb inequalities, for which a result parallel to Theorem 11 holds.
However, the notion of “geometricity” in the context of [8] does not easily
pass through the aforementioned limit, so it seems the simplest proof of
Theorem 11 is a more direct one, as given here.

##  4 Characterization of extremizers

The goal of this section is to give a complete characterization of the
extremizers in (3). In view of Theorem 11, it suffices to consider geometric
instances of the AJN inequality; indeed, the extremizers of any other
extremizable instance of the AJN inequality will be linear transformations of
the extremizers for an equivalent AJN-geometric datum.

Toward this end, let $(\mathbf{c},\mathbf{d},\mathbf{B})$ be AJN-geometric,
and regard $(E_{i})_{i=1}^{k}$ and $(E^{j})_{j=1}^{m}$ as subspaces of
$E_{0}$, as in the discussion preceding Corollary 9. We now extend definitions
found in Valdimarsson [23] to the present setting. A nonzero subspace
$K\subset E_{0}$ is said to be independent if it can be written as

$K=E_{i}\cap\bigcap_{j=1}^{m}V_{j},$

for some $i\in\\{1,\dots,k\\}$, and each $V_{j}$ equal to $E^{j}$ or
${E^{j}}^{\perp}$ (the latter equal to the orthogonal complement of $E^{j}$ in
$E_{0}$). Each independent subspace is contained in some $E_{i}$, and distinct
independent subspaces are orthogonal by construction. So, if
$K^{i}_{1},\dots,K^{i}_{n_{i}}$ is an enumeration of independent subspaces of
$E_{i}$, then we can uniquely decompose

$\displaystyle E_{i}=K^{i}_{0}\oplus K^{i}_{1}\oplus\cdots\oplus
K^{i}_{n_{i}},$ (21)

where $K^{i}_{0}$ is defined to be the orthogonal complement of
$\oplus_{\ell=1}^{n_{i}}K^{i}_{\ell}$ in $E_{i}$. Now, we can uniquely define
the dependent subspace $K_{dep}$ as the product-form subspace

$\displaystyle K_{dep}:=\oplus_{i=1}^{k}K^{i}_{0}.$ (22)

######  Proposition 15.

If $K_{dep}$ is nonzero, there is an orthogonal decomposition

$\displaystyle K_{dep}=\oplus_{\ell=1}^{n}K^{\ell}_{dep},$ (23)

where each $K^{\ell}_{dep}$ is critical for the datum
$(\mathbf{c},\mathbf{d},\mathbf{B})$.

A decomposition of the form (23) is said to be a critical decomposition; we
remark that critical decompositions are not necessarily unique. Together with
Theorem 11, the following completely characterizes the extremizers in the AJN
inequality (3). In the statement, we let $\Pi_{V}:E_{0}\to E_{0}$ denote the
orthogonal projection onto the indicated subspace $V$.

######  Theorem 16.

Let $(\mathbf{c},\mathbf{d},\mathbf{B})$ be AJN-geometric, and decompose each
$E_{i}$ as in (21). Independent $X_{i}\sim\mathcal{P}(E_{i})$, $1\leq i\leq k$
and $X=(X_{1},\dots,X_{k})$ satisfy (3) with equality iff

  1. (i)

$\Pi_{K^{i}_{0}}(X),\dots,\Pi_{K^{i}_{n_{i}}}(X)$ are independent for each
$1\leq i\leq k$; and

  2. (ii)

there is a critical decomposition $K_{dep}=\oplus_{\ell=1}^{n}K^{\ell}_{dep}$
such that $\Pi_{K^{1}_{dep}}(X)$, …, $\Pi_{K^{n}_{dep}}(X)$ are independent
isotropic Gaussians on their respective subspaces.

In words, (i) says that each random vector $X_{i}$ splits into independent
factors on the orthogonal decomposition of $E_{i}$ given by (21). Condition
(ii) tells us that the factor of $X$ supported on $K_{dep}$ is Gaussian with
$\operatorname{Cov}(\Pi_{K_{dep}}(X))=\sum_{\ell=1}^{n}\sigma_{\ell}^{2}\Pi_{K^{\ell}_{dep}}$,
for some critical decomposition (23) and choice of variances
$(\sigma_{\ell}^{2})_{\ell=1}^{n}$. In effect, this links the covariances of
the Gaussian factors of the $X_{i}$’s.

######  Remark 17.

In the case of $k=1$, the above characterization of extremizers is compatible
with that articulated by Valdimarsson for the functional Brascamp–Lieb
inequalities [23]. As noted in Remark 14, the AJN inequality is formally
implied by the Euclidean forward-reverse Brascamp–Lieb inequalities. A
characterization of extremizers for the latter remains unknown at the moment,
but will necessarily involve a new ingredient of log-concavity (since, e.g.,
the Prékopa–Leindler inequality is realized as a special case, and the
extremizers are log-concave [9]).

Before giving the proof, let us consider a few quick examples to demonstrate
the result.

######  Example 18.

Consider the Shannon–Stam inequality on $E_{1}=E_{2}=\mathbb{R}^{n}$ with
$\lambda\in(0,1)$, stated as

$\lambda h(X_{1})+(1-\lambda)h(X_{2})\leq
h(\lambda^{1/2}X_{1}+(1-\lambda)^{1/2}X_{2}),$

for independent $X_{1},X_{2}$ with finite entropies and second moments. There
are no independent subspaces, and every maximal critical decomposition of
$K_{dep}=E_{0}=\mathbb{R}^{n}\oplus\mathbb{R}^{n}$ can be written as

$\mathbb{R}^{n}\oplus\mathbb{R}^{n}=\bigoplus_{\ell=1}^{n}(\operatorname{span}\\{e_{\ell}\\}\oplus\operatorname{span}\\{e_{\ell}\\}),$

with $(e_{\ell})_{\ell=1}^{n}$ an orthonormal basis of $\mathbb{R}^{n}$. Thus,
(ii) is equivalent to the assertion that $X_{1}$ and $X_{2}$ must be Gaussian,
with identical covariances.

######  Example 19.

In the toy inequality (6), the subspace on which $X$ is supported is the only
independent subspace. So, if equality is achieved in (6), then condition (i)
of the theorem tells us that $X$ and $Y$ must be independent; and condition
(ii) implies that $Y$ and $Z$ are Gaussian with identical covariances, as in
the previous example.

######  Example 20.

The Zamir–Feder inequality [24] can be stated as follows (see, e.g., [18]). If
a matrix $B\in\mathbb{R}^{k\times n}$ satisfying
$BB^{T}=\operatorname{id}_{\mathbb{R}^{n}}$ has columns
$(b_{i})_{i=1}^{k}\subset\mathbb{R}^{n}$, then any random vector
$X=(X_{1},\dots,X_{k})\in\mathcal{P}(\mathbb{R}^{k})$ with independent
coordinates satisfies

$\displaystyle h(BX)\geq\sum_{i=1}^{k}|b_{i}|^{2}h(X_{i}).$ (24)

Observe that this is a geometric instance of the AJN inequality, with
$B_{1}=B$, $d_{1}=1$, and $c_{i}=|b_{i}|^{2}$. Letting $(e_{i})_{i=1}^{k}$
denote the natural basis for $\mathbb{R}^{k}$, it follows by definitions that
any independent subspace must be equal to $\operatorname{span}\\{e_{i}\\}$ for
some $1\leq i\leq k$, and $\operatorname{span}\\{e_{i}\\}$ is an independent
subspace iff $e_{i}\in\ker(B)\cup\ker(B)^{\perp}$. Hence, any
$X\in\mathcal{P}(\mathbb{R}^{k})$ with independent coordinates meeting (24)
with equality has the following form:

  1. 1.

If $e_{i}\in\ker(B)\cup\ker(B)^{\perp}$, then $X_{i}$ can have any
distribution in $\mathcal{P}(\mathbb{R})$.

  2. 2.

Otherwise, $X_{i}$ is Gaussian.

Observe that $e_{i}\in\ker(B)\Leftrightarrow b_{i}=0$; in this case,
coordinate $X_{i}$ is not present in (24). If $e_{i}\in\ker(B)^{\perp}$, then
$X_{i}$ is recoverable from $BX$ in the sense that there exists
$u\in\mathbb{R}^{n}$ such that $u^{T}BX=X_{i}$. Hence, we might say that the
extremizers in (24) are characterized by all present non-recoverable
components being Gaussian. This is precisely the statement given by Rioul and
Zamir in their recent work [19, Theorem 1], which gave the first
characterization of extremizers in the Zamir–Feder inequality.

To give an application that yields a new result, consider the following
inequality proposed in [1]:

$\displaystyle c_{1}h(Z_{1},Z_{2})+c_{2}h(Y)\leq
h(Z_{1}+Y,Z_{2}+Y)+d_{2}h(Z_{1})+d_{3}h(Z_{2})+C_{g},$ (25)

where the $Z_{1},Z_{2},Y$ are random variables with $(Z_{1},Z_{2})$
independent of $Y$, and all coefficients are assumed to be strictly positive.
An immediate consequence of Theorem 4 is that the sharp constant $C_{g}$ can
be computed by considering only Gaussians, and conditions on the coefficients
$\mathbf{c},\mathbf{d}$ ensuring finiteness of $C_{g}$ can be deduced from (4)
and (5). Using Theorem 16, we can further conclude that if $\mathbf{c}$ and
$\mathbf{d}$ are such that (25) is extremizable, then it admits only Gaussian
extremizers.

To see that this is the case, let $(\mathbf{c},\mathbf{d},\mathbf{B})$ denote
the datum corresponding to (25). In matrix notation with respect to the
natural choice of basis, we have

$B_{1}=\begin{bmatrix}1&0&1\\\
0&1&1\end{bmatrix},~{}~{}B_{2}=\begin{bmatrix}1&0&0\end{bmatrix},~{}~{}B_{3}=\begin{bmatrix}0&1&0\end{bmatrix}.$

Assuming $(\mathbf{c},\mathbf{d},\mathbf{B})$ is extremizable, let $C$ and
$(A_{j})_{j=1}^{3}$ be the matrices in (1) that transform
$(\mathbf{c},\mathbf{d},\mathbf{B})$ to an AJN-geometric datum
$(\mathbf{c},\mathbf{d},\mathbf{B^{\prime}})$. By rescaling, we can assume
without loss of generality that $C=\operatorname{diag}(C_{1},1)$, where
$C_{1}$ is an invertible $2\times 2$ matrix. In order to show (25) admits only
Gaussian extremizers, we need to show that
$(\mathbf{c},\mathbf{d},\mathbf{B^{\prime}})$ admits no independent subspaces.
To do this, we will show the stronger claim that

$\bigcap_{j=1}^{3}V_{j}=\\{0\\}$

for any choice of $V_{j}$ equal to $E^{j}$ or ${E^{j}}^{\perp}$, where we
identify
$E^{j}=\operatorname{col}(C^{-T}B_{j}^{T}A_{j}^{-T})=\operatorname{col}(C^{-T}B_{j}^{T})$,
with $\operatorname{col}(\cdot)$ denoting the columnspace of its argument.
Explicitly, we have

$\displaystyle E^{1}=\operatorname{col}\left(\begin{bmatrix}\,C_{1}^{-T}\,\\\
1~{}~{}~{}1\end{bmatrix}\right),~{}~{}E^{2}=\operatorname{col}\left(\begin{bmatrix}\,C_{1}^{-T}\begin{bmatrix}1\\\
0\end{bmatrix}\,\\\
0\end{bmatrix}\right),~{}~{}E^{3}=\operatorname{col}\left(\begin{bmatrix}\,C_{1}^{-T}\begin{bmatrix}0\\\
1\end{bmatrix}\,\\\ 0\end{bmatrix}\right).$

Direct computation shows

$\displaystyle{E^{1}}^{\perp}=\operatorname{col}\left(\begin{bmatrix}\,C_{1}\begin{bmatrix}1\\\
1\end{bmatrix}\,\\\
-1\end{bmatrix}\right),~{}~{}{E^{2}}^{\perp}=\operatorname{col}\left(\begin{bmatrix}\begin{matrix}0\\\
0\end{matrix}&C_{1}\begin{bmatrix}0\\\ 1\end{bmatrix}\,\\\
1&0\end{bmatrix}\right),~{}~{}{E^{3}}^{\perp}=\operatorname{col}\left(\begin{bmatrix}\begin{matrix}0\\\
0\end{matrix}&C_{1}\begin{bmatrix}1\\\ 0\end{bmatrix}\,\\\
1&0\end{bmatrix}\right).$

The problem now reduces to casework. By inspection, we have
${E^{1}}^{\perp}\cap E^{2}={E^{1}}^{\perp}\cap E^{3}=\\{0\\}$. Next, since
$C_{1}$ is invertible, we have $E^{2}\cap E^{3}=\\{0\\}$, and it similarly
follows that $E^{1}\cap E^{2}=E^{1}\cap
E^{3}={E^{1}}^{\perp}\cap{E^{2}}^{\perp}=\\{0\\}$. It only remains to show
that $E^{1}\cap{E^{2}}^{\perp}\cap{E^{3}}^{\perp}=\\{0\\}$. To this end,
invertibility of $C_{1}$ allows us to write

${E^{2}}^{\perp}\cap{E^{3}}^{\perp}=\operatorname{col}\left(\begin{bmatrix}0\\\
0\\\ 1\end{bmatrix}\right).$

However, the only vector in $E^{1}$ that is zero in the first two components
is the all-zero vector (again, by invertibility of $C_{1}$), so it follows
that $E^{1}\cap{E^{2}}^{\perp}\cap{E^{3}}^{\perp}=\\{0\\}$, and we conclude
that the datum $(\mathbf{c},\mathbf{d},\mathbf{B^{\prime}})$ admits no
independent subspaces.

Although the above shows (25) can only admit Gaussian extremizers, it does not
tell us whether any exist, or their structure if they do. This is, however,
the content of Theorem 11. Namely, the covariances of Gaussian extremizers are
characterized completely by solutions $K$ to (14) for the datum
$(\mathbf{c},\mathbf{d},\mathbf{B})$; see Remark 12. This emphasizes the
complementary nature of Theorems 16 and 11.

###  4.1 Proof of Theorem 16

The remainder of this section is dedicated to the proof of Theorem 16. We
establish the assertion of sufficiency first, and necessity second. The
assumption that the datum $(\mathbf{c},\mathbf{d},\mathbf{B})$ is AJN-
geometric prevails throughout. Accordingly we will regard $E^{j}$ as a
subspace of $E_{0}$, with $\Pi_{E^{j}}=B_{j}^{T}B_{j}$ denoting the orthogonal
projection onto $E^{j}$.

######  Lemma 21.

Let the notation of (21) and (22) prevail. For each $1\leq j\leq m$, we have
the orthogonal decomposition

$\displaystyle
E^{j}=(\Pi_{E^{j}}K_{dep})\oplus\left(\bigoplus_{i=1}^{k}\bigoplus_{\begin{subarray}{c}1\leq\ell\leq
n_{i}:\\\ K^{i}_{\ell}\subset E^{j}\end{subarray}}K^{i}_{\ell}\right).$ (26)

Moreover, for any critical decomposition
$K_{dep}=\oplus_{\ell=1}^{n}K^{\ell}_{dep}$, we have the orthogonal
decomposition

$\displaystyle\Pi_{E^{j}}K_{dep}=\oplus_{\ell=1}^{n}\Pi_{E^{j}}K^{\ell}_{dep}.$
(27)

###### Proof of Proposition 15 and Lemma 21.

We first note that $\Pi_{E^{j}}K_{dep}$ is orthogonal to $\Pi_{E^{j}}K$, for
any independent subspace $K$. Indeed, by definition of an independent
subspace, we either have $\Pi_{E^{j}}K=\\{0\\}$ or $\Pi_{E^{j}}K=K$. The
former is trivially orthogonal to $\Pi_{E^{j}}K_{dep}$, and the latter is
orthogonal to $\Pi_{E^{j}}K_{dep}$ since $K_{dep}$ is orthogonal to $K$ by
definition and $\Pi_{E^{j}}$ is self-adjoint. Indeed,

$(\Pi_{E^{j}}x)^{T}y=(\Pi_{E^{j}}x)^{T}y=x^{T}(\Pi_{E^{j}}y)=x^{T}y=0,~{}~{}\forall
x\in K_{dep},y\in K.$

This establishes (26).

Now, using the decomposition (21) and the scaling condition (4) (which holds
by AJN-geometricity), we have

$\displaystyle\sum_{i=1}^{k}c_{i}\sum_{\ell=0}^{n_{i}}\dim(K_{\ell}^{i})=\sum_{i=1}^{k}c_{i}\dim(E_{i})$
$\displaystyle=\sum_{j=1}^{m}d_{j}\dim(E^{j})$
$\displaystyle=\sum_{j=1}^{m}d_{j}\dim(\Pi_{E^{j}}K_{dep})+\sum_{j:K_{\ell}^{i}\subset
E^{j}}^{m}d_{j}\dim(K^{i}_{\ell}).$

To summarize,

$\displaystyle\sum_{i=1}^{k}c_{i}\sum_{\ell=0}^{n_{i}}\dim(K_{\ell}^{i})$
$\displaystyle=\sum_{j=1}^{m}d_{j}\dim(\Pi_{E^{j}}K_{dep})+\sum_{j:K_{\ell}^{i}\subset
E^{j}}^{m}d_{j}\dim(K^{i}_{\ell}).$ (28)

Since each independent subspace is of product form, the dimension condition
(5) implies, for each $1\leq i\leq k$ and $1\leq\ell\leq n_{i}$,

$\displaystyle c_{i}\dim(K_{\ell}^{i})$
$\displaystyle\leq\sum_{j:K_{\ell}^{i}\subset
E^{j}}^{m}d_{j}\dim(K^{i}_{\ell}).$ (29)

Likewise, since $K_{dep}=\oplus_{i=1}^{k}K_{0}^{i}$ is of product form, (5)
also implies

$\displaystyle\sum_{i=1}^{k}c_{i}\dim(K_{0}^{i})$
$\displaystyle\leq\sum_{j=1}^{m}d_{j}\dim(\Pi_{E^{j}}K_{dep}).$ (30)

Comparing against (28), we necessarily have equality in (29) and (30), which
proves that $K_{dep}$ is critical. Thus, there exists at least one critical
decomposition of $K_{dep}$ (the trivial one), and Proposition 15 follows.

It remains to show (27). By induction, it suffices to show if $K\subset E_{0}$
is a critical subspace, and $K=K_{1}\oplus K_{2}$ is a critical decomposition,
then $\Pi_{E^{j}}K_{1}$ and $\Pi_{E^{j}}K_{2}$ are orthogonal complements in
$\Pi_{E^{j}}K$. The proof is similar to that of [3, Lemma 7.12]. Letting
$\Pi_{K_{1}}:E_{0}\to E_{0}$ denote the orthogonal projection onto $K_{1}$, we
have that $\Pi_{E^{j}}\Pi_{K_{1}}$ is a contraction in $E_{0}$, so
$\operatorname{Tr}(\Pi_{E^{j}}\Pi_{K_{1}})\leq\dim(\Pi_{E^{j}}K_{1})$. Since
$K_{1}$ is critical, it is product-form by definition and therefore
$\Pi_{K_{1}}=\sum_{i=1}^{k}\Pi_{E_{i}}\Pi_{K_{1}}\Pi_{E_{i}}$. From (7), this
implies

$\displaystyle\sum_{i=1}^{k}c_{i}\dim(\Pi_{E_{i}}K_{1})$
$\displaystyle=\sum_{i=1}^{k}c_{i}\operatorname{Tr}(\Pi_{E_{i}}\Pi_{K_{1}})=\sum_{j=1}^{m}d_{j}\operatorname{Tr}(\Pi_{E^{j}}\Pi_{K_{1}})\leq\sum_{j=1}^{m}d_{j}\dim(\Pi_{E^{j}}K_{1}).$

Since $K_{1}$ is critical, we have equality throughout, implying
$\operatorname{Tr}(\Pi_{E^{j}}\Pi_{K_{1}})=\dim(\Pi_{E^{j}}K_{1})$ for each
$j$. From this, we can conclude that $\Pi_{K_{1}}\Pi_{E^{j}}$ is an isometry
from $\Pi_{E^{j}}K_{1}$ into $K_{1}$, and similarly $\Pi_{K_{2}}\Pi_{E^{j}}$
is an isometry from $\Pi_{E^{j}}K_{2}$ into $K_{2}$. Since $K_{1}$ and $K_{2}$
are orthogonal complements in $K$, it follows that $\Pi_{E^{j}}K_{1}$ and
$\Pi_{E^{j}}K_{2}$ are orthogonal complements in $\Pi_{E^{j}}K$. ∎

###### Sufficiency of conditions (i)-(ii) in Theorem 16.

Let $X_{i}\sim\mathcal{P}(E_{i})$, $1\leq i\leq k$ be independent and satisfy
(i)-(ii), and let $X=(X_{1},\dots,X_{k})$. By the orthogonal decomposition
(26) and the independence assumptions imposed by (i), we can decompose

$\displaystyle
h(B_{j}X)=h(B_{j}\Pi_{K_{dep}}(X))+\sum_{i=1}^{k}\sum_{\begin{subarray}{c}1\leq\ell\leq
n_{i}:\\\ K^{i}_{\ell}\subset
E^{j}\end{subarray}}h(\Pi_{K^{i}_{\ell}}(X_{i})),$ (31)

where all entropies are computed with respect to the subspace being projected
upon. In the proof of Lemma 21, we found (29) was met with equality. So,
whenever $E_{i}$ contains an independent subspace (i.e., $n_{i}\geq 1$), we
have

$c_{i}=\sum_{j:K_{\ell}^{i}\subset E^{j}}^{m}d_{j}.$

Now, using the decomposition (21) and the independence assumptions imposed by
(i), an application of the above identity followed by (31) reveals

$\displaystyle\sum_{i=1}^{k}c_{i}h(X_{i})$
$\displaystyle=\sum_{i=1}^{k}\sum_{\ell=0}^{n_{i}}c_{i}h(\Pi_{K_{\ell}^{i}}(X_{i}))$
$\displaystyle=\sum_{i=1}^{k}c_{i}h(\Pi_{K_{0}^{i}}(X_{i}))+\sum_{i=1}^{k}\sum_{\ell=1}^{n_{i}}\sum_{j:K_{\ell}^{i}\subset
E^{j}}^{m}d_{j}h(\Pi_{K_{\ell}^{i}}(X_{i}))$
$\displaystyle=\sum_{i=1}^{k}c_{i}h(\Pi_{K_{0}^{i}}(X_{i}))+\sum_{j=1}^{m}d_{j}\sum_{i=1}^{k}\sum_{\begin{subarray}{c}1\leq\ell\leq
n_{i}:\\\ K^{i}_{\ell}\subset
E^{j}\end{subarray}}h(\Pi_{K_{\ell}^{i}}(X_{i}))$
$\displaystyle=\sum_{i=1}^{k}c_{i}h(\Pi_{K_{0}^{i}}(X_{i}))+\sum_{j=1}^{m}d_{j}\left(h(B_{j}X)-h(B_{j}\Pi_{K_{dep}}(X))\right).$

In summary,

$\displaystyle\sum_{i=1}^{k}c_{i}h(X_{i})-\sum_{j=1}^{m}d_{j}h(B_{j}X)=\sum_{i=1}^{k}c_{i}h(\Pi_{K_{0}^{i}}(X_{i}))-\sum_{j=1}^{m}d_{j}h(B_{j}\Pi_{K_{dep}}(X)),$
(32)

where any entropies over the trivial subspace $\\{0\\}$ are to be neglected.

It remains to show the RHS is zero. By (ii) and translation invariance of
entropy, we can assume that $\Pi_{K^{\ell}_{dep}}(X)\sim
N(0,\sigma_{\ell}^{2}\operatorname{id}_{K^{\ell}_{dep}})$ for each
$1\leq\ell\leq n$. Using the independence assumption in (ii) and the
decomposition (27), we can express

$h(B_{j}\Pi_{K_{dep}}(X))=\sum_{\ell=1}^{n}\frac{\dim(B_{j}K_{dep}^{\ell})}{2}\log(2\pi
e\sigma_{\ell}^{2}).$

Since each $K_{dep}^{\ell}$ is critical by definition, we have

$\displaystyle\sum_{j=1}^{m}d_{j}h(B_{j}\Pi_{K_{dep}}(X))$
$\displaystyle=\sum_{\ell=1}^{n}\frac{1}{2}\log(2\pi
e\sigma_{\ell}^{2})\sum_{j=1}^{m}d_{j}\dim(B_{j}K_{dep}^{\ell})$
$\displaystyle=\sum_{\ell=1}^{n}\frac{1}{2}\log(2\pi
e\sigma_{\ell}^{2})\sum_{i=1}^{k}c_{i}\dim(\pi_{E_{i}}K_{dep}^{\ell})$
$\displaystyle=\sum_{i=1}^{k}c_{i}\sum_{\ell=1}^{n}\frac{\dim(\pi_{E_{i}}K_{dep}^{\ell})}{2}\log(2\pi
e\sigma_{\ell}^{2})$
$\displaystyle=\sum_{i=1}^{k}c_{i}h(\Pi_{K_{0}^{i}}(X_{i})),$

where we used the independence assumption in (ii) for the last line. Putting
everything together shows

$\sum_{i=1}^{k}c_{i}h(X_{i})=\sum_{j=1}^{m}d_{j}h(B_{j}X),$

so that (i) and (ii) are sufficient conditions for the $X_{i}$’s to be
extremal, since $C_{g}(\mathbf{c},\mathbf{d},\mathbf{B})=0$ by Proposition 7.
∎

As we turn our attention to the necessity part of Theorem 16, we record
several technical lemmas for convenience. We define $\mathbf{S}_{0}^{+}(E)$ to
be the closure of $\mathbf{S}^{+}(E)$ (i.e., the positive semidefinite
symmetric linear operators on $E$). For $A_{i}\in\mathbf{S}_{0}^{+}(E_{i})$,
$1\leq i\leq k$, we define the set
$\Pi(A_{1},\dots,A_{k})\subset\mathbf{S}_{0}^{+}(E_{0})$ to be the set of
symmetric positive semidefinite linear maps $A:E_{0}\to E_{0}$ satisfying

$\pi_{E_{i}}A\pi_{E_{i}}^{T}=A_{i},~{}~{}~{}1\leq i\leq k.$

In terms of matrices, this means $A\in\Pi(A_{1},\dots,A_{k})$ iff $A$ is a
positive semidefinite matrix with diagonal blocks $A_{1},\dots,A_{k}$.

######  Lemma 22.

Let $(\mathbf{c},\mathbf{d},\mathbf{B})$ be AJN-geometric, and
$A_{i}\in\mathbf{S}_{0}^{+}(E_{i})$, $1\leq i\leq k$. For any
$A\in\Pi(A_{1},\dots,A_{k})$, we have

$\displaystyle\sum_{i=1}^{k}c_{i}\operatorname{Tr}\left((A_{i}-\operatorname{id}_{E_{i}})^{2}\right)\geq\sum_{j=1}^{m}d_{j}\operatorname{Tr}\left(((B_{j}A^{2}B_{j}^{T})^{1/2}-\operatorname{id}_{E^{j}})^{2}\right),$
(33)

with equality if and only if
$(\operatorname{id}_{E_{0}}-\Pi_{E^{j}})A\Pi_{E^{j}}=0$ for each $1\leq j\leq
m$.

###### Proof.

Using the block-diagonal structure of $A$ and the definition of AJN-
geometricity, we have

$\displaystyle\sum_{i=1}^{k}c_{i}\operatorname{Tr}\left((A_{i}-\operatorname{id}_{E_{i}})^{2}\right)$
$\displaystyle=\sum_{j=1}^{m}d_{j}\operatorname{Tr}(B_{j}(A-\operatorname{id}_{E_{0}})^{2}B_{j})$
$\displaystyle=\sum_{j=1}^{m}d_{j}\operatorname{Tr}(B_{j}A^{2}B_{j}^{T}-2B_{j}AB_{j}^{T}+\operatorname{id}_{E^{j}})$
$\displaystyle\geq\sum_{j=1}^{m}d_{j}\operatorname{Tr}(B_{j}A^{2}B_{j}^{T}-2(B_{j}A^{2}B_{j}^{T})^{1/2}+\operatorname{id}_{E^{j}})$
$\displaystyle=\sum_{j=1}^{m}d_{j}\operatorname{Tr}\left(((B_{j}A^{2}B_{j}^{T})^{1/2}-\operatorname{id}_{E^{j}})^{2}\right),$

where the inequality follows because square root is operator monotone. More
precisely, AJN-geometricity implies

$(B_{j}AB_{j}^{T})^{2}=B_{j}AB_{j}^{T}B_{j}AB_{j}^{T}\leq
B_{j}A^{2}B_{j}^{T},$

so that operator monotonicity of square root gives
$B_{j}AB_{j}^{T}\leq(B_{j}A^{2}B_{j}^{T})^{1/2}$. Equality in (33) is
therefore equivalent to equality above, which can be rewritten as

$B_{j}A(\operatorname{id}_{E_{0}}-B_{j}^{T}B_{j})AB_{j}^{T}=0~{}~{}\Leftrightarrow~{}~{}(\operatorname{id}_{E_{0}}-\Pi_{E^{j}})A\Pi_{E^{j}}=0.$

∎

The following is due to [11]; we sketch the proof for completeness.

######  Lemma 23.

Fix a Euclidean space $E$. Consider a filtered probability space carrying an
$E$-valued Brownian motion $(W_{t})_{\geq 0}$, and let $(F_{t})_{\geq 0}$ be
an adapted process taking values in $\mathbf{S}^{+}(E)$. If
$\int_{0}^{1}F_{t}dW_{t}\sim\mu$, then

$D(\mu\|\gamma_{E})\leq\frac{1}{2}\int_{0}^{1}\frac{\mathbb{E}\operatorname{Tr}\left((F_{t}-\operatorname{id}_{E})^{2}\right)}{1-t}dt.$

###### Proof.

Define the drift

$u_{t}=\int_{0}^{t}\frac{F_{s}-\operatorname{id}_{E}}{1-s}dW_{s}.$

We claim that $W_{1}+\int_{0}^{1}u_{t}dt\sim\mu$. To see this, write

$\displaystyle\int_{0}^{1}F_{t}dW_{t}=\int_{0}^{1}\operatorname{id}_{E}dW_{t}+\int_{0}^{1}(F_{t}-\operatorname{id}_{E})dW_{t}$
$\displaystyle=W_{1}+\int_{0}^{1}\int_{t}^{1}\frac{F_{t}-\operatorname{id}_{E}}{1-t}dsdt$
$\displaystyle=W_{1}+\int_{0}^{1}u_{s}ds,$

where we used the stochastic Fubini theorem. Now, by Proposition 26 and the
data processing inequality, Itô’s isometry, and Fubini’s theorem, we have

$\displaystyle
D(\mu\|\gamma_{E})\leq\frac{1}{2}\int_{0}^{1}\mathbb{E}|u_{t}|^{2}dt$
$\displaystyle=\frac{1}{2}\int_{0}^{1}\int_{0}^{t}\frac{\mathbb{E}\operatorname{Tr}\left((F_{s}-\operatorname{id}_{E})^{2}\right)}{(1-s)^{2}}dsdt$
(34)
$\displaystyle=\frac{1}{2}\int_{0}^{1}\frac{\mathbb{E}\operatorname{Tr}\left((F_{s}-\operatorname{id}_{E})^{2}\right)}{1-s}ds.$

∎

######  Lemma 24.

Let $(P_{t})_{t\geq 0}$ be the heat semigroup, and let
$X\sim\mu\in\mathcal{P}(E)$ have density $d\mu=fd\gamma_{E}$. For each
$0<t<1$, there is a constant $C$ depending only on $t$ and the second moments
of $X$ such that

$|\nabla\log P_{1-t}f(x)|\leq C(|x|+1),~{}~{}~{}x\in E.$

If, moreover, $\mu$ is of the form $\mu=\nu*\gamma_{E}$, then

$\nabla^{2}\log(P_{1-t}f(x))\in\mathbf{S}_{0}^{+}(E),~{}~{}~{}x\in E,0<t<1.$

###### Proof.

Let $\rho$ denote the density of $X$ with respect to Lebesgue measure on $E$.
By direct calculation, we can reparametrize $P_{1-t}f$ in terms of $\rho$ as

$P_{1-t}f(x)=\left(\frac{2\pi}{t}\right)^{\dim(E)/2}e^{\frac{|x|^{2}}{2t}}P_{\frac{1-t}{t}}\rho(x/t)$

Hence,

$\displaystyle\nabla\log P_{1-t}f(x)=\frac{1}{t}x+\frac{1}{t}\nabla(\log
P_{\frac{1-t}{t}}\rho)(x/t).$ (35)

Regularity estimates for evolution of densities under $(P_{t})_{t\geq 0}$
imply

$|\nabla\log P_{s}\rho(x)|\leq c_{s}(|x|+1),~{}~{}~{}s>0$

for some finite constant $c_{s}$ depending only on $s$ and the second moments
of $\rho$ (see, e.g., [17, Proposition 2]). Hence, the first claim follows.

For the second claim, we have $\rho=P_{1}\rho_{0}$ for some density
$\rho_{0}$. Hence, by the semigroup property combined with (35), we have

$\nabla^{2}\log
P_{1-t}f(x)=\frac{1}{t}\operatorname{id}_{E}+\frac{1}{t^{2}}\nabla^{2}(\log
P_{\frac{1}{t}}\rho_{0})(x/t).$

By a simple convexity calculation [10, Lemma 1.3], it holds that
$\nabla^{2}(\log P_{s}g)\geq-\frac{1}{s}\operatorname{id}_{E}$ for any density
$g$ and $s>0$, so we find

$\nabla^{2}\log
P_{1-t}f(x)\geq\left(\frac{1}{t}-\frac{1}{t}\right)\operatorname{id}_{E}=0.$

∎

###### Necessity of conditions (i)-(ii) in Theorem 16.

Let $\mu_{i}\in\mathcal{P}(E_{i})$, $1\leq i\leq k$ satisfy

$\displaystyle\sum_{i=1}^{k}c_{i}D(\mu_{i}\|\gamma_{E_{i}})$
$\displaystyle=\sum_{j=1}^{m}d_{j}D(B_{j}\sharp(\mu_{1}\otimes\cdots\otimes\mu_{k})\|\gamma_{E^{j}})$
(36)

under the prevailing assumption of AJN-geometricity; this is the same as
equality in (11). Without loss of generality, we can assume each $\mu_{i}$ is
centered. Moreover, since the extremizers of the AJN inequality are closed
under convolutions (Proposition 13) and standard Gaussians are extremal in the
geometric AJN inequality (Proposition 7), we can assume without loss of
generality that each $\mu_{i}$ is of the form

$\displaystyle\mu_{i}=\widetilde{\mu}_{i}*\gamma_{E_{i}}$ (37)

for some extremal $\widetilde{\mu}_{i}\in\mathcal{P}(E_{i})$, $1\leq i\leq k$.
Indeed, $X\sim\otimes_{i=1}^{k}\mu_{i}$ satisfies (i)-(ii) if and only if
$X+Z$ satisfies (i)-(ii) for $Z\sim\gamma_{E_{0}}$, independent of $X$.

Necessity of condition (i): In the proof of Proposition 7, the sole inequality
is (10). Hence, properties of the drift $u_{t}$ warrant a closer inspection;
we follow the approach developed in [11]. Toward this end, let $f$ denote the
density of $\mu_{1}\otimes\cdots\otimes\mu_{k}$ with respect to
$\gamma_{E_{0}}$, and define the function

$u_{t}(x):=\nabla\log P_{1-t}f(x),~{}~{}~{}~{}x\in E_{0},~{}0\leq t\leq 1,$

where $(P_{t})_{t\geq 0}$ denotes the heat semigroup. Define the matrix-valued
function

$\displaystyle\Gamma_{t}(x):=(1-t)\nabla
u_{t}(x)+\operatorname{id}_{E_{0}},~{}~{}~{}~{}x\in E_{0},~{}0\leq t\leq 1,$
(38)

which, for each $0\leq t\leq 1$, takes the block-diagonal form
$\Gamma_{t}=\operatorname{diag}(\Gamma^{1}_{t},\dots,\Gamma^{k}_{t})$ with
$\Gamma_{t}^{i}\in\mathbf{S}^{+}(E_{i})$ due to the product form of the
density $f$ and Lemma 24 applied to (37).

Now, consider the Wiener space of continuous functions
$\mathbb{W}=\\{\omega:[0,1]\to E_{0};~{}\omega(0)=0\\}$, equipped with the
uniform norm, the Borel sets $\mathcal{B}$, and the Wiener measure $\gamma$.
Let $X_{t}(\omega)=\omega(t)$ be the coordinate process, and
$\mathcal{F}=(\mathcal{F}_{t})_{0\leq t\leq 1}$ be the natural filtration of
$(X_{t})_{0\leq t\leq 1}$. We’ll work on the filtered probability space
$(\mathbb{W},\mathcal{B},\nu,\mathcal{F})$, where $\nu$ is the Brownian bridge

$\frac{d\nu}{d\gamma}(\omega):=f(\omega(1)),~{}~{}~{}\omega\in\mathbb{W}.$

By the representation theorem for Brownian bridges, we have

$\displaystyle X_{t}\overset{\text{law}}{=}tX+\sqrt{t(1-t)}Z,$ (39)

where $X\sim\mu_{1}\otimes\cdots\otimes\mu_{k}$ and $Z\sim\gamma_{E_{0}}$ are
independent. Writing $u_{t}\equiv u_{t}(X_{t})$, the classical de Bruijn
identity, parametrized with respect to the bridge (39), gives

$\displaystyle
D(\mu_{i}\|\gamma_{E_{i}})=\frac{1}{2}\int_{0}^{1}\mathbb{E}|\pi_{E_{i}}(u_{t})|^{2}dt,~{}~{}~{}1\leq
i\leq k,$ (40)

where the expectation is with respect to $\nu$. Moreover, if we define the
$\mathcal{F}_{t}$-adapted process $(W_{t})_{0\leq t\leq 1}$ by the equation

$\displaystyle W_{t}:=X_{t}-\int_{0}^{t}u_{s}(X_{s})ds,~{}~{}~{}~{}0\leq t\leq
1,$ (41)

then $(W_{t})_{0\leq t\leq 1}$ is a Brownian motion by an application of
Girsanov’s theorem [14, 10]. Using the SDE (41) and the heat equation

$\partial_{t}P_{1-t}f(x)=-\frac{1}{2}\Delta P_{1-t}f(x),$

we can apply Itô’s formula to $u_{t}$ to reveal the relationship

$du_{t}=\nabla
u_{t}dW_{t}=\frac{\Gamma_{t}-\operatorname{id}_{E_{0}}}{1-t}dW_{t},$

with $\Gamma_{t}\equiv\Gamma_{t}(X_{t})$. Rearranging and integrating gives

$\displaystyle\int_{0}^{1}\Gamma_{t}dW_{t}=W_{1}+\int_{0}^{1}u_{t}dt\sim\mu_{1}\otimes\cdots\otimes\mu_{k}.$
(42)

In particular, equality in (40) together with the computation in (34) gives
the following representation for the entropies in terms of the
$\Gamma_{t}^{i}$ processes:

$\displaystyle D(\mu_{i}\|\gamma_{E_{i}})$
$\displaystyle=\frac{1}{2}\int_{0}^{1}\frac{\mathbb{E}\operatorname{Tr}\left((\Gamma^{i}_{t}-\operatorname{id}_{E_{i}})^{2}\right)}{1-t}dt,~{}~{}~{}1\leq
i\leq k.$ (43)

Next, positive-definiteness of $\Gamma_{t}$ and the assumption that
$B_{j}B_{j}^{T}=\operatorname{id}_{E^{j}}$ together justify the definition of
a new process $(\widetilde{W}^{j}_{t})_{0\leq t\leq 1}$ via

$d\widetilde{W}^{j}_{t}=(B_{j}\Gamma_{t}^{2}B_{j}^{T})^{-1/2}B_{j}\Gamma_{t}dW_{t},~{}~{}~{}1\leq
j\leq m.$

By Lévy’s characterization, this process is a Brownian motion, since it has
quadratic covariation

$[\widetilde{W}^{j}]_{t}=\int_{0}^{t}(B_{j}\Gamma_{s}^{2}B_{j}^{T})^{-1/2}B_{j}\Gamma_{s}^{2}B_{j}^{T}(B_{j}\Gamma_{s}^{2}B_{j}^{T})^{-1/2}ds=t\operatorname{id}_{E^{j}}.$

Putting things together, observe that definitions and (42) give

$\int_{0}^{1}(B_{j}\Gamma_{t}^{2}B_{j}^{T})^{1/2}d\widetilde{W}^{j}_{t}=B_{j}\int_{0}^{1}\Gamma_{t}dW_{t}\sim
B_{j}\sharp(\mu_{1}\otimes\cdots\otimes\mu_{k}).$

Thus, by (43) and an application of Lemmas 22 and 23, we have

$\displaystyle\sum_{i=1}^{k}c_{i}D(\mu_{i}\|\gamma_{E_{i}})$
$\displaystyle=\frac{1}{2}\int_{0}^{1}\frac{\sum_{i=1}^{k}c_{i}\mathbb{E}\operatorname{Tr}\left((\Gamma^{i}_{t}-\operatorname{id}_{E_{i}})^{2}\right)}{1-t}dt$
$\displaystyle\geq\frac{1}{2}\int_{0}^{1}\frac{\sum_{j=1}^{m}d_{j}\mathbb{E}\operatorname{Tr}\left(((B_{j}\Gamma_{t}^{2}B_{j}^{T})^{1/2}-\operatorname{id}_{E^{j}})^{2}\right)}{1-t}dt$
$\displaystyle\geq\sum_{j=1}^{m}d_{j}D(B_{j}\sharp(\mu_{1}\otimes\cdots\otimes\mu_{k})\|\gamma_{E^{j}}).$

We have equality throughout due to (36). Since $X_{t}$ has full support for
each $0<t\leq 1$ and $(t,x)\mapsto\Gamma_{t}(x)$ is smooth by the regularizing
properties of the heat semigroup, Lemma 22 and the above equality implies that

$\displaystyle(\operatorname{id}_{E_{0}}-\Pi_{E^{j}})\Gamma_{t}(x)\Pi_{E^{j}}=0,~{}~{}x\in
E_{0},~{}0<t<1,~{}1\leq j\leq m.$ (44)

By definition, this implies that, for each $t\in(0,1)$, we have

$\displaystyle(\operatorname{id}_{E_{0}}-\Pi_{E^{j}})\ \nabla^{2}\log
P_{1-t}f(x)\Pi_{E^{j}}=0,~{}~{}x\in E_{0},~{}1\leq j\leq m.$

Since $f$ is assumed regular by virtue of (37), the above also holds for $t=1$
by continuity of the derivatives of the heat semigroup. Since
$f=\prod_{i=1}^{k}f_{i}$ by definition, where each $f_{i}$ is a density on
$E_{i}$ with respect to $\gamma_{E_{i}}$, the above imposes a block-diagonal
structure on the Hessian of $\log f_{i}$, which can be summarized as

$D^{2}(\log f_{i})(x,y)=0,$

whenever $x,y$ are vectors from distinct spaces in the decomposition (21).
This implies, for each $1\leq i\leq k$, that the density $f_{i}$ has product
form

$\displaystyle f_{i}(x)=\prod_{\ell=0}^{n_{i}}\
f_{i,{\ell}}(\Pi_{K_{\ell}^{i}}(x)),~{}~{}x\in E_{i},$ (45)

establishing necessity of (i).

######  Remark 25.

The above proof can be viewed as a modification of Eldan and Mikulincer’s
argument for bounding the deficit in the Shannon–Stam inequality [11],
suitable for setting of the AJN inequality. The emergence of the factorization
(45) is new, and results from AJN-geometricity via the matrix inequality in
Lemma 22. Although Valdimarsson’s arguments in the context of the functional
Brascamp–Lieb inequalities are slightly different, the same basic
factorization emerges in [23, Lemma 13]. Hence, the above might be regarded as
a combination of ideas from both [11] and [23]. In the next step, the Fourier
analytic argument is effectively the same as that found in [23, Lemma 14],
with the drift $u_{t}$ playing the role of what Valdimarsson calls $\nabla\log
F$.

Necessity of condition (ii): Having established necessity of (i), the initial
calculations in the proof of sufficiency hold, leading to the conclusion (32).
The reduced datum $(\mathbf{c},\mathbf{d},\mathbf{B}_{K_{dep}})$ obtained by
restricting the maps in $\mathbf{B}$ to domain $K_{dep}$ remains AJN-
geometric, so without loss of generality, we can assume for simplicity that
there are no independent subspaces henceforth; i.e., $K_{dep}\equiv E_{0}$. As
in the previous step, we let $f$ denote the density of
$X\sim\mu_{1}\otimes\cdots\otimes\mu_{k}$ with respect to $\gamma_{E_{0}}$.

Letting definitions from the previous step prevail, Lemma 24 implies that
$u_{t}$ has linear growth in $x$ for each $0<t<1$. Hence, we are justified in
taking the Fourier transform, which we denote by $\hat{u}_{t}$. By (45),
$u_{t}$ is additively separable in the variables $\Pi_{E_{j}}x$ and
$(\operatorname{id}_{E_{0}}-\Pi_{E_{j}})x$, and therefore $\hat{u}_{t}$ is
supported on $H^{j}\cup(H^{j})^{\perp}$ for each $1\leq j\leq m$ (where
$H^{j}$ denotes the complex Hilbert space $E^{j}+\mathbf{i}E^{j}$). Similarly,
since $u_{t}$ is additively separable in the variables
$\pi_{E_{1}}(x),\dots,\pi_{E_{k}}(x)$, it follows that $\hat{u}_{t}$ is
supported on $\cup_{i=1}^{k}H_{i}$ (where, $H_{i}:=E_{i}+\mathbf{i}E_{i}$).
Taking intersections, we find $\hat{u}_{t}$ is supported on the set

$(H_{1}\cup\cdots\cup
H_{k})\cap\bigcap_{j=1}^{m}(H^{j}\cup(H^{j})^{\perp})=\\{0\\},$

where the equality follows by the assumption that there are no independent
subspaces. A tempered distribution with Fourier transform supported at the
origin is a polynomial [20, p. 194], so the linear growth estimate in Lemma 24
implies that $x\mapsto u_{t}(x)$ is affine for each $0<t<1$. As a consequence
of its defnition, $\Gamma_{t}$ is therefore deterministic for each $0<t<1$, in
the sense that $\Gamma_{t}(x)$ does not depend on $x$. Using the Itô isometry,
we conclude from the representation
$\int_{0}^{1}\Gamma_{t}dW_{t}\overset{\text{law}}{=}X$ that $X$ is Gaussian
with covariance

$\Sigma:=\operatorname{Cov}(X)=\int_{0}^{1}(\Gamma_{t})^{2}dt.$

Note that $\Sigma$ has diagonal form

$\displaystyle\Sigma=\Pi(\Sigma_{1},\dots,\Sigma_{k}),~{}~{}~{}\Sigma_{i}\in\mathbf{S}_{0}^{+}(E_{i}),1\leq
i\leq k$ (46)

due to independence of the coordinates of $X$.

From (44), we have $\Pi_{E^{j}}\Sigma=\Pi_{E^{j}}\Sigma\Pi_{E^{j}}$ for each
$1\leq j\leq m$. This implies that if $v\in E_{0}$ is an eigenvector of
$\Sigma$ with eigenvalue $\lambda$, then $\Pi_{E^{j}}v$ is an eigenvector of
$\Pi_{E^{j}}\Sigma\Pi_{E^{j}}$ with eigenvalue $\lambda$. In particular, if we
consider the spectral decomposition
$\Sigma=\sigma_{1}^{2}\Pi_{K^{1}_{dep}}+\cdots\sigma_{n}^{2}\Pi_{K^{n}_{dep}}$
with $\sigma^{2}_{1},\dots,\sigma^{2}_{n}$ distinct, then we have the
orthogonal decomposition

$\displaystyle B_{j}E_{0}=\oplus_{\ell=1}^{n}B_{j}K^{\ell}_{dep},\hskip
14.22636pt1\leq j\leq m,$ (47)

where we note each $K^{\ell}_{dep}$ is product-form due to (46). To see that
$E_{0}=\oplus_{\ell=1}^{n}K^{\ell}_{dep}$ is a critical decomposition, observe
that

$\displaystyle\sum_{i=1}^{k}c_{i}h(X_{i})$
$\displaystyle=\sum_{j=1}^{m}d_{j}h(B_{j}X)$ (48)
$\displaystyle=\sum_{\ell=1}^{n}\frac{1}{2}\log(2\pi
e\sigma_{\ell}^{2})\sum_{j=1}^{m}d_{j}\dim(B_{j}K_{dep}^{\ell})$ (49)
$\displaystyle\geq\sum_{\ell=1}^{n}\frac{1}{2}\log(2\pi
e\sigma_{\ell}^{2})\sum_{i=1}^{k}c_{i}\dim(\pi_{E_{i}}K_{dep}^{\ell})$ (50)
$\displaystyle=\sum_{i=1}^{k}c_{i}\sum_{\ell=1}^{n}\frac{\dim(\pi_{E_{i}}K_{dep}^{\ell})}{2}\log(2\pi
e\sigma_{\ell}^{2})=\sum_{i=1}^{k}c_{i}h(X_{i}),$ (51)

where (48) is the extremality assumption; (49) is due to (47) and the spectral
decomposition of $\Sigma$; (50) is the dimension condition (5); and (51)
follows due to the orthogonal decomposition
$E_{i}=\oplus_{\ell=1}^{n}\pi_{E_{i}}K_{dep}^{\ell}$ for each $1\leq i\leq k$,
because each $K_{dep}^{\ell}$ is of product-form. Since we have equality
throughout, this implies $K_{dep}\equiv
E_{0}=\oplus_{\ell=1}^{n}K_{dep}^{\ell}$ is a critical decomposition, as
desired. Since $K_{dep}^{1},\dots,K_{dep}^{n}$ are eigenspaces of $\Sigma$,
(ii) holds. ∎

### Acknowledgement

T.C. thanks Dan Mikulincer for his explanations of the properties of the
Föllmer drift and the martingale embedding used in [11]. This work was
supported in part by NSF grant CCF-1750430 (CAREER).

## References

  * [1] V. Anantharam, V. Jog, and C. Nair.  Unifying the Brascamp-Lieb inequality and the entropy power inequality.  arXiv preprint arXiv:1901.06619, 2019.
  * [2] A. R. Barron.  Entropy and the central limit theorem.  The Annals of probability (1986): 336-342.
  * [3] J. Bennett, A. Carbery, M. Christ, and T. Tao.  The Brascamp-Lieb inequalities: finiteness, structure and extremals.  Geometric and Functional Analysis, 17(5):1343–1415, 2008.
  * [4] H. J. Brascamp, E. H. Lieb, and J. M. Luttinger.  A general rearrangement inequality for multiple integrals.  Journal of functional analysis, 17(2):227–237, 1974.
  * [5] H. J. Brascamp and E. H. Lieb.  Best constants in Young’s inequality, its converse, and its generalization to more than three functions.  Advances in Mathematics, 20(2):151–173, 1976.
  * [6] E. Carlen and A. Soffer.  Entropy production by block variable summation and central limit theorems.  Commun. Math. Phys., vol. 140, no. 2, pp. 339–371, 1991.
  * [7] E. A. Carlen and D. Cordero-Erausquin.  Subadditivity of the entropy and its relation to Brascamp–Lieb type inequalities.  Geometric and Functional Analysis, 19(2):373–405, 2009.
  * [8] T. A. Courtade and J. Liu.  Euclidean forward-reverse Brascamp–Lieb inequalities: Finiteness, structure, and extremals.  The Journal of Geometric Analysis 31.4 (2021): 3300–3350.
  * [9] S. Dubuc.  Critères de convexité et inégalit s integralés.
Ann. Inst. Fourier Grenoble 27 (1) (1977) 135–165.

  * [10] R. Eldan and J. R. Lee.  Regularization under diffusion and anti-concentration of temperature.  arXiv preprint arXiv:1410.3887, 2014.
  * [11] R. Eldan and D. Mikulincer.  Stability of the Shannon–Stam inequality via the Föllmer process.  Probability Theory and Related Fields 177.3 (2020): 891-922.
  * [12] H. Föllmer.  An entropy approach to the time reversal of diffusion processes, in _Stochastic differential systems (Marseille-Luminy, 1984)_.  Lecture Notes in Control and Inform. Sci. 69, Springer (1985) 156–163.
  * [13] H. Föllmer.  Time reversal on Wiener space, in _Stochastic processes – mathematics and physics (Bielefeld, 1984)_.  Lecture Notes in Math. 1158, Springer (1986) 119–129.
  * [14] J. Lehec.  Representation formula for the entropy and functional inequalities.  In Annales de l’IHP Probabilités et statistiques, volume 49, pages 885–899, 2013.
  * [15] E. H. Lieb.  Gaussian kernels have only Gaussian maximizers.  Inventiones mathematicae, 102(1):179–208, 1990.
  * [16] J. Liu, T. A. Courtade, P. Cuff, and S. Verdú.  A forward-reverse Brascamp-Lieb inequality: Entropic duality and Gaussian optimality.  Entropy (special issue on information inequalities), 20(6):418, 2018\.
  * [17] Y. Polyanskiy and Y. Wu.  Wasserstein continuity of entropy and outer bounds for interference channels.  IEEE Trans. Inf. Theory, vol. 62, no. 7, pp. 3992–4002, 2016.
  * [18] O. Rioul  Information theoretic proofs of entropy power inequalities.  IEEE Transactions on Information Theory 57.1 (2010): 33–55.
  * [19] O. Rioul and R. Zamir.  Equality in the matrix entropy-power inequality and blind separation of real and complex sources.  in Proc. of the IEEE International Symposium on Information Theory (ISIT). IEEE, 2019.
  * [20] W. Rudin.  Functional Analysis.  Second edition. New York: McGraw-Hill, 1991. Print.
  * [21] C. E. Shannon.  A mathematical theory of communication.  The Bell system technical journal 27.3: 379–423, 1948.
  * [22] A. J. Stam.  Some inequalities satisfied by the quantities of information of Fisher and Shannon.  Information and Control 2.2: 101–112, 1959.
  * [23] S. I. Valdimarsson.  Optimisers for the Brascamp-Lieb inequality.  Israel J. Math., 168:253–274, 2008.
  * [24] R. Zamir and M. Feder.  A generalization of the entropy power inequality with applications.  IEEE transactions on Information Theory, 39.5: 1723–1728, 1993.

##  Appendix A Föllmer’s drift

The material in this appendix can be found in [14], and interested readers are
referred there for more details. We summarize the required results for
completeness, since it plays an important role in the proofs of Proposition 7
and Theorem 16.

For a Euclidean space $E$, let $\mathbb{W}$ denote the classical Wiener space
of continuous functions $C^{0}([0,1],E):=\\{\omega:[0,1]\to E;\omega(0)=0\\}$
equipped with the topology of uniform convergence, and the Borel
$\sigma$-algebra $\mathcal{B}$. Let $\gamma$ denote the Wiener measure on
$(\mathbb{W},\mathcal{B})$. Let $X_{t}:\omega\mapsto\omega(t)$ be the
coordinate process, and $\mathcal{G}=(\mathcal{G}_{t})_{0\leq t\leq 1}$ be the
natural filtration of $X=(X_{t})_{0\leq t\leq 1}$. It is a fact that
$\mathcal{B}$ is the $\sigma$-algebra generated by $\mathcal{G}$.

Given a filtered probability space
$(\Omega,\mathcal{A},\mathbb{P},\mathcal{F})$, where $\mathcal{A}$ is the
Borel $\sigma$-algebra of a Polish topology on $\Omega$, a drift is any
adapted process $U:[0,1]\to E$ such that there exists $u\in L^{1}([0,1];E)$
satisfying

$U_{t}=\int_{0}^{t}u_{s}ds,~{}~{}0\leq t\leq 1$

and $\int_{0}^{1}|u_{s}|^{2}ds<\infty$ almost surely. By definition and
Cauchy–Schwarz, any drift $U$ belongs to $\mathbb{W}$ almost surely.

A process $B=(B_{t})_{t\geq 0}$ taking values in $E$ is said to be a standard
Brownian motion if it is a Brownian motion with $B_{0}=0$ and
$\operatorname{Cov}(B_{1})=\operatorname{id}_{E}$. The following is a
consequence of Girsanov’s theorem; it can be found in [14, Proposition 1].

######  Proposition 26.

Let a standard Brownian motion $B$, taking values in $E$, be defined on a
filtered probability space $(\Omega,\mathcal{A},\mathbb{P},\mathcal{F})$, and
let $U_{t}=\int_{0}^{t}u_{s}ds$ be a drift. If $\nu$ is the law of the process
$(B_{t}+U_{t})_{0\leq t\leq 1}$, then

$D(\nu\|\gamma)\leq\frac{1}{2}\int_{0}^{1}\mathbb{E}|u_{s}|^{2}ds.$

It turns out that the upper bound given on the relative entropy above can be
met with equality. The result is due to Föllmer [12, 13]; the statement given
can be found in [14, Theorem 2].

######  Proposition 27 (Föllmer’s drift).

Let $\nu\ll\gamma$ be a probability measure on $(\mathbb{W},\mathcal{B})$ with
$D(\nu\|\gamma)<\infty$. There exists an adapted process $u$ such that, under
$\nu$, the following holds:

  1. 1.

The process $U_{t}=\int_{0}^{t}u_{s}ds$ is a drift.

  2. 2.

The process $B_{t}=X_{t}-U_{t}$ is standard Brownian motion.

  3. 3.

We have $D(\nu\|\gamma)=\frac{1}{2}\int_{0}^{1}\mathbb{E}_{\nu}|u_{s}|^{2}ds$.

Let $\mu\in\mathcal{P}(E)$ have density $d\nu=fd\gamma_{E}$. By defining the
Brownian bridge $\nu$ on $(\mathbb{W},\mathcal{B})$ via

$\displaystyle\frac{d\nu}{d\gamma}(\omega)=f(\omega(1)),\hskip
14.22636pt\omega\in\mathbb{W},$ (52)

we have $D(\mu\|\gamma_{E})=D(\nu\|\gamma)$, which follows by data processing
and the observation that $X_{1}\sim\gamma_{E}$ under $\gamma$. This gives the
following convenient representation for the entropy. For $\mu\ll\gamma_{E}$
with $D(\mu\|\gamma_{E})<\infty$, let $\nu$ be the bridge in (52). On the
filtered probability space $(\mathbb{W},\mathcal{B},\nu,\mathcal{G})$, we have

$\displaystyle
D(\mu\|\gamma_{E})=\min_{U}\frac{1}{2}\int_{0}^{1}\mathbb{E}|u_{s}|^{2}ds,$
(53)

where the minimum is over all drifts $U_{t}=\int_{0}^{t}u_{s}ds$ such that
$\mu\sim B_{1}+U_{1}$ for a standard Brownian motion $B$ carried by
$(\mathbb{W},\mathcal{B},\nu,\mathcal{G})$. Moreover, since the process
$(X_{t})_{0\leq t\leq 1}$ under $\nu$ is the Brownian bridge

$X_{t}\sim tX_{1}+\sqrt{t(1-t)}Z,$

with $Z\sim\gamma_{E}$ independent of $X_{1}\sim\mu$, we can take expectations
in Proposition 27(ii) to find, with the help of Fubini’s theorem, that the
minimum-energy process $(u_{t})_{0\leq t\leq 1}$ in (53) satisfies

$\displaystyle\mathbb{E}[u_{t}]=\int_{E}xd\mu(x),\hskip
14.22636pt\mbox{a.e.~{}}0\leq t\leq 1.$ (54)

We now record a simple application of the above, which will suit our needs.

######  Theorem 28.

Fix probability measures $\mu_{i}\ll\gamma_{E_{i}}$ on $E_{i}$ satisfying
$D(\mu_{i}\|\gamma_{E_{i}})<\infty$ for each $1\leq i\leq k$. There is a
filtered probability space $(\Omega,\mathcal{A},\mathbb{P},\mathcal{F})$
carrying a Brownian motion $B$ with
$\operatorname{Cov}(B_{1})=\operatorname{id}_{E_{0}}$ and a drift
$U_{t}=\int_{0}^{t}u_{s}ds$, $u\in L^{1}([0,1];E_{0})$ such that, for each
$1\leq i\leq k$,

  1. 1.

$\mu_{i}\sim\pi_{E_{i}}(B_{1}+U_{1})$.

  2. 2.

$D(\mu_{i}\|\gamma_{E_{i}})=\frac{1}{2}\int_{0}^{1}\mathbb{E}|\pi_{E_{i}}(u_{s})|^{2}ds$.

Moreover, the processes

$(B^{i},u^{i})=\big{(}\pi_{E_{i}}(B_{t}),\pi_{E_{i}}(u_{t})\big{)}_{0\leq
t\leq 1},~{}~{}1\leq i\leq k$

are independent.

###### Proof.

For each $1\leq i\leq k$, let $\mathbb{W}_{i}=C^{0}([0,1];E_{i})$,
$\mathcal{G}_{i}$ be its natural filtration, $\mathcal{B}_{i}$ be the
corresponding Borel $\sigma$-algebra, and $\gamma_{i}$ the Wiener measure.
Define measure $\nu_{i}\ll\gamma_{i}$ on $(\mathbb{W}_{i},\mathcal{B}_{i})$ by

$\frac{d\nu_{i}}{d\gamma_{i}}(\omega)=\frac{d\mu_{i}}{d\gamma_{E_{i}}}(\omega(1)),\hskip
14.22636pt\omega\in\mathbb{W}_{i}.$

By Proposition 27 and the subsequent discussion, there exists a drift
$U_{t}^{i}=\int_{0}^{t}u^{i}_{s}ds$ and a standard Brownian motion $B^{i}$,
both carried on $(\mathbb{W}_{i},\mathcal{B}_{i},\nu_{i},\mathcal{G}_{i})$,
such that $\mu_{i}\sim B^{i}_{1}+U^{i}_{1}$ and

$D(\mu_{i}\|\gamma_{E_{i}})=D(\nu_{i}\|\gamma_{i})=\frac{1}{2}\int_{0}^{1}\mathbb{E}|u^{i}_{s}|^{2}ds,\hskip
14.22636pt1\leq i\leq k.$

Now, put everything together on the product space
$\Omega=\prod_{i=1}^{k}(\mathbb{W}_{i}\times\mathbb{W}_{i})$ equipped with its
natural filtration, the Borel sets, and the product measure
$\mathbb{P}=\otimes_{i=1}^{k}P_{B^{i}U^{i}}$. ∎