$q^{\ast}$ satisfies (2.15) pointwise at each
$(t,{\bm{\varphi}})\in(0,t_{0})\times\mathbb{R}^{p}$. Furthermore, the uniform
boundedness and continuity of $q_{t}^{\ast}({\bm{\varphi}})$ and weak
convergence $q_{t}^{\ast}\Rightarrow q_{0}^{\ast}$ imply that
$q_{t}^{\ast}({\bm{\varphi}})\to q_{0}^{\ast}({\bm{\varphi}})$ pointwise over
${\bm{\varphi}}\in\mathbb{R}^{p}$, so $\\{q_{t}^{\ast}\\}_{t\in[0,t_{0}]}$ is
a classical solution to (2.15).

Conversely, for any solution $(q,g)$ to (2.16–2.15), Lemma C.3-(2) applied
with this $g$ shows that we must have $q\in\mathcal{L}_{q}^{\prime}(t_{0})$.
Then $(q,g)\in\mathcal{L}_{q}^{\prime}(t_{0})\times\mathcal{L}_{g}(t_{0})$ is
also a fixed point of $\Gamma$, so by the uniqueness guarantees in Lemmas
C.1-(1) and C.3-(1), $q=q^{\ast}$ and $g=g^{\ast}$ as equalities of elements
in $\mathcal{L}_{q}^{\prime}(t_{0})$ and $\mathcal{L}_{g}(t_{0})$. Since
$(q,g)$ are continuous, this implies $g_{t}(\theta)=g^{\ast}_{t}(\theta)$ and
$q_{t}({\bm{\varphi}})=q^{\ast}_{t}({\bm{\varphi}})$ pointwise, so the
solution $(q^{\ast},g^{\ast})$ to (2.15–2.16) is unique.

We now extend this solution to $[0,T]$, by applying this argument
sequentially. Let $\\{g_{t},q_{t}\\}_{t\in[0,t_{0}]}$ be the above solution
over $[0,t_{0}]$. For any $T>0$ and extension of $\\{g_{t}\\}_{t\in[0,t_{0}]}$
to $\\{g_{t}\\}_{t\in[0,t_{0}+T]}\in\mathcal{L}_{g}(t_{0}+T)$, Lemma C.3 shows
that there is a unique solution to the diffusion (C.10) over $[0,t_{0}+T]$
with initialization $q_{0}$, which satisfies (C.14) for all $t\in[0,t_{0}+T]$
with $K(q_{0})=K$ for the above value $K>0$ defining (C.25). The portion of
this solution on $[t_{0},t_{0}+T]$ must coincide with the unique solution to
(C.11) over $[0,T]$ that is initialized at $q_{t_{0}}$ and defined by
$\\{g_{t}\\}_{t\in[t_{0},t_{0}+T]}$. So for the initialization $q_{t_{0}}$, we
must have that (C.14) holds with $K(q_{t_{0}})=K$ and the same value $K$ as
above. Then we may apply the above argument to $[t_{0},2t_{0}]$ with
$(q_{t_{0}},g_{t_{0}})$ in place of $(q_{0},g_{0})$, and then sequentially to
$[2t_{0},3t_{0}]$, etc. This proves the existence and uniqueness of the
solution $\\{(q_{t},g_{t})\\}_{t\in[0,T]}$ to (2.15–2.16), as claimed. The
properties $g_{t}(\theta)>0$ and
$\sup_{t\in[0,T]}\mathbb{E}_{{\bm{\varphi}}_{t}\sim
q_{t}}e^{{\bm{\lambda}}^{\top}{\bm{\varphi}}_{t}}<\infty$ follow from the
properties of the solutions guaranteed by Lemmas C.1-(1) and C.3-(2), and this
concludes the proof. ∎

###  C.4 Proof of Lemma B.4

###### Proof of Lemma B.4.

Recall the form of $F_{n}$ in (2.11). To show $\operatorname*{lim\,sup}_{t\to
0}F_{n}(q_{t},g_{t})\leq F_{n}(q_{0},g_{0})$, recall from (C.22) that for any
fixed $T>0$, $q_{t}({\bm{\varphi}})$ is uniformly bounded over $t\in[0,T]$.
Since $U_{g_{t}}({\bm{\varphi}})$ is also uniformly bounded over
$\|{\bm{\varphi}}\|_{2}\leq R$, for any $R>0$,

$\lim_{t\to 0}\int\mathbf{1}\\{\|{\bm{\varphi}}\|_{\infty}\leq
R\\}U_{g_{t}}({\bm{\varphi}})q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}=\int\mathbf{1}\\{\|{\bm{\varphi}}\|_{\infty}\leq
R\\}U_{g_{0}}({\bm{\varphi}})q_{0}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}$

by the bounded convergence theorem. Applying
$U_{g_{t}}({\bm{\varphi}})<C(\|{\bm{\varphi}}\|_{2}^{2}+1)$ and Cauchy-
Schwarz,

$\int\mathbf{1}\\{\|{\bm{\varphi}}\|_{\infty}>R\\}|U_{g_{t}}({\bm{\varphi}})|q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}\leq\mathbb{P}_{q_{t}}[\|{\bm{\varphi}}_{t}\|_{\infty}>R]^{1/2}(\mathbb{E}_{q_{t}}[\|{\bm{\varphi}}_{t}\|^{4}]+1)^{1/2}.$

By Lemma C.3-(2), $\mathbb{E}_{q_{t}}[\|{\bm{\varphi}}_{t}\|^{4}]<C$ and
$\mathbb{P}_{q_{t}}[|\varphi_{t,j}|>R]\leq
e^{-R}\mathbb{E}[e^{|\varphi_{t,j}|}]<Ce^{-R}$ for a constant $C>0$, uniformly
over $j=1,\ldots,p$ and $t\in[0,T]$. Thus the above display converges to 0
uniformly over $t\in[0,T]$ as $R\to\infty$, so

$\lim_{t\to 0}\int
U_{g_{t}}({\bm{\varphi}})q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}=\int
U_{g_{0}}({\bm{\varphi}})q_{0}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}.$

Applying the uniform upper bound $\log q_{t}({\bm{\varphi}})\leq C_{0}$ and
Fatou’s lemma, we have also

$\int[C_{0}-\log
q_{0}({\bm{\varphi}})]q_{0}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}\leq\operatorname*{lim\,inf}_{t\to
0}\int[C_{0}-\log
q_{t}({\bm{\varphi}})]q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}},$

hence $\operatorname*{lim\,sup}_{t\to 0}\int(\log q_{t})q_{t}\leq\int(\log
q_{0})q_{0}$. Combining these statements gives $\operatorname*{lim\,sup}_{t\to
0}F_{n}(q_{t},g_{t})\leq F_{n}(q_{0},g_{0})$.

To show the form (B.14) for the derivative of $F_{n}$, we will appeal to the
following properties of $\\{q_{t}\\}_{t\in[0,T]}$: For any fixed $0<t<T$ and
compact subset $K\subset\mathbb{R}^{p}$,

$\displaystyle\sup_{s\in[t,T]}\sup_{{\bm{\varphi}}\in\mathbb{R}^{p}}\frac{|\log
q_{s}({\bm{\varphi}})|}{\|{\bm{\varphi}}\|^{2}+1}$ $\displaystyle<\infty,$
(C.26) $\displaystyle\sup_{s\in[t,T]}\sup_{{\bm{\varphi}}\in K}\|\nabla
q_{s}({\bm{\varphi}})\|_{2}$ $\displaystyle<\infty,$ (C.27)
$\displaystyle\sup_{s\in[t,T]}\sup_{{\bm{\varphi}}\in
K}\|\nabla^{2}q_{s}({\bm{\varphi}})\|_{\mathrm{F}}$ $\displaystyle<\infty,$
(C.28) $\displaystyle\int_{t}^{T}\int_{\mathbb{R}^{p}}\|\nabla\log
q_{s}({\bm{\varphi}})\|_{2}^{2}q_{s}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}\mathrm{d}s$
$\displaystyle<\infty.$ (C.29)

The upper bound for $\log q_{s}({\bm{\varphi}})$ in (C.26) follows from
(C.22), and the lower bound follows from [BKRS22, Example 8.2.3], where we may
take $\beta=1$ in light of the form (C.16) for the drift $\nabla
U_{g_{t}}({\bm{\varphi}})$. The next two statements (C.27–C.28) follow from
the $C^{\infty}$ smoothness of $(t,{\bm{\varphi}})\mapsto
q_{t}({\bm{\varphi}})$. By Lemma C.3-(2), we must have
$\mathbb{E}_{q_{s}}[\|{\bm{\varphi}}_{s}\|_{2}^{2}]<C$ for some $C>0$
uniformly over $s\in[0,T]$. Then (C.26) implies $\int|\log
q_{t}({\bm{\varphi}})|q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}<\infty$,
(C.16) implies

$\int\|\nabla
U_{g_{s}}({\bm{\varphi}})\|_{2}^{2}q_{s}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}<C$
(C.30)

for some $C>0$ uniformly over $s\in[0,T]$, and the last statement (C.29) then
follows from [BKRS22, Theorem 7.4.1].

Let ${\bm{\eta}}({\bm{\varphi}})=\prod_{j=1}^{p}\eta(\varphi_{j})$ where
$\eta$ is a smooth, compactly supported bump function on $\mathbb{R}$. Define
a truncated version of $F_{n}$ by

$F_{n}^{\eta}(q_{t},g_{t})=\frac{1}{p}\int{\bm{\eta}}({\bm{\varphi}})[U_{g_{t}}({\bm{\varphi}})+\log
q_{t}({\bm{\varphi}})]q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}+\frac{n}{2p}\log(2\pi)+\frac{1}{2p}\log\det{\bm{\Sigma}}.$

Then for any $t>0$, we may decompose

$\frac{\mathrm{d}}{\mathrm{d}t}F_{n}^{\eta}(q_{t},g_{t})=\mathrm{I}^{\eta}(t)+\mathrm{II}^{\eta}(t).$

Here, we set

$\displaystyle\mathrm{I}^{\eta}(t)$
$\displaystyle=\frac{1}{p}\int{\bm{\eta}}({\bm{\varphi}})\cdot\frac{\mathrm{d}}{\mathrm{d}t}U_{g_{t}}({\bm{\varphi}})\cdot
q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}$
$\displaystyle=\int\eta(\varphi)\cdot\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}{-}\log[\mathcal{N}_{\tau}*g_{t}](\varphi)\Big{)}\cdot\bar{q}_{t}(\varphi)\mathrm{d}\varphi={-}\alpha\int\bigg{(}\bigg{[}\mathcal{N}_{\tau}*\frac{\eta\bar{q}_{t}}{\mathcal{N}_{\tau}*g_{t}}\bigg{]}(\theta)-1\bigg{)}^{2}g_{t}(\theta)\mathrm{d}\theta,$

where these calculations and argument for differentiating under the integral
are the same as in (B.4). We also set

$\displaystyle\mathrm{II}^{\eta}(t)$
$\displaystyle=\frac{1}{p}\int{\bm{\eta}}({\bm{\varphi}})[U_{g_{t}}({\bm{\varphi}})+\log
q_{t}({\bm{\varphi}})+1]\cdot\frac{\mathrm{d}}{\mathrm{d}t}q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}$
$\displaystyle=\frac{1}{p}\int{\bm{\eta}}({\bm{\varphi}})[U_{g_{t}}({\bm{\varphi}})+\log
q_{t}({\bm{\varphi}})+1]\cdot\big{(}\nabla\cdot[q_{t}({\bm{\varphi}})\nabla
U_{g_{t}}({\bm{\varphi}})]+\Delta
q_{t}({\bm{\varphi}})\big{)}\mathrm{d}{\bm{\varphi}},$

where we may differentiate under the integral in the first line using the
dominated convergence theorem and boundedness of $U_{g_{t}},\nabla
U_{g_{t}},\log q_{t},\|\nabla q_{t}\|,|\Delta q_{t}|$ in compact subsets of
$(0,T)\times\mathbb{R}^{p}$ as follows from (C.26–C.28). Applying integration
by parts in ${\bm{\varphi}}$, we then have
$\mathrm{II}^{\eta}(t)=\mathrm{III}^{\eta}(t)+\mathrm{IV}^{\eta}(t)$ where

$\displaystyle\mathrm{III}^{\eta}(t)$
$\displaystyle={-}\frac{1}{p}\int{\bm{\eta}}({\bm{\varphi}})\Big{\|}\nabla\Big{(}U_{g_{t}}({\bm{\varphi}})+\log
q_{t}({\bm{\varphi}})\Big{)}\Big{\|}_{2}^{2}q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}},$
$\displaystyle\mathrm{IV}^{\eta}(t)$
$\displaystyle={-}\frac{1}{p}\int[U_{g_{t}}({\bm{\varphi}})+\log
q_{t}({\bm{\varphi}})+1]\cdot\nabla{\bm{\eta}}({\bm{\varphi}})^{\top}\nabla[U_{g_{t}}({\bm{\varphi}})+\log
q_{t}({\bm{\varphi}})]q_{t}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}.$

Take $\eta\equiv\eta_{R}$ such that $\eta_{R}(\varphi)=1$ for $|\varphi|\leq
R$, $\eta_{R}(\varphi)=0$ for $|\varphi|\geq R+1$, and
$\eta_{R}^{\prime}(\varphi)$ is bounded by a constant. Then, recalling from
the arguments preceding (B.2) that
$\mathcal{N}_{\tau}*\frac{\bar{q}_{t}}{\mathcal{N}_{\tau}*g}$ is uniformly
bounded, a direct application of the dominated convergence theorem yields

$\lim_{R\to\infty}\int_{t}^{T}\mathrm{I}^{\eta_{R}}(s)\mathrm{d}s={-}\alpha\int_{t}^{T}\int_{-M}^{M}\bigg{(}\bigg{[}\mathcal{N}_{\tau}*\frac{\bar{q}_{s}}{\mathcal{N}_{\tau}*g_{s}}\bigg{]}(\theta)-1\bigg{)}^{2}g_{s}(\theta)\mathrm{d}s.$

Applying the dominated convergence theorem using (C.30) and (C.29), we also
have

$\lim_{R\to\infty}\int_{t}^{T}\mathrm{III}^{\eta_{R}}(s)\mathrm{d}s={-}\frac{1}{p}\int_{t}^{T}\int\Big{\|}\nabla\Big{(}U_{g_{s}}({\bm{\varphi}})+\log
q_{s}({\bm{\varphi}})\Big{)}\Big{\|}_{2}^{2}q_{s}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}\mathrm{d}s.$

Finally, applying again (C.30) and (C.29), the bounds
$|U_{g_{s}}({\bm{\varphi}})|,|\log q_{s}({\bm{\varphi}})|\leq
C(\|{\bm{\varphi}}\|_{2}^{2}+1)$ uniformly over $s\in[t,T]$, and Holder’s
inequality, for some $C,C^{\prime}>0$ we have

$\displaystyle\int_{t}^{T}|\mathrm{IV}^{\eta_{R}}(s)|\mathrm{d}s$
$\displaystyle\leq
C\left[\int_{t}^{T}\int\mathbf{1}\\{\nabla{\bm{\eta}}_{R}({\bm{\varphi}})\neq
0\\}q_{s}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}\mathrm{d}s\right]^{1/4}\cdot\left[\int_{t}^{T}\int(\|{\bm{\varphi}}\|_{2}^{8}+1)q_{s}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}\mathrm{d}s\right]^{1/4}$
$\displaystyle\hskip
72.26999pt\cdot\left[\int_{t}^{T}\int\Big{\|}\nabla\Big{(}U_{g_{s}}({\bm{\varphi}})+\log
q_{s}({\bm{\varphi}})\Big{)}\Big{\|}_{2}^{2}q_{s}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}\mathrm{d}s\right]^{1/2}$
$\displaystyle\leq
C^{\prime}\sup_{s\in[t,T]}\mathbb{P}_{q_{s}}[\nabla{\bm{\eta}}_{R}({\bm{\varphi}}_{s})\neq
0]^{1/4}\cdot\sup_{s\in[t,T]}(\mathbb{E}_{q_{s}}[\|{\bm{\varphi}}_{s}\|_{2}^{8}]+1)^{1/4}.$

By Lemma C.3-(2), $\mathbb{E}_{q_{s}}[\|{\bm{\varphi}}_{s}\|^{8}]<C$ and
$\mathbb{P}_{q_{s}}[|\varphi_{s,j}|>R]\leq
e^{-R}\mathbb{E}[e^{|\varphi_{s,j}|}]<Ce^{-R}$ for a constant $C>0$, uniformly
over $j=1,\ldots,p$ and $s\in[0,T]$. Thus

$\lim_{R\to\infty}\int_{t}^{T}|\mathrm{IV}^{\eta_{R}}(s)|\mathrm{d}s=0$

Combining these statements,

$\displaystyle F_{n}(q_{T},g_{T})-F_{n}(q_{t},g_{t})$
$\displaystyle=\lim_{R\to\infty}F_{n}^{\eta_{R}}(q_{T},g_{T})-F_{n}^{\eta_{R}}(q_{t},g_{t})$
$\displaystyle=\lim_{R\to\infty}\int_{t}^{T}\Big{(}\mathrm{I}^{\eta_{R}}(s)+\mathrm{III}^{\eta_{R}}(s)+\mathrm{IV}^{\eta_{R}}(s)\Big{)}\mathrm{d}s$
$\displaystyle=\int_{t}^{T}\left({-}\alpha\int\bigg{(}\bigg{[}\mathcal{N}_{\tau}*\frac{\bar{q}_{s}}{\mathcal{N}_{\tau}*g_{s}}\bigg{]}(\theta)-1\bigg{)}^{2}g_{s}(\theta)\mathrm{d}s{-}\frac{1}{p}\int\Big{\|}\nabla\Big{(}U_{g_{s}}({\bm{\varphi}})+\log
q_{s}({\bm{\varphi}})\Big{)}\Big{\|}_{2}^{2}q_{s}({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}\right)\mathrm{d}s$

where the first equality follows from applying the dominated convergence
theorem to $\lim_{R\to\infty}F_{n}^{\eta_{R}}$. Differentiating both sides in
$T$ shows that $T\mapsto F_{n}(q_{T},g_{T})$ is differentiable at any $T>0$,
with derivative given by (B.14). ∎

##  Appendix D Derivations of gradient flows

We discuss the derivations of the Wasserstein-2 and Fisher-Rao gradient flow
equations (2.15–2.16) used in this paper. This discussion is non-rigorous and
only at the level of formal derivations, and is not used in our proofs.

###  D.1 General background

Let $\mathcal{P}$ be the space of probability density functions on
$\mathcal{X}$ with finite second moment. For a smooth curve
$\\{\rho_{t}\\}_{t\in[0,1]}$ in $\mathcal{P}$, we identify its tangent vector
$\dot{\rho}_{t}=\partial_{t}\rho_{t}(\cdot)$ at $\rho_{t}$ as a function
$\dot{\rho}_{t}:\mathcal{X}\to\mathbb{R}$. We equip $\mathcal{P}$ with the
formal structure of a Riemannian manifold, associating to each
$\rho\in\mathcal{P}$ a tangent space
$\operatorname{\text{Tan}}_{\rho}\mathcal{P}$ of such tangent vectors, and a
metric tensor
$g_{\rho}(\cdot,\cdot):\operatorname{\text{Tan}}_{\rho}\mathcal{P}\times\operatorname{\text{Tan}}_{\rho}\mathcal{P}\rightarrow\mathbb{R}$.
This induces a metric, for any $\mu,\nu\in\mathcal{P}$,

$\displaystyle\mathsf{d}(\mu,\nu)=\inf\Big{\\{}\int_{0}^{1}\lVert\dot{\rho}_{t}\rVert_{\rho_{t}}:\rho_{0}=\mu,\rho_{1}=\nu\Big{\\}}$
(D.1)

where the infimum is over all smooth curves $\\{\rho_{t}\\}_{t\in[0,1]}$ in
$\mathcal{P}$ from $\mu$ to $\nu$, and
$\lVert\zeta\rVert_{\rho}=\sqrt{g_{\rho}(\zeta,\zeta)}$ for any
$\zeta\in\operatorname{\text{Tan}}_{\rho}\mathcal{P}$. Given a functional
$F:\mathcal{P}\rightarrow\mathbb{R}$, the gradient of $F$ at
$\rho\in\mathcal{P}$, denoted by $\operatorname{grad}F[\rho]$, is the unique
element in $\operatorname{\text{Tan}}_{\rho}\mathcal{P}$ such that, for all
curves $\\{\rho_{t}\\}_{t\in[0,1]}$ with $\rho_{0}=\rho$ and
$\dot{\rho}_{0}=\zeta\in\operatorname{\text{Tan}}_{\rho}\mathcal{P}$,

$\displaystyle
g_{\rho}\big{(}\operatorname{grad}F[\rho],\zeta\big{)}=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}F(\rho_{t})=\int\delta
F[\rho](x)\zeta(x)\mathrm{d}x$ (D.2)

where $\delta F[\rho]:\mathbb{R}^{d}\to\mathbb{R}$ is the first variation of
$F$. We may then define the gradient flow
$\frac{\mathrm{d}}{\mathrm{d}t}\rho_{t}={-}\operatorname{grad}F[\rho_{t}]$.

For any $\mu,\nu\in\mathcal{P}$ if the infimum (D.1) is achieved by a curve
$\\{\rho_{t}\\}_{t\in[0,1]}$, then $\\{\rho_{t}\\}$ is a geodesic from $\mu$
to $\nu$. We say the functional $F$ is geodesically convex at $\rho$ if for
all geodesics $\\{\rho_{t}\\}_{t\in[0,1]}$ with $\rho_{0}=\rho$, we have
$\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}|_{t=0}\mathcal{F}(\rho_{t})\geq 0$.

We now specialize to the Wasserstein-2 and Fisher-Rao geometries, following
closely the presentation in [YWR23].

_Wasserstein-2 geometry._ For the formal Riemannian structure underlying the
Wasserstein-2 geometry on $\mathcal{P}_{*}(\mathbb{R}^{p})$, the tangent space
at $\rho\in\mathcal{P}_{*}(\mathbb{R}^{p})$ is

$\displaystyle\operatorname{\text{Tan}}_{\rho}^{W_{2}}\mathcal{P}_{*}(\mathbb{R}^{p})=\Big{\\{}\zeta:\mathbb{R}^{p}\to\mathbb{R}\;\Big{|}\;\zeta=-\nabla\cdot(\rho\nabla
u)\text{ for some $u$ satisfying}\int\lVert\nabla
u\rVert^{2}\mathrm{d}\rho<\infty\Big{\\}},$

equipped with the metric tensor
$g^{W_{2}}_{\rho}(\zeta_{1},\zeta_{2})=\int\left\langle\nabla u_{1},\nabla
u_{2}\right\rangle\rho(x)\mathrm{d}x$. To compute the Wasserstein-2 gradient,
let $\\{\rho_{t}\\}_{t\in[0,1]}$ be a curve with $\rho_{0}=\rho$ and
$\dot{\rho}_{0}=\zeta=-\nabla\cdot(\rho\nabla v)$. Then (D.2) reads

$\displaystyle
g_{\rho}^{W_{2}}\big{(}\operatorname{grad}^{W_{2}}F[\rho],\zeta\big{)}$
$\displaystyle=\int\delta F[\rho]\cdot\zeta\,\mathrm{d}x=\int{-}\delta
F[\rho]\Big{(}\nabla\cdot(\rho\nabla
v)\Big{)}\mathrm{d}x=\int\left\langle\nabla\delta F[\rho],\nabla
v\right\rangle\rho\,\mathrm{d}x,$

the last equality applying integration by parts. Comparing with the definition
of $g_{\rho}^{W_{2}}$, this identifies the Wasserstein-2 gradient as

$\displaystyle\operatorname{grad}^{W_{2}}F[\rho]=-\nabla\cdot(\rho\nabla\delta
F[\rho]).$ (D.3)

_Fisher-Rao geometry._ For the formal Riemannian structure underlying the
Fisher-Rao geometry on $\mathcal{P}_{*}(M)$, the tangent space at
$\rho\in\mathcal{P}_{*}(M)$ is

$\displaystyle\operatorname{\text{Tan}}_{\rho}^{{\operatorname{FR}}}\mathcal{P}=\Big{\\{}\zeta:[-M,M]\to\mathbb{R}\;\Big{|}\;\zeta=\rho\Big{(}\alpha-\int_{-M}^{M}\alpha\mathrm{d}\rho\Big{)}\text{
for some $\alpha$
satisfying}\int_{-M}^{M}\alpha^{2}\mathrm{d}\rho<\infty\Big{\\}},$

equipped with the metric tensor
$g^{{\operatorname{FR}}}_{\rho}(\zeta_{1},\zeta_{2})=\int\frac{\zeta_{1}\cdot\zeta_{2}}{\rho^{2}}\mathrm{d}\rho=\int(\alpha_{1}-\int\alpha_{1}\mathrm{d}\rho)(\alpha_{2}-\int\alpha_{2}\mathrm{d}\rho)\mathrm{d}\rho$.
To compute the Fisher-Rao gradient, let $\\{\rho_{t}\\}_{t\in[0,1]}$ be a
curve with $\rho_{0}=\rho$ and
$\dot{\rho}_{0}=\zeta=\rho(\alpha-\int\alpha\,\mathrm{d}\rho)$. Then (D.2)
reads

$\displaystyle
g^{{\operatorname{FR}}}_{\rho}(\operatorname{grad}^{{\operatorname{FR}}}F[\rho],\zeta)=\int_{-M}^{M}\delta
F[\rho]\cdot\zeta\,\mathrm{d}x$ $\displaystyle=\int_{-M}^{M}\delta
F[\rho]\cdot\Big{(}\alpha-\int_{-M}^{M}\alpha\mathrm{d}\rho\Big{)}\mathrm{d}\rho$
$\displaystyle=\int_{-M}^{M}\Big{(}\delta F[\rho]-\int_{-M}^{M}\delta
F[\rho]\mathrm{d}\rho\Big{)}\cdot\Big{(}\alpha-\int_{-M}^{M}\alpha\mathrm{d}\rho\Big{)}\mathrm{d}\rho.$

Comparing with the definition of $g_{\rho}^{{\operatorname{FR}}}$, this
identifies the Fisher-Rao gradient as

$\displaystyle\operatorname{grad}^{{\operatorname{FR}}}F[\rho]=\rho\Big{(}\delta
F[\rho]-\int\delta F[\rho]\mathrm{d}\rho\Big{)}.$ (D.4)

###  D.2 Derivatives of $\bar{F}_{n}(g)$ and $F_{n}(q,g)$

From the form (1.4) of $\bar{F}_{n}(g)$, for any positive density
$g\in\mathcal{P}_{*}(M)$ and bounded $\chi:[-M,M]\to\mathbb{R}$ with
$\int_{-M}^{M}\chi(\theta)\mathrm{d}\theta=0$, we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\Big{|}_{\varepsilon=0}\bar{F}_{n}(g+\varepsilon\chi)$
$\displaystyle={-}\frac{1}{p}\sum_{j=1}^{p}\frac{\int_{[-M,M]^{p}}\exp(-\frac{\|\mathbf{y}-\mathbf{X}{\bm{\theta}}\|^{2}}{2\sigma^{2}})\frac{\chi(\theta_{j})}{g(\theta_{j})}\prod_{i=1}^{p}g(\theta_{i})\mathrm{d}{\bm{\theta}}}{\int_{[-M,M]^{p}}\exp(-\frac{\|\mathbf{y}-\mathbf{X}{\bm{\theta}}\|^{2}}{2\sigma^{2}})\prod_{i=1}^{p}g(\theta_{i})\mathrm{d}{\bm{\theta}}}$
$\displaystyle=-\frac{1}{p}\sum_{j=1}^{p}\int_{-M}^{M}\frac{\chi(\theta_{j})}{g(\theta_{j})}P_{g}({\bm{\theta}}\mid\mathbf{y})=\int_{-M}^{M}{-}\frac{\chi(\theta)}{g(\theta)}\bar{\mu}[g](\theta)\mathrm{d}\theta=\int_{-M}^{M}{-}\frac{\bar{\mu}[g](\theta)}{g(\theta)}\chi(\theta)\mathrm{d}\theta,$

where $\bar{\mu}[g]$ is the average marginal density of
$\mu[g]({\bm{\theta}})=P_{g}({\bm{\theta}}\mid\mathbf{y})$ as defined in
(2.7). Similarly, from the equivalent form (2.10), we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\Big{|}_{\varepsilon=0}\bar{F}_{n}(g+\varepsilon\chi)$
$\displaystyle={-}\frac{1}{p}\sum_{j=1}^{p}\int\frac{[\mathcal{N}_{\tau}*\chi](\varphi_{j})}{[\mathcal{N}_{\tau}*g](\varphi_{j})}P_{g}({\bm{\varphi}}\mid\mathbf{y})$
$\displaystyle=\int{-}\frac{[\mathcal{N}_{\tau}*\chi](\varphi)}{[\mathcal{N}_{\tau}*g](\varphi)}\bar{\nu}[g](\varphi)\mathrm{d}\varphi=\int_{-M}^{M}{-}\left[\mathcal{N}_{\tau}*\frac{\bar{\nu}[g]}{\mathcal{N}_{\tau}*g}\right](\theta)\chi(\theta)\mathrm{d}\theta,$

where the last equality uses the identity $\int
f(\varphi)[\mathcal{N}_{\tau}*\chi](\varphi)\mathrm{d}\varphi=\int_{-M}^{M}[\mathcal{N}_{\tau}*f](\theta)\chi(\theta)\mathrm{d}\theta$.
Thus we identify two equivalent forms for the first variation of
$\bar{F}_{n}(g)$,

$\displaystyle\delta\bar{F}_{n}[g](\theta)={-}\frac{\bar{\nu}[g](\theta)}{g(\theta)}+1={-}\mathcal{N}_{\tau}*\frac{\bar{\nu}[g]}{\mathcal{N}_{\tau}*g}(\theta)+1,$
(D.5)

where the constant $+1$ is chosen to center $\delta\bar{F}_{n}[g]$ to have
mean 0 under $g$. The second form shows (3.12), as well as (2.3) when
specialized to $\mathbf{X}=\operatorname{Id}$ with $\sigma^{2}=\tau^{2}$ and
$\bar{\nu}[g]=n^{-1}\sum_{i=1}^{n}\delta_{y_{i}}$. The first form shows (2.6),
in light of the general definition (D.4) for the Fisher-Rao gradient. More
generally, for $g\in\mathcal{P}(M)$, the last expression of (D.5) still holds
and equals to

$\displaystyle\delta\bar{F}_{n}[g](\theta)=-\frac{1}{p}\sum_{j=1}^{p}\frac{\int_{[-M,M]^{p-1}}\exp(-\frac{\|\mathbf{y}-\mathbf{X}(\theta,{\bm{\theta}}_{-j})\|^{2}}{2\sigma^{2}})\prod_{i:i\neq
j}g(\mathrm{d}\theta_{i})}{\int_{[-M,M]^{p}}\exp(-\frac{\|\mathbf{y}-\mathbf{X}{\bm{\theta}}\|^{2}}{2\sigma^{2}})\prod_{i=1}^{p}g(\mathrm{d}\theta_{i})}+1,$

which satisfies
$\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\Big{|}_{\varepsilon=0}\bar{F}_{n}(g+\varepsilon\chi)=\int_{[-M,M]}\delta\bar{F}_{n}[g](\theta)\chi(\theta)\mathrm{d}\theta$
for any $\chi$ on $[-M,M]$ such that $g+\varepsilon\chi$ remains a valid
distribution for $\varepsilon$ in a neighborhood of $0$.

Similarly, for the functional $F_{n}(q,g)$ in (2.11), we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\Big{|}_{\varepsilon=0}F_{n}(q,g+\varepsilon\chi)$
$\displaystyle=-\frac{1}{p}\sum_{j=1}^{p}\int\frac{[\mathcal{N}_{\tau}\ast\chi](\varphi_{j})}{[\mathcal{N}_{\tau}\ast
g](\varphi_{j})}q({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}=\int_{-M}^{M}{-}\left[\mathcal{N}_{\tau}\ast\frac{\bar{q}}{\mathcal{N}_{\tau}\ast
g}\right](\theta)\chi(\theta)\mathrm{d}\theta,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\Big{|}_{\varepsilon=0}F_{n}(q+\varepsilon\chi,g)$
$\displaystyle=\frac{1}{p}\int\big{[}U_{g}({\bm{\varphi}})+\log
q({\bm{\varphi}})+1\big{]}\chi({\bm{\varphi}})\mathrm{d}{\bm{\varphi}},$

where $U_{g}({\bm{\varphi}})$ is the negative log-posterior-density (3.5).
Then, applying $\int_{-M}^{M}\chi(\theta)\mathrm{d}\theta=0$ and
$\int\chi({\bm{\varphi}})\mathrm{d}{\bm{\varphi}}=0$, the first variations of
$F_{n}$ are

$\delta_{g}F_{n}[q,g](\theta)=-\mathcal{N}_{\tau}\ast\frac{\bar{q}}{\mathcal{N}_{\tau}\ast
g}(\theta)+1,\qquad\delta_{q}F_{n}[q,g]({\bm{\varphi}})=\frac{1}{p}\big{[}U_{g}({\bm{\varphi}})+\log
q({\bm{\varphi}})\big{]},$

where, for notational convenience, we have centered this first quantity
$\delta_{g}F_{n}[q,g](\theta)$ to have mean 0 under $g$. This leads to the
forms (2.15–2.16) for the gradient flows (1.7), in light of the general
definitions (D.3) and (D.4) for the Wasserstein-2 and Fisher-Rao gradients.

###  D.3 Non-convexity

We provide a concrete example to show that even for the identity design
$\mathbf{X}=\operatorname{Id}$, the functional $\bar{F}_{n}(g)$ is not
geodesically convex under either the Wasserstein-2 geometry or the Fisher-Rao
geometry, even though it is convex in the sense (3.4) in the linear geometry.

For the Wasserstein-2 geometry, consider $g_{0}=g_{\ast}=\delta_{0}$,
$g_{1}=\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1}$, and
$y_{1},\ldots,y_{n}\overset{iid}{\sim}\mathcal{N}(0,1)*g_{\ast}$. The
constant-speed Wasserstein-2 geodesic connecting $g_{0}$ and $g_{1}$ is
$g_{t}=\frac{1}{2}\delta_{-t}+\frac{1}{2}\delta_{t}$ for $t\in[0,1]$. Explicit
computation shows

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\bar{F}_{n}(g_{t})=1-\frac{1}{n}\sum_{i=1}^{n}\frac{y_{i}^{2}}{\cosh^{2}(ty_{i})},$

yielding that
$\mathbb{P}_{g_{\ast}}(\frac{\mathrm{d}^{2}}{\mathrm{d}^{2}t}|_{t=0}\bar{F}_{n}(g_{t})\geq
0)=\mathbb{P}_{g_{\ast}}(n^{-1}\sum_{i=1}^{n}Y_{i}^{2}\leq 1)\approx 1/2$ for
large $n$.

For the Fisher-Rao geometry, let $g_{*}=\operatorname{Bernoulli}(1/2)$,
$g_{0}=\operatorname{Bernoulli}(1/4)$, $g_{1}=\operatorname{Bernoulli}(3/4)$,
and $y_{1},\ldots,y_{n}\overset{iid}{\sim}\mathcal{N}(0,1)*g_{\ast}$. Using
the fact [MMC23] that the geodesic distance in (D.1) is
$d_{{\operatorname{FR}}}(\text{Bernoulli}(\theta_{1}),\text{Bernoulli}(\theta_{2}))=2|\arcsin(\theta_{1})-\arcsin(\theta_{2})|$
for $\theta_{1},\theta_{2}\in(0,1)$, we have
$d_{{\operatorname{FR}}}(g_{0},g_{1})=\pi/3$. The constant-speed Fisher-Rao
geodesic connecting $g_{0}$ and $g_{1}$ is
$g_{t}=\operatorname{Bernoulli}(p_{t})$ with
$p_{t}=\sin^{2}(\frac{\pi}{6}(t+1))$, from equating
$d_{{\operatorname{FR}}}(g_{0},g_{t})=t\cdot\pi/3$. Explicit computation shows
that

$\displaystyle\mathbb{E}_{g_{\ast}}[\bar{F}_{n}(g_{t})]=-\frac{1}{2}\int\Big{(}\mathcal{N}(x)+\mathcal{N}(x-1)\Big{)}\cdot\log\Big{[}\cos^{2}\Big{(}\frac{\pi}{6}(t+1)\Big{)}\mathcal{N}(x)+\sin^{2}\Big{(}\frac{\pi}{6}(t+1)\Big{)}\mathcal{N}(x-1)\Big{]}\mathrm{d}x$

is not convex in $t$, where $\mathcal{N}(x)$ is the standard
$\mathcal{N}(0,1)$ density at $x$.

##  Appendix E Details of algorithms

CAVI. We consider the Gibbs variational representation

$\bar{F}_{n}(g)=\min_{q}\widetilde{F}_{n}(q,g):=\min_{q}\frac{1}{p}\int_{[-M,M]^{p}}\bigg{[}\frac{\lVert\mathbf{y}-\mathbf{X}{\bm{\theta}}\rVert^{2}}{2\sigma^{2}}-\sum_{j=1}^{p}\log
g(\theta_{j})+\log
q({\bm{\theta}})\bigg{]}q({\bm{\theta}})\mathrm{d}{\bm{\theta}}$

in the parametrization by ${\bm{\theta}}$; in other words,
$\widetilde{F}_{n}(q,g)$ coincides with ${F}_{n}(q,g)$ in (2.11) for $\tau=0$.
Mean-field variational inference optimizes $\widetilde{F}_{n}(q,g)$ over
product distributions $q({\bm{\theta}})=\prod_{j=1}^{p}q_{j}(\theta_{j})$. It
is easy to check that, for fixed $g$ and
$q_{-j}=(q_{1},\ldots,q_{j-1},q_{j+1},\ldots,q_{p})$, the minimizer over
$q_{j}$ has the explicit form

$\displaystyle
q_{j}(\theta_{j})\propto\exp\Big{(}{-}\frac{\|\mathbf{x}_{j}\|^{2}}{2\sigma^{2}}\theta_{j}^{2}+\frac{1}{\sigma^{2}}(\mathbf{y}-\mathbf{X}_{-j}\mathbb{E}_{q_{-j}}[{\bm{\theta}}_{-j}])^{\top}\mathbf{x}_{j}\cdot\theta_{j}\Big{)}g(\theta_{j})$
(E.1)

where $\mathbf{X}_{-j}\in\mathbb{R}^{n\times(p-1)}$ is $\mathbf{X}$ without
the $j$-th column, and ${\bm{\theta}}_{-j}\in\mathbb{R}^{p-1}$ is
${\bm{\theta}}$ without the $j$-th coordinate. Over a discrete support
$b_{1},\ldots,b_{K}\in[-M,M]$, we represent the prior $g$ by probability
weights $w=(w_{1},\ldots,w_{K})$ and each $q_{j}$ by weights
$w_{j}=(w_{j,1},\ldots,w_{j,K})$. Then $w_{j}$ is updated as

$\displaystyle
w_{j,k}=\frac{w_{k}\exp\Big{(}-\frac{\|\mathbf{x}_{j}\|^{2}}{2\sigma^{2}}b_{k}^{2}+\frac{1}{\sigma^{2}}(\mathbf{y}-\mathbf{X}_{-j}\mathbb{E}_{q_{-j}}[{\bm{\theta}}_{-j}])^{\top}\mathbf{x}_{j}\cdot
b_{k}\Big{)}}{\sum_{i=1}^{K}w_{i}\exp\Big{(}-\frac{\|\mathbf{x}_{j}\|^{2}}{2\sigma^{2}}b_{i}^{2}+\frac{1}{\sigma^{2}}(\mathbf{y}-\mathbf{X}_{-j}\mathbb{E}_{q_{-j}}[{\bm{\theta}}_{-j}])^{\top}\mathbf{x}_{j}\cdot
b_{i}\Big{)}}\quad\mbox{for all }k=1,\ldots,K,$ (E.2)

where
$\mathbb{E}_{q_{-j}}[{\bm{\theta}}_{-j}]=(\sum_{k=1}^{K}b_{k}w_{j^{\prime},k})_{j^{\prime}\neq
j}$.

Fixing $q=\prod_{j=1}^{p}q_{j}$, we update $g$ to minimize

$\widetilde{F}_{n}(q,g)+\frac{\lambda}{2}\int_{-M}^{M}g^{\prime\prime}(\theta)^{2}\mathrm{d}\theta=\int_{-M}^{M}[{-}\log
g(\theta)]\bar{q}(\theta)\mathrm{d}\theta+\frac{\lambda}{2}\int_{-M}^{M}g^{\prime\prime}(\theta)^{2}\mathrm{d}\theta+\text{constant}$
(E.3)

where $\bar{q}=p^{-1}\sum_{j}q_{j}$. This is implemented by the weight update

$w=\arg\,\min_{w\geq
0,\|w\|_{1}=1}{-}\sum_{k=1}^{K}\frac{\sum_{j=1}^{p}w_{j,k}}{p}\log
w_{k}+\frac{\lambda\Delta}{2}\sum_{k=1}^{K}\left[\frac{Dw}{\Delta}\right]_{k}^{2}$

where $D$ is the differential quadrature matrix described following (2.25).
When $\lambda=0$, this takes the closed form
$w_{k}=\frac{1}{p}\sum_{j=1}^{p}w_{j,k}$; when $\lambda>0$, we solve this
convex program using CVXR [FNB20]. Each iteration of CAVI iteratively updates
$(q_{1},\ldots,q_{p},g)$ once in this order.

We initialize CAVI from
$q_{1}=\ldots=q_{p}=g=\operatorname{Uniform}(\\{b_{1},\ldots,b_{K}\\})$.

Gibbs-MCEM. Under the prior $g(\theta)$, the conditional posterior density of
$\theta_{j}$ given ${\bm{\theta}}_{-j}$ is

$\displaystyle
P_{g}(\theta_{j}\mid\mathbf{y},{\bm{\theta}}_{-j})\propto\exp\Big{(}{-}\frac{\|\mathbf{x}_{j}\|^{2}}{2\sigma^{2}}\theta_{j}^{2}+\frac{1}{\sigma^{2}}(\mathbf{y}-\mathbf{X}_{-j}{\bm{\theta}}_{-j})^{\top}\mathbf{x}_{j}\cdot\theta_{j}\Big{)}g(\theta_{j}).$
(E.4)

Over a discrete support $b_{1},\ldots,b_{K}\in[-M,M]$, this is represented by
the weight vector $w_{j}=(w_{j,1},\ldots,w_{j,K})$ with weights

$w_{j,k}=\frac{w_{k}\exp\Big{(}-\frac{\|\mathbf{x}_{j}\|^{2}}{2\sigma^{2}}b_{k}^{2}+\frac{1}{\sigma^{2}}(\mathbf{y}-\mathbf{X}_{-j}{\bm{\theta}}_{-j})^{\top}\mathbf{x}_{j}\cdot
b_{k}\Big{)}}{\sum_{i=1}^{K}w_{i}\exp\Big{(}-\frac{\|\mathbf{x}_{j}\|^{2}}{2\sigma^{2}}b_{i}^{2}+\frac{1}{\sigma^{2}}(\mathbf{y}-\mathbf{X}_{-j}{\bm{\theta}}_{-j})^{\top}\mathbf{x}_{j}\cdot
b_{i}\Big{)}}\quad\mbox{for all }k=1,\ldots,K$ (E.5)

where $w=(w_{1},\ldots,w_{k})$ are weights for the prior.

Gibbs-MCEM iteratively resamples the coordinates
$\theta_{1},\ldots,\theta_{p}$ of ${\bm{\theta}}$ in sequence from the
conditional law (E.5) over the support points $b_{1},\ldots,b_{K}$, fixing the
priors weights $w=(w_{1},\ldots,w_{K})$. Having drawn $T_{\text{iter}}$
samples $\theta_{j}^{(1)},\ldots,\theta_{j}^{(T_{\text{iter}})}$ of each
$j$-th coordinate, it then approximates $\bar{q}$ in (E.3) by the empirical
distribution of these samples, yielding the update for prior weights

$w=\arg\,\min_{w\geq
0,\|w\|_{1}=1}{-}\frac{\sum_{j=1}^{p}\sum_{t=1}^{T_{\text{iter}}}\sum_{k=1}^{K}1\\{\theta_{j}^{(t)}=b_{k}\\}}{pT_{\text{iter}}}\log
w_{k}+\frac{\lambda\Delta}{2}\sum_{k=1}^{K}\left[\frac{Dw}{\Delta}\right]_{k}^{2}.$

We initialize Gibbs-MCEM from ${\bm{\theta}}=0$ and
$g=\operatorname{Uniform}(\\{b_{1},\ldots,b_{K}\\})$, and run the first 200
sampling iterations without updating prior weights.

Langevin-MCEM. Langevin-MCEM applies the ULA iterations (4.3) to sample from
$P_{g}({\bm{\varphi}}\mid\mathbf{y})$, fixing the prior weights $w=w_{t}$
between every $T_{\text{iter}}$ iterations. After every $T_{\text{iter}}$
iterations, denoting by
${\bm{\varphi}}^{(1)},\ldots,{\bm{\varphi}}^{(T_{\text{iter}})}$ the sampled
iterates, we sample $S=10000$ coordinates $\varphi_{1},\ldots,\varphi_{S}$
uniformly at random from the $pT_{\text{iter}}$ total coordinates of
${\bm{\varphi}}^{(1)},\ldots,{\bm{\varphi}}^{(T_{\text{iter}})}$ (for reasons
of computational efficiency), and approximate the exact M-step update (2.22)
in EM by the prior weight update

$w=\arg\,\min_{w\geq
0,\|w\|_{1}=1}{-}\frac{1}{S}\sum_{s=1}^{S}\log\left(\sum_{k=1}^{K}w_{k}\mathcal{N}_{\tau}(b_{k}-\varphi_{s})\right)+\frac{\lambda\Delta}{2}\sum_{k=1}^{K}\left[\frac{Dw}{\Delta}\right]_{k}^{2}.$

In other words, this is a discretized NPMLE with sample size $S$ with a
smoothness penalty. This convex program is solved using CVXR.

We initialize Langevin-MCEM from ${\bm{\varphi}}=0$ and
$q=\operatorname{Uniform}(\\{b_{1},\ldots,b_{K}\\})$, and run the first 200
sampling iterations at large step size $\eta_{t}^{\varphi}=1.0$ without
updating prior weights, before transitioning to the fixed step size
$\eta_{t}^{\varphi}\in\\{0.1,1.0\\}$ with prior weight updates.

EBflow. EBflow alternately applies the ULA iteration (4.3) and the $w$
iteration (2.26) with spline smoothing, or (4.5) and (2.26) in the
preconditioned setting. We initialize EBflow also from ${\bm{\varphi}}=0$ and
$g=\operatorname{Uniform}(\\{b_{1},\ldots,b_{K}\\})$, and again run the first
200 sampling iterations at large step size $\eta_{t}^{\varphi}=1.0$ without
updating prior weights, before transitioning to the log-linear step size decay
schedule (4.4).

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