Approximation, Kernelization, and Entropy-Dissipation of
Gradient Flows: from Wasserstein to Fisher-Rao
Jia-Jie Zhu
Alexander Mielke

zhu@wias-berlin.de
mielke@wias-berlin.deWeierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39
10117 Berlin, Germany

Abstract
Motivated by various machine learning applications, we present a principled investigationof gradient flow dissipation geometry, emphasizing the Fisher-Rao type gradient flows andthe interplay with Wasserstein space. Using the dynamic Benamou-Brenier formulation, we
reveal a few precise connections between those flow dissipation geometries and commonlyused machine learning tools such as Stein flows, kernel discrepancies, and nonparametric
regression. In addition, we present analysis results in terms of Lojasiewicz type functional
inequalities, with an explicit threshold condition for a family of entropy dissipation along the
Fisher-Rao flows. Finally, we establish rigorous evolutionary Γ-convergence for the FisherRao type gradient flows obtained via regression, justifying the approximation beyond static
point-wise convergence.
Keywords: optimal transport, kernel methods, gradient flow, partial differential equation, Wasserstein, Fisher-Rao, Hellinger, optimization, calculus of variations, variational
inference, sampling, generative models,Contents
1 Introduction

2

2 Preliminaries
2.1 Gradient-flow systems and geodesics . . . . . . . . . . . . . . . . . . . . . .
2.2 Reproducing kernel Hilbert space . . . . . . . . . . . . . . . . . . . . . . . .

10
10
13

3 Fisher-Rao setting
3.1 Kernelization and approximation of Fisher-Rao gradient flows . . . . . . . .
3.2 MMD as de-kernelized Fisher-Rao distance . . . . . . . . . . . . . . . . . .
3.3 Flattened Fisher-Rao, Allen-Cahn, and φ-divergences . . . . . . . . . . . . .

15
15
17
19

4 Wasserstein setting
4.1 Gradient structure for the (regularized) Stein gradient flow . . . . . . . . .
4.2 De-Stein: de-kernelized Wasserstein geometry . . . . . . . . . . . . . . . . .
4.3 Flattened Wasserstein, Cahn-Hilliard, and Sobolev discrepancy . . . . . . .

20
20
21
22

5 Wasserstein-Fisher-Rao setting
5.1 Kernelization and approximation of the WFR gradient flow . . . . . . . . .

23
23©2022 Jia-Jie Zhu and Alexander Mielke.
License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/.

Zhu and Mielke

5.2
5.3

De-kernelized Wasserstein-Fisher-Rao and Kernel Sobolev-Fisher . . . . . .
Flattened Wasserstein-Fisher-Rao and kernel-Sobolev-Fisher . . . . . . . . .

24
25

6 Analysis
6.1 Entropy dissipation and functional inequalities . . . . . . . . . . . . . . . .
6.2 Kernel discrepancies and entropy dissipation in kernelized geometries . . . .
6.3 Explicit connection between nonparametric regression and Rayleigh Principle
6.4 Energy dissipation in kernel-approximate flows: formal arguments . . . . .
6.5 Evolutionary Γ-convergence at the approximation limit . . . . . . . . . . . .

26
26
32
35
37
38

7 Other related works

41

8 Further proofs

42

9 Discussion

46

1 Introduction
We adopt a perspective rooted in the series of works from the 1990s that pioneered thestudy of Wasserstein gradient flows, as eloquently articulated by Otto:
The merit of the right gradient flow formulation of a dissipative evolution
equation is that it separates energetics and kinetics: The energetics endow
the state space with a functional, the kinetics endow the state space with a(Riemannian) geometry via the metric tensor.
In essence, the seminal works such as (Otto; Jordan et al., 1998) enabled a systematic
perspective of studying the PDE such as the typeδF∂t µ = − div µ∇[µ]δµas gradient flows of the energy functional F , i.e., the solution paths of the measure optimization problem
min F (µ)µ∈P(Rd )(1.1)

in the Wasserstein space P(Rd ), Wp .
While much recent research in machine learning applications has predominantly focusedon modifying the energy functionals (i.e., F above) of the pure Wasserstein gradient flows,e.g., (Arbel et al., 2019; Chewi et al., 2020; Glaser et al., 2021; Korba et al., 2021; Carrillo
et al., 2019; Lu et al., 2023; Javanmard et al., 2020; Craig et al., 2023), we delve into various
approximations and kernelizations of the gradient flow geometry, beyond the confines of thepure Wasserstein and Fisher-Rao. In doing so, our investigation also reveals precise relations
between various previously proposed geometries over probability measures, such as the Stein
distance (Duncan et al., 2019; Liu and Wang, 2019), kernel Stein discrepancy (Liu et al.),
Sobolev discrepancy (Mroueh and Rigotti, 2020), and maximum mean discrepancy (Gretton
et al., 2012).
2

Approximation and Kernelization of Gradient Flows

By working with the different gradient flow geometries and leaving the energy functional
to be chosen for specific applications, we provide a measure optimization (1.1) framework
to be adapted to various applications beyond the confines of ad-hoc energy functionals,
evidenced in the following examples.
Sampling in Stein geometry Suppose a statistician wishes to generate samples from1a probability distribution π, whose density is in the form π(x) = R −V (x) e−V (x) , whereedx
V is the potential energy. In this case, using the fact that π is the invariant distribution of
the Langevin stochastic differential equation√(1.2)
dXt = −∇V (Xt )dt + 2dZt ,where Zt is the standard Brownian motion. From a PDE perspective, this LangevinSDE (1.2) describes the same dynamical system as the deterministic drift-diffusion FokkerPlanck PDE∂t µ = − div (µ∇ (V + log µ))(1.3)

for probability measure µ, which is also the gradient-flow equation of a Wasserstein gradientflow. Therefore, instead of relying on the stochastic simulation of (1.2), one can forward
simulate the deterministic PDE (1.3). Liu and Wang (2019) have proposed a deterministic
discrete-time update algorithm referred to as Stein variational gradient descent (SVGD).
This algorithm has been related to the Stein PDE by Duncan et al. (2019)∂t µ = − div (µKµ ∇ (V + log µ))(1.4)

where Kµ is the integral operator. The gradient flow equation (1.4) can be viewed as thekernelization of the pure Wasserstein gradient flow equation (1.3).
Variational inference and natural gradient descent One major topic in machinelearning research is inferring the posterior distribution of the model parameters θ ∈ Θ, given
the observed data, i.e., finding π(θ|Data). In practice, the exact posterior distribution isoften intractable, and one must resort to approximate variational inference methods (Jordan
et al., 1999; Wainwright and Jordan, 2008; Blei et al., 2017). This amounts to finding theapproximate posterior probability measure µ by solving
min DKL (µ|π).
µ∈P

(1.5)

A large class of variational inference methods is based on optimizing a parameterized distribution µη , e.g., the Gaussian distribution family. Then, the optimization problem (1.5)
is solved by minimizing with respect to the parameter η, often picked as the natural parameters of exponential families. In such cases, an efficient approach is the natural gradientdescent (Amari, 1998; Khan and Nielsen, 2018; Hoffman et al., 2013; Khan and Rue, 2023)
on η that respects the geometry of the parameterized probability space. In practice, theupdate rule is an Riemannian gradient descent schemeη k+1 ← argmin ∇η F (µηk )(η − η k ) +η31(η − η k )⊤ G(η k )(η − η k ),
2τ

(1.6)

Zhu and Mielke

where F (µη ) = DKL (µη |π) in the variational inference context and ∇η F (µηk ) its gradient⊤Rwith respect to η. The matrix G(η k ) = µηk (x) · ∇η log µηk (x) ∇η log µηk (x) dx isreferred to as the Fisher information matrix. In this paper’s context, the last (proximal)
term in the update rule (1.6) is a quadratic approximation to the squared Fisher-Rao
distance between the probability measures when η and η k are close, i.e.,(η − η k )⊤ G(η k )(η − η k ) ≈ FR2 (µηk , µη ).
Therefore, those methods correspond to gradient flows in the Fisher-Rao geometry, which
is a central topic in this paper. See (Chen et al., 2023) for more details on this connectionand (Lambert et al., 2022) for the Wasserstein flow for Gaussian variational inference.
Deep generative models A recent application of gradient flows is generative models.
One particular relevant class of algorithms is the score-based deep diffusion generativemodels (Song et al., 2020; Song and Ermon, 2020; Ho et al., 2020; Sohl-Dickstein et al.,2015; De Bortoli, 2023; Oko et al., 2023). The goal of the so-called score-matching task is
to compute the vector field ∇ log µt to simulate a backward SDE√
dXt = (Xt + 2∇ log µt (Xt )) dt + 2dWt ,(1.7)
The term ∇ log µt (Xt ) can be approximated via regression in practice,Z T
inf∥f (·, t) − ∇ log µt ∥2L2µ dt,
f ∈F

(1.8)

t

0

where µt is the state distribution of a diffusion process at time t, e.g., Ornstein–Uhlenbeck
process. Another class of generative models that has shown improved efficiency and stability
is the flow-based generative models (Lipman et al., 2022). They learn the solution u to the
ODE u̇ = −vt (u) for some velocity field vt by solving the regression problem with explicit
target velocity vt
Z T
(1.9)
inf
∥f (·, t) − vt ∥2L2µ dt.
f ∈F

t

0

Furthermore, they observed that, by choosing vt to be the velocity field for the optimaltransport between Gaussian distributions, they obtained more efficient and stable training
than previous generative models.
The hope of those learning algorithms is to approximate some vector field of the original
target flows. Note that in practice, f is often parameterized using a time-dependent neuralnetwork and training is further done over various initial conditions. From this paper’s
perspective, it is crucial to note that the new flow following the learned velocity field, in theabove formulations, has a new geometry that is different from the original flow even if the
training error is close to zero.
Motivated by those applications, we first study in detail the gradient flows in the FisherRao geometry, generated by the Fisher-Rao distance, also known as the Hellinger distance,
between two nonnegative measures µ, ν. It is defined as
s !2
Z s
δµ
δν
FR2 (µ, ν) = 4 ·−dγ(1.10)
δγ
δγ
4

Approximation and Kernelization of Gradient Flowsfor a reference measure γ, µ, ν << γ. Recall its dynamic formulation 1 (see, e.g., (Gallouëtand Monsaingeon, 2017), (Liero et al., 2018))
FR2 (µ, ν) = min
µ,ξt

Z 1
0

∥ξt ∥2L2µ dtµ̇ = −µ · ξt , µ(0) = µ, µ(1) = ν .

(1.11)

Using the tools from kernel methods and the Benaou-Brenier dynamic formulation, we firstinvestigate new gradient systems centered around the Fisher-Rao geometry. Our emphasis ison the precise relation in terms of the kernelization of gradient flows, stated in Definition 3.1.
Motivated by the kernelization in the Stein geometry (see, e.g., (1.4)), we provide thegeodesic and gradient structures of the kernelized Fisher-Rao geometry in Section 3, whose
gradient flow equation is a reaction equation with a kernelized growth fieldµ̇ = −µKµ

δF
[µ] .
δµ

Furthermore, we find that the kernelization of the kernel MMD (Gretton et al., 2012),
commonly used in machine learning applications, results precisely in the pure Fisher-Rao
geometry. We summarize the relations below.

MMDde-kernelize δFδµ [µ]
Section 3.2

kernelize δFδµ [µ]
Fisher-Rao

Section 3.1

Kernelized Fisher-Rao
1

The arrows denote kernelization operations of the gradient flow by the operator Kρ2 , asin Definition 3.1. Analogously, for the Wasserstein setting, we first continue the work of
Duncan et al. (2019) on Stein geometry, of which we briefly provide the gradient structurein Section 4.1. We then establish the kernelization and de-kernelization relation in the
diagram below.

De-Steinde-kernelize ∇ δF
δµ [µ]
Section 4.2

Wassersteinkernelize ∇ δF
δµ [µ]
Section 4.1

Stein

For example, we de-kernelize the Wasserstein geometry to obtain the De-Stein distance,
which results in a flat distance in the form of a weak norm
Z

1
2
2
De-Stein (µ, ν) = supζ d(µ − ν) − ∥∇ζ∥H ,4
ζwhich is the transport analog of the MMD.
Motivated by the approximation of velocity fields in generative models via score-matching (1.8)
and flow-based models (1.9), we investigate the gradient flow structure that generates theR
1. One may replace the dynamics by µ̇ = −µ · ξt + µ · µξt for a flow over probability measures, i.e.,
spherical Hellinger (Laschos and Mielke, 2019), instead of non-negative measures. We do not consider
this flow in this paper mainly due to technicality and additional steps in the analysis. However, manyof our results generalize directly to the spherical version.

5

Zhu and Mielke

reaction equation whose growth field is obtained by nonparametric regression, e.g., thekernel ridge regression,µ̇t = −µt · rt ,



2
δF
2
rt = argmin f −+ λ∥f ∥F .
[µt ]
δµ
f ∈F
L2µ

(1.12)tNote that the nonparametric regression can be cast in the more general maxium likelihood
estimation (MLE) form (6.26). We refer to the (gradient) system that generates the equation
above as an approximate Fisher-Rao (gradient) system.
The analog in the Wasserstein setting has been studied under the name of (regularized)
Stein gradient flow by Duncan et al. (2019); He et al. (2022). Combining the Fisher-Rao
and Wasserstein settings, we find the approximate Wasserstein-Fisher-Rao flowµ̇t = div(µt · vt ) − µt · rt ,



2
δF
2
rt = argmin f −+ λ∥f ∥F ,
[µt ]
δµ
f ∈F
L2µt


δF
2
2
[µt ] ∥L2µ + λ∥f ∥F .
vt = argmin ∥f − ∇
t
δµ
f ∈F

The connection between nonparametric regression and gradient flows can be seen in theHelmholtz-Rayleigh Principle (Rayleigh, 1873); see Section 6.3. The intuition is that the
principle ofmin energy + dissipation potential
is equivalent to the nonparametric regression formulation (1.12) agnostic of the dissipationgeometry, i.e., in both the Fisher-Rao and Wasserstein settings. This implies different
training objectives, such as matching the score function or the log density, can be unified
using the same formalism of the Rayleigh Principle.
Historically, the entropy dissipation method is used in proving the Bakry-Émery theorem, e.g., in (Arnold et al., 2001). It gives rise to a sufficient condition closely related to
the Polyak-Lojasiewicz inequality in optimization∗R(µ, µ̇) + R (µ, −DF ) ≥ c · F (µ(t)) − inf F (µ) ,(1.13)µ

where the dissipation potential terms R, R∗ must be adapted to the specific gradient-flowgeometry. Beyond standard KL-dissipation in the Wasserstein geometry, researchers havealso studied specialized versions of the log-Sobolev inequality (LSI), e.g., the Stein-logSobolev inequality (Duncan et al., 2019, Lemma 35). In Table 1, we provide generalized
Lojasiewicz inequalities specialized to the various geometries studied in this paper. In
addition, one immediately obtains other functional inequalities, such as the LSI-type by
setting F (µ) = DKL (µ|π) in Wasserstein type geometries.
A commonly used energy functional in applications is the φ-divergences (Csiszár, 1967)
Zdµ
Dφ (µ|ν) := φ(σ(x)) dν, σ =.
(1.14)dν
6

Approximation and Kernelization of Gradient Flows

Gradient system

Lojasiewicz-type inequality (c > 0)

Kernelized FR2∥Kµ2 δFδµ [µ]∥L2 ≥ c · (F (µ(t)) − inf µ F (µ))

Kernel-approx. FR

2∥(Kµ + λ Id)− 2 Kµ δFδµ [µ]∥L2 ≥ c · (F (µ(t)) − inf µ F (µ))

MMD

2
∥ δF
δµ [µ]∥H ≥ c · (F (µ(t)) − inf µ F (µ))

Stein

2
∥Kµ2 ∇ δF
δµ [µ]∥L2 ≥ c · (F (µ(t)) − inf µ F (µ))

1

µ

1

µ

1

µ

1
2

+ λ Id)−1 K

De-SteinδF2∥(Kµµ ∇ δµ [µ]∥L2 ≥ c · (F (µ(t)) − inf µ F (µ))µ
2 + ∥∇ δF [µ] ∥2 ≥ c · (F (µ(t)) − inf F (µ))
∥ δF
[µ]
∥
µ
2
2
δµ
δµ
Lµ
Lµ
δF
2
∥∇ δµ [µ]∥H ≥ c · (F (µ(t)) − inf µ F (µ))

Kernelized WFR

δF
2
2
2
∥Kµ2 δFδµ [µ]∥L2 + ∥Kµ ∇ δµ [µ]∥L2 ≥ c · (F (µ(t)) − inf µ F (µ))

Kernel-approx. W
WFR

1

K-approx. WFR

1µ∥(Kµ + λ Id)

− 12

µ

− 21
2
2
Kµ ∇ δFKµ δF
δµ [µ]∥L2µ + ∥(Kµ + λ Id)δµ [µ]∥L2µ≥ c · (F (µ(t)) − inf µ F (µ))
Table 1: Summary of the Lojasiewicz inequalities for different gradient systems
where φ : [0, +∞) → [0, +∞] is a convex entropy generator function. We delve specificallyinto the concrete instantiations of the Lojasiewicz inequality for the following power-likeentropy generator (see e.g. (Liero et al., 2018)).
1
sp − p(s − 1) − 1 , p ∈ R \ {0, 1},p(p − 1)
φ0 (s) := s − 1 − log s, φ1 (s) := φKL = s log s − s + 1.
φp (s) :=(1.15)

Note that by this definition and our scaling of the Fisher-Rao distance (1.10), we have
 
Z
1 2
δµ
FR (µ, ν) = φ 1dν.
(1.16)
22
δν
Slightly abusing the terminology, we still refer to power-like entropy generated by φ 1 as
2
the squared FR distance. We plot the corresponding entropy generator functions in Figure 1. Alternatively, one may use Hellinger’s integral to define the α-divergence Dα (µ|ν) :=R 1+α 1−α4(1−µ 2 ν 2 ), from which one obtains the KL, reverse KL, and the Fisher-Rao as21−αspecial cases.
First, we examine a few concrete cases of the Lojasiewicz inequality for the Fisher-Rao
and Wasserstein geometry, e.g., the lack of global Lojasiewicz inequality for the Fisher-Rao
flow of KL energy, and the unconditionally satisfied Lojasiewicz inequality for the squared
Fisher-Rao energy. From the perspective of optimization, the global Lojasiewicz inequality
is arguably more nontrivial to characterize than the local version since we need to create
enough “slope” for the gradient flow to escape the initial birth from zero mass. See theillustration in Figure 4. To that end, we extract an explicit condition for global convergencein terms of a power threshold when the energy is chosen as the power-like entropy functional
φp in (1.15). See the summary in Table 2 for our current knowledge of the global Lojasiewicz
inequality in the Fisher-Rao and Wasserstein setting. Regarding the Wasserstein-FisherRao geometry, the Lojasiewicz condition of our results is significantly weaker than the7Zhu and Mielke

p=0
p=0.5
p=0.25
p=1
p=2

2.5

Value

2.0
1.5
1.0
0.5
0.0
0.0

0.2

0.4

0.6
s

0.8

1.0

1.2

Figure 1: The plot illustrates the power-like entropy generator functions φp for different
values of p. The purple curve represents p = 0 (reverse KL), the green curve
represents p = 0.25, the blue curve represents p = 0.5 (FR), the red curve represents p = 1 (KL), and the orange curve represents p = 2 (χ2 ). The large red dot
represents the equilibrium at s = 1. The plot helps visualize the behavior of thefunctions for different values of p and provides insights into their convexity andslopes.

Order p of entropy Dφp[1, 2]d−1[ d , ∞)d
[ d+2 , 21 ] ∪ (1, ∞)Gradient-flow geometry
Wasserstein (Bakry-Émery)
Wasserstein (McCann cond.)
WFR (Liero et al., 2023)(−∞, 12 ]Fisher-Rao (Corollary 6.5)(−∞, 12 ]WFR (Corollary 6.7)

Functional inequality(BE), c > 0 =⇒ L, c > 0
geod. cvx with c ≥ 0
1geod. cvx with c ≥ 1−p
1(L-FR) with c∗ =1−p1
L with c∗ ≥1−pTable 2: Summary of Lojasiewicz inequalities for power-like entropy energy functionals in
different geometries. The Bakry-Émery case for p = 1 is the well-known (LSI).

geodesic convexity condition in (Liero et al., 2023). Consequently, we have shown that the
Lojasiewicz inequality holds for p ∈ (−∞, 12 ] rather than the dimension-dependent lower
threshold by Liero et al. (2023) as in Table 2. Our perspective here also established that
1) the Fisher-Rao energy functional creates enough slope for the birth of mass from zero8

Approximation and Kernelization of Gradient Flowsin the Fisher-Rao geometry; 2) the reverse KL entropy (i.e., 0-th order power-like entropy)
creates a singularity near zero and thus a large slope for mass creation.
In the kernelized geometry, it is clear that energies that fail our threshold condition will
not satisfy the global Lojasiewicz inequality, as discussed in Lemma 6.8 and Corollary 6.9
for bounded kernels. While sufficient conditions, likely placing restrictions on the kernels,
are unclear, we uncover a few interesting kernel discrepancy functionals that are interactionenergies obtained by dissipating entropies along the kernelized FR and Stein flow. Two
prominent cases are the MMD (Gretton et al., 2012) and the KSD (Liu et al.). They areboth generated by the dissipation in kernelized geometries of the reverse KL entropy.
Finally, we concern ourselves with the quality of the approximation in our approximate
gradient flows. For example, in regression problems that appeared in generative models, e.g.,
the flow-matching problem (1.9) and the score-matching problem (1.8), does the learned
flow exist, is it a gradient flow? If so, what is the gradient structure, e.g., energy and dissipation geometry? Furthermore, in nonparametric regression (1.12), one typically bounds
the prediction error, i.e., quantities such as ∥rt − log µt ∥L2µ for a fixed time t. However, intgradient flows, we are also interested in the behavior of the system that follows the flowµ̇t = −µt · rt , e.g., its variational structure, solution existence, and convergence behavior.
For those reasons, we establish the evolutionary Γ-convergence in the kernel-approximate
Fisher-Rao geometries, i.e., we use tools from the calculus of variations and functional
analysis instead of statistical bounds. In a nutshell, we establish
Γthe system generating µ̇t = −µt · rt −→ Fisher-Rao gradient system.
Thus, we provide a rigorous justification for the quality of approximation using nonparametric regression in the Fisher-Rao flows. This also differs from the perspective of local
regression and local smoothing analyzed in the works by, e.g., Lu et al. (2023); Carrillo
et al. (2019).
Organization of the paper In Section 2, we provide background on gradient systems
and optimal transport, with a focus on dynamic formulation and geodesics. Then, we
provide background on reproducing kernel Hilbert spaces (RKHS). Those two topics are
married through our concrete investigation in the next four sections. In Section 3, 4, and 5,
we provide gradient structures for a few new and existing gradient systems of interest. They
are motivated by two types of geometries, namely, the Fisher-Rao and Wasserstein space.
We characterize their precise relations with other kenelized and approximating geometries.
Section 6 is dedicated to the analysis of evolutionary behaviors in the gradient systems
using the celebrated Polyak-Lojasiewicz inequalities. Then, we analyze the approximation
quality, by proving the evolutionary Γ-convergence of the kernel-approximate Fisher-Rao
gradient systems. Additional proofs are given in Section 8. In Section 9, we conclude thepaper and mention several future directions and implications on practical computational
and learning algorithms.
Notation We use the notation P(Ω̄), M+ (Ω̄) to denote the space of probability and nonnegative measures on the closure of a set Ω ⊂ Rd . The base space symbol Ω is often dropped
if there is no ambiguity in the context. We express the standard integral
R operator weighted
by measure ρ as a weighted convolution Kρ : L2 (ρ) → L2 (ρ), f 7→ k(x, ·)f (x) dρ(x); cf.
9Zhu and Mielke

Theorem 2.2. The measure ρ is omitted if it is the Lebesgue measure. In this paper, thefirst variation of a functional F at µ ∈ M+ is defined as a function δFδµ [µ]ZδFd
F (µ + ϵ · v)|ϵ=0 =[µ](x) dv(x)(1.17)dϵ
δµfor any perturbation in measure v such that µ + ϵ · v ∈ M+ . The Fréchet (sub-)differential
on a Banach space (X, ∥∥X ) is defined as a set in the dual space
DX F := {ξ ∈ X ∗ | F (µ) ≥ Fν + ⟨ξ, µ − ν⟩X + o (∥µ − ν∥X ) for µ → ν} ,where the small-o notation signifies that the term vanishes more rapidly than the term
inside the parentheses. We use superscripts for differential derivatives to emphasize thecorresponding space of those operations, i.e., we distinguish between DX F and DY F . For
simplicity, we carry out the Fenchel-conjugation calculation in the un-weighted L2 space.
Replacing that with duality pairing in the weighted L2ρ space does not alter the results.
Common acronyms, such as partial differential equation (PDE) and integration by parts(IBP), are used without further specifications. We often omit the time index t to lessen
the notational burden, e.g., the measure at time t, µ(t, ·), is written as µ. The infimal
convolution (inf-convolution) of two functions f, g on Banach spaces is defined as (f □g)(x) =inf y {f (y) + g(x − y)}.
R In our formal calculation, we often use measures and their density
interchangeably, i.e., f · µ means the integral w.r.t. the measure µ. This is based on thestandard rigorous generalization from flows over continuous measures to discrete measures(Ambrosio et al., 2005).

2 Preliminaries
2.1 Gradient-flow systems and geodesics
Intuitively, a gradient flow describes a dynamical system that is driven towards the fastest
dissipation of certain energy, through a geometric structure measuring dissipation. In thiswork we restrict to the case that the dissipation law is linear, which means it can be given
in terms of a (pseudo) Riemannian metric. Such a system is called a gradient system.
For example, the dynamical system described by an ordinary differential equation in the
Euclidean space u̇(t) = −∇F (u(t)), u(t) ∈ Rd is a simple gradient system.
In this paper, we take the perspective of variational modeling and principled mathematical analysis, i.e., we study the underlying dynamical systems modeled as gradient systems
specified by the underlying space X, energy functional F , and the dissipation geometry
specified by the potential R. Given a smooth state space X, a dissipation potential is afunction on the tangent bundle TX, i.e. R = R(u, u̇), where, for all u ∈ X, the functionalR(u, ·) is non-negative, convex, and satisfies R(u, 0) = 0. We denote byR∗ (u, ξ) = sup ⟨ξ, v⟩ − R(u, v) v ∈ Tu X
(2.1)
the (partial) Legendre transform of R and call it the dual dissipation potential. Throughout
this work, we will restrict to the case that R(u, ·) is quadratic, i.e.
11R(u, u̇) = ⟨G(u)u̇, u̇⟩ or equivalently R∗ (u, ξ) = ⟨ξ, K(u)ξ⟩.
22
10

Approximation and Kernelization of Gradient Flows

Definition 2.1 (Gradient system) A triple (X, F, R) is called a generalized gradient system, if X is a manifold or a subset of a Banach space, F : X → R is a differentiable function,
and R is a dissipation potential. The associated gradient-flow equation has the primal anddual form
0 = Du̇ R(u, u̇) + DF (u) ⇐⇒ u̇ = Dξ R∗ u, −DF (u) .
(2.2)
If R is quadratic, we simply call (X, F, R) a gradient system and obtain the gradient flow
equations
0 = G(u)u̇ + DF (u) ⇐⇒ u̇ = −K(u)DF (u).
G = K−1 is called the Riemannian tensor, and K = G−1 is called the Onsager operator.
Of particular interest to this paper is the gradient flow in the Fisher-Rao metric space,also called Hellinger-Kakutani or simple Hellinger space in (Liero et al., 2018; Laschos andMielke, 2019), which is the gradient system that generates the following reaction equation
as its gradient flow equation
δF∂t µ = −µ ·
[µ] ,
(2.3)
δµ
where δFδµ [µ] is the first variation (1.17). Alternatively, one can also view the whole r.h.s as
the Fisher-Rao metric gradient using the weighted tangent space L2µ .
The Fisher-Rao gradient system is a special case of general gradient flow in metricspaces Ambrosio et al. (2005), which has gained significant attention in recent machinelearning literature due to the study of Wasserstein gradient flow (WGF), originated fromthe seminal works of Otto and colleagues, e.g., Otto (1996); Jordan et al. (1998); Otto.
Rigorous characterizations of general metric gradient systems have been carried out in
PDE literature, for which we refer to Ambrosio et al. (2005) for complete treatments andSantambrogio (2015); Peletier (2014); Mielke (2023) for a first principles’ introduction. To
get a concrete intuition, the gradient structure of the following two classical PDEs will
become relevant in later discussions about Fisher-Rao and Wasserstein respectively.
Example 2.1 (Classical PDE: Allen-Cahn and Cahn-Hilliard) Recall the Allen-Cahn
PDE∂t µ = ∆µ − ∇V,(2.4)

and the Cahn-Hilliard PDE∂t µ = ∆ (−∆µ + ∇V ) .

(2.5)
R
1

They are the gradient flows of the energy functional F (µ) = 2 |∇µ|2 + V (µ) in two
different Hilbert spaces, where V is a potential function, e.g., the double-well potential
V (x) = 41 (1 − x2 )2 . Allen-Cahn is the Hilbert-space gradient flow of the energy F in unweighted L2 , i.e.,
1
1
(2.6)
RAC (µ, u̇) = ∥u̇∥2L2 , R∗AC (µ, ξ) = ∥ξ∥2L2 .
22
Cahn-Hilliard is the gradient flow of F in unweighted H −1 , i.e.,11
RCH (ρ, u̇) = ∥u̇∥2H −1 , R∗CH (ρ, ξ) = ∥∇ξ∥2L2 .
(2.7)
2
2

11

R

Zhu and Mielke

Geodesics and their Hamiltonian formulation. For many considerations of gradient
flows, the geodesic curves play an important role. These curves are obtained as minimizersof the length of all curves connecting two points:
Z 11
γu0 →u1 ∈ argmin⟨G(u(s))u̇(s), u̇(s)⟩ ds.
2
u0
In the sense of classical mechanics, one may the consider the dissipation potential R(u, u̇) =1∗
2 ⟨G(u)u̇, u̇⟩ as a “Lagrangian” L(u, u̇) = R(u, u̇) and the dual dissipation potential R (u, ξ) =1∗
2 ⟨ξ, K(u)ξ⟩ as a Hamiltonian H(u, ξ) = R (u, ξ). Then, minimizing the integral over L isequivalent to solving the Hamiltonian system()
u̇ = ∂ξ H(u, ξ) = ∂ξ R∗ (u, ξ) = K(u)ξ,, u(0) = u0 , u(1) = u1 .
(H)ξ˙ = −Du H(u, ξ) = −Du R∗ (u, ξ),Here, the conditions for u at s = 0 and s = 1 indicate that finding geodesic curves leads tosolving a two-point boundary value problem.
The theory for geodesics becomes particularly interesting in the case that R∗ is linear in
the state u, because then Du R∗ (u, ξ) no longer depends on u. This means that the system(H) decouples in the sense that the equation for ξ no longer depends on u. This particularcase occurs in the Wasserstein, Fisher-Rao, and consequently Wasserstein-Fisher-Rao space.
This structure allows for the derivation of the following characterizations of the geodesic
curves and static formulations of the associated Riemannian distances.
Example 2.2 (Wasserstein geodesics in Hamiltonian formulation) For the Wasserstein case, the dual dissipation potential takes the form 2
Z
1
1
∗
2
H(µ, ξ) = RW2 (µ, ξ) = ∥∇ξ∥L2µ =|∇ξ|2 dµ.
22
The Onsager operator is given by K(µ)ξ = − div(µ∇ξ) and the geodesic curves are characterized byµ̇ = − div (µ∇ξ) ,(Geod-W) ξ˙ = − 1 |∇ξ|2 .
2Here, the first equation is the continuity equation that implies that µ is transported along thevector field (t, x) 7→ ∇ξ(t, x), and the second equation is the Hamilton-Jacobi equation, which
is notably independent of µ. The Hopf-Lax formula then gives the explicit characterizationof the solution1
ξ(s, x) = inf ξ(0, y) + |x−y|2 ,y2syielding the celebrated dual Kantorovich formulation of the Wasserstein distance. See Ambrosio et al. (2005) for details.
2. For ease of calculation, we always consider the 12 scaling for quadratic dissipation potentials. That is,
this case corresponds to the geodesics of the 21 W22 .

12Approximation and Kernelization of Gradient Flows

Example 2.3 (Fisher-Rao (or Hellinger) geodesics in Hamiltonian formulation)
For the Fisher-Rao case in (1.11), the primal and dual dissipation potential takes the form
1 δ u̇
RFR (µ, u̇) = ∥ ∥2L2µ ,2 δµ1H(µ, ξ) = R∗FR (µ, ξ) =

2

ξ

2
=
L2µ

Z

1 2
ξ dµ.
2

(2.8)

The Onsager operator is given by K(µ)ξ = ξµ and the geodesic curves are characterized byµ̇ = −µξ,(Geod-FR) ξ˙ = − 1 |ξ|2 .
2
Different from the Hamilton-Jacobi setting, this system can be solved in the explicit form
ξ(s, x) =ξ(0, x)1+sξ(0, x)/2and

2µ(s, dx) = 1+sξ(0, x)/2 µ0 (dx),where we already used the initial condition µ(0) = µ0 . Applying the final condition µ(1) = µ1
we arrive at the explicit representation of the Fisher-Rao geodesic
√ 2
√
√
(2.9)
ω(s) = (1−s) µ0 + s µ1 = (1−s)2 µ0 + 2s(1−s) µ0 µ1 + s2 µ1 .
See (Laschos and Mielke, 2019) for details. Formally, one can also obtain a static dual
Kantorovich type formulation using the closed-form solution above
ZZ
FR2 (µ0 , µ1 ) =supψ dµ1 − ϕ dµ0 .
(2+ϕ)(2−ψ)=4

2.2 Reproducing kernel Hilbert space
We first remember some basic facts about the reproducing kernel Hilbert spaces (RKHS),
which are a class of Hilbert spaces that are widely used in approximation theory (Wendland,
2004; Cucker and Zhou, 2007) and machine learning (Steinwart and Christmann, 2008).
In this paper, we refer to a bi-variate function k : Ω × Ω → R as a symmetric positive
definite kernel
if k is symmetric and, for all n ∈ N, α1 , . . . , αn ∈ R and all x1 , . . . , xn ∈ Ω,
P P
we have ni=1 nj=1 αi αj k (xj , xi ) ≥ 0. If the inequality is strict, then k is called strictly
positive definite. Here, the space Ω can be a subset of Rd . k is a reproducing kernel if itsatisfies the reproducing property, i.e., for all x ∈ X and all functions in a Hilbert spacef ∈ H, we have f (x) = ⟨f, k(·, x)⟩H . Furthermore, the space H is an RKHS if the Diracmap δx : H 7→ R, δx (f ) := f (x) is continuous. It can be shown that there is a one-to-one
correspondence between the RKHS H and the reproducing kernel k. The following fact
regarding the RKHS, whose statement is adapted from (Steinwart and Christmann, 2008,
Theorem 4.27), is instrumental to this paper.
Theorem 2.2 (Integral operator) Suppose the kernel is square-integrable ∥k∥2L2 :=ρR
k(x, x)dρ(x) < ∞ w.r.t. a probability measure ρ. Then the inclusion from the the associated RKHS H to L2ρ , Id : H → L2ρ , is continuous. Moreover, its adjoint is the operator
13

Zhu and Mielke

Tk,ρ : L2ρ → H defined byZ
Tk,ρ g(x) :=k x, x′ g x′ dρ x′ ,g ∈ L2ρ

Tk,ρ is Hilbert-Schmidt (i.e., singular values are square-summable). The integral operator
Kρ := Id ◦Tk,ρ , L2 (ρ) → L2 (ρ)
is compact, positive, self-adjoint, and nuclear (i.e., singular values are summable).
Pα
We define the power of the integral operator Kρ as, for α > 0, Kρα := ∞i=1 λi ⟨·, ϕi ⟩L2ρ ϕi ,where λi and ϕi are the eigenvalues and eigenfunctions of Kρ given by the spectral theorem.
1

The image of the square root integral operator is the RKHS, i.e., H = Kρ2 (L2ρ ) for someprobability measure ρ. See, e.g., (Cucker and Zhou, 2007, Chapter 4). Then, for any g ∈ H,
1

∃f ∈ L2ρ such that g = Kρ2 f, ∥g∥H = ∥f ∥L2ρ . Therefore, we can conveniently write downsome formal relations between the RKHS and L2 norm useful for our later analysis−1∥g∥2H = ∥Kρ 2 g∥2L2ρ = ⟨g, Kρ−1 g⟩L2ρ ,−1⟨g, Tk,ρ f ⟩H = ⟨Id g, f ⟩L2ρ ,(2.10)

1

where Kρ 2 denotes the inverse of Kρ2 . The power of the kernel integral operator is prominently manifested in the following nonparametric regression problem.
Lemma 2.3 (Kernel ridge regression estimator) Given the target function ξ ∈ L2ρ ,the kernel ridge regression (KRR) problem for λ > 022
inf ∥g − ξ∥L2ρ + λ∥g∥H ,(2.11)gadmits the closed-form solutiong ∗ = (Kρ + λ Id)−1 Kρ ξ.

(2.12)To set the stage for our derivation later, we establish below an alternative optimization
formulation of the KRR solution.
Lemma 2.4 (Alternative optimization problem of KRR estimation) The KRR solution (2.12) coincides with the solution of the optimization problem
2
inf ⟨f − ξ, Kρ (f − ξ)⟩L2 (ρ) + λ∥f ∥L2 (ρ) .
f

One prominent applications of kernel methods to machine learning is the kernel maximum
mean discrepancy (MMD) (Gretton et al., 2012), for measuring the discrepancy between
probability measures. It is a special case of the integral probability metric (IPM) family
that is defined as a weak norm
Z
MMD(µ, ν) := supf d(µ − ν),(2.13)∥f ∥H ≤1

14

Approximation and Kernelization of Gradient Flows

where H is the RKHS associated with the kernel k. MMD is a metric on the space of
probability measures if the kernel is positive definite. Furthermore, its advatage lies in its
simple structure as a Hilbert space norm
Z Z
MMD2 (µ, ν) = ∥K (µ − ν) ∥2H =k(x, x′ ) d(µ − ν)(x) d(µ − ν)(x′ ).
(2.14)
This linear-in-measure form allows for efficient computation of the MMD via Monte Carlosampling. It can also be viewed as a form of interaction energy (Ambrosio et al., 2005) thatdissipates in gradient-flow geometry, e.g., the Wasserstein flow of the MMD energy (Arbel et al., 2019). One of our contributions is to provide a precise relation between the
MMD and the Fisher-Rao geometry from two different perspectives in Theorem 3.7 and
Proposition 6.10.

3 Fisher-Rao settingThis section addresses one of the main subjects of this paper, the Fisher-Rao-type gradient
flow geometry. We first study the kernelized Fisher-Rao geometry in Section 3.1 and provide
the gradient structure of the resulting kernelized Fisher-Rao gradient flow. Its growth field
is an approximation to that of the Fisher-Rao. In Section 3.2, we perform the inverse
operation to “de-kernelize” the Fisher-Rao geometry. Consequently, we obtain a flat (in thesense of Riemannian manifold) gradient-flow geometry, which we show is equivalent to the
MMD.
3.1 Kernelization and approximation of Fisher-Rao gradient flows
In the machine learning literature, the term kernelization has been used in many contexts.
We first make precise what kernelization entails in this paper through the following operationon the dual dissipation potentials of gradient systems.
Definition 3.1 (Kernelization of gradient systems) Given a gradient system definedby (X, F, R∗ ), where R∗ is the dual dissipation potential, we say its force-kernelization
counterpart is (X, F, R∗F −k ), where the force-kernelized dual dissipation potential is definedby1R∗F −k (u, ξ) := R(u, Ku2 ξ).

(3.1)

If the original R∗ depends on the generalized force ξ only through its gradient ∇ξ, denotedf∗ (∇ξ) = R∗ (ξ). Then, its velocity-kernelization is (X, F, R∗ ),
by R
V −k
1

f∗ (u, Ku2 ∇ξ).
R∗V −k (u, ξ) := R

(3.2)

Equivalently for the Fisher-Rao and Wasserstein type flows, we can define kernelization
using the operator Tk,u in Theorem 2.2 as a change the dissipation potentials
1
1
Fisher-Rao: R∗ (ρ, ξ) = ⟨ξ, ξ⟩L2ρ −→ R∗k (ρ, ξ) = ⟨Tk,ρ ξ, Tk,ρ ξ⟩H ,
2
2
1
1
∗
∗
Wasserstein: R (ρ, ξ) = ⟨∇ξ, ∇ξ⟩L2ρ −→ Rk (ρ, ξ) = ⟨Tk,ρ ∇ξ, Tk,ρ ∇ξ⟩H .
2215

Zhu and Mielke

This kernelization relation can also be stated using the Riemannian tensor G and the Onsager operator K; see Corollary 3.8. Note that velocity-kernelization of Wasserstein gradientflow, under the name of Stein geometry, has already been investigated, e.g., by Duncan et al.
(2019). As overviewed in Section 1, we will construct systems in the kernelization relation
illustrated in Figure 1.
Using the above definition for kernelization, we now construct a new geometry by forcekernelizing the Fisher-Rao gradient system.
Definition 3.2 (Dynamic formulation of kernelized Fisher-Rao distance) The kernelized Fisher-Rao distance is defined by the following dynamic formulation
FR2k (µ, ν) = min
µ,ξt

Z 1

∥Kµ ξt ∥2H dtµ̇ = −µKµ ξt , µ(0) = µ, µ(1) = ν .

(3.3)

0

From the Hamiltonian perspective in (H), we can derive the geodesic equation(µ̇ = −µξ,ξ˙ = −ξ · Kµ ξ.

(Geod-FR-k)t

However, is important to note that the geodesic equation above is only the necessary condition for optimality. That is, we have not proved whether its solution exists, in contrast
to the Hamiltonian system (Geod-FR). This is due to the coupling introduced by thestate-dependent integral operator Kµt . Furthermore, as a technical point, the integral operator used here is defined w.r.t a non-negative measure rather than a probability measure;see, e.g., (Conway, 1985). In the gradient structure of the kernelized Fisher-Rao gradientsystem, the corresponding primal and dual dissipation potentials are
1 δu
RFRk (ρ, u) = ∥ ∥2H ,
2 δρ

1
1
R∗FRk (ρ, ξ) = ⟨ξ, Kρ ξ⟩L2ρ = ∥Kρ ξ∥2H .
2
2

(3.4)

Therefore, the gradient-flow equation of the FRk gradient system (M+ , F, FRk ) is the reaction equation kernelized growth fieldµ̇ = −µKµ

δF
[µ] .
δµ

(3.5)

Going beyond kernelization, we derive the approximation to the original Fisher-Rao dynamics by constructing the following regularized dissipation geometry, i.e., adding the kernelized
Fisher-Rao dissipation potential RFRk (3.4) to that of the pure Fisher-Rao RFR (2.8)
Rλ-FRk (ρ, u̇) := RFR + λ · RFRk =1 δ u̇ 2
λ δ u̇ 2
1 δ u̇δ u̇
= ⟨ , Kρ−1 (Kρ +λ Id) ⟩L2ρ ,
2 +
L
H
ρ
2 δρ
2 δρ
2 δρ
δρ

1
R∗λ-FRk (ρ, ξ) = ⟨ξ, (Kρ +λ Id)−1 Kρ ξ⟩L2ρ .
2

(3.6)

Then, we obtain the following approximate gradient-flow equation with an approximate
growth field given by the kernel ridge regression solution (Kρ + λ Id)−1 Kρ δF
δµ [µ].
16

Approximation and Kernelization of Gradient Flows

Proposition 3.3 (Kernel-approximate Fisher-Rao gradient flow) The generalized gradient system (M+ , F, Rλ-FRk ) generates the gradient flow equation where the approximate
growth field r is given by the KRR solutionµ̇t = −µt · rt ,



2
δF
2
rt = argmin f −[µt ]+ λ∥f ∥H .
δµ
f
L2µ

(3.7)

t

Specifically, the closed-form solution for the growth field isµ̇ = −µ · (Kµ + λ Id)−1 Kµ

δF
[µ] .
δµ

(3.8)

Remark 3.4 (Relation between KRR and infimal convolution) The above optimization problem (3.7) is not equivalent to the infimal convolution of Rλ -FRk and R∗λ-FRk definedin (3.6). However, it is easy to check using Lemma 2.4 thatrt = argmin

g

g

1
2
+
L2µ
λ

2
Kµ1/2 (g−ξ) L2µ.

This quadratic form matches the definition of R∗λ-FRk (3.6) as an infimal convolution, i.e.,
o
n
2
2
1/2
1
Kµ (g−ξ) L2 .
R∗λ-FRk (µ, ξ) = inf g 21 g L2 + 2λµµ3.2 MMD as de-kernelized Fisher-Rao distance
We now take a different direction from the previous subsection to de-kernelize the FisherRao geometry (1.11). The result is somewhat surprising to us: the resulting gradient-flowgeometry is equivalent to the MMD geometry.
We now develop the gradient flow structure of the MMD3 . The primal and dual dissipation potentials are trivial due to the flatness of the MMD geometry
1
δµ1
RMMD (u) = ∥Kρ ∥2H = ∥Kµ∥2H ,
2
δρ
2

1
1
R∗MMD (ξ) = ⟨ξ, Kρ−1 ξ⟩L2ρ = ⟨ξ, K−1 ξ⟩L2 . (3.9)
2
2

It is important to note that the MMD dissipation potentials are state-independent, i.e., they
are not functions of the measure ρ since ρ · Kρ−1 v = K−1 v.
Proposition 3.5 (Gradient flow equation in the MMD geometry) The gradient-flow
equation in the MMD geometry is given byµ̇ = −K−1

δF
[µ] .
δµ

(3.10)

3. We follow the naming convention of the Wasserstein gradient flow and refer to the gradient flows in the
MMD dissipation geometry as the MMD gradient flow. This is distinct from the setting considered in(Arbel et al., 2019), which is a WGF with the MMD energy.

17Zhu and Mielke

Proposition 3.5 gives the intuition that the MMD gradient-flow equation is equivalent to areaction equation with the de-kernelized growth field µ̇ = −µKµ−1 δFδµ [µ]. It is worth notingdthat the gradient-flow equation can be stated in the dual space, dt Kµ = − δFδµ [µ], where Kµ
is the kernel-mean embedding (Smola et al., 2007).
We now derive the main result of this section using the dynamic formulation
MMD2 (µ, ν) = minZ 1u̇ = −K−1 ξt , u(0) = µ, u(1) = ν .

∥ξt ∥2H dt(3.11)

0

Because of its flat structure, the adjoint equation for the MMD is simply ξ˙ = 0. It is alsoeasy to verify the following lemma.
Lemma 3.6 (Unconstrained dual formulation of squared-MMD) The squared-MMD
admits the unconstrained dual representation
Z
1
MMD2 (µ, ν) = sup h d(µ − ν) − ∥h∥2H .
(3.12)4h∈H
The MMD geodesic curve is simply the straight line between µ and ν, u(t) = (1 − t)µ + tν.
Therefore, when both µ and ν are probability measures, the solution along the MMD
geodesic remains a probability measure. This is significantly simplified compared to theFisher-Rao setting as remarked in footnote 1. To be consistent with the FR setting, we also
consider the MMD between non-negative measures instead of only probability measures.
Summarizing, we present our main result regarding the MMD-Fisher-Rao relation.
Theorem 3.7 The dynamic formulation of the force-kernelized (Definition 3.1) squared
MMD (3.11) coincides with that of the squared Fisher-Rao distance (1.11).
Riemannian metric perspective Using the perspective in Section 2.1, we show anotherperspective of the kernelization of Fisher-Rao and MMD following Definition 3.1. Using thedissipation geometry (2.8), one can easily show that the Fisher-Rao Riemannian tensor (see,e.g., Chen et al. (2023)) and Onsager operator are
GFR (ν) =1·,
νKFR (ν) = ν · .

Using the RKHS-L2 relation (2.10), the state-independent counterparts for the MMD are
GMMD = K,

KMMD = K−1 .

Therefore, following Theorem 3.7, we conclude:
Corollary 3.8 (Kernelization of Fisher-Rao Riemannian tensor) The MMD and
Fisher-Rao Riemannian tensors and Onsager operators are related by the integral operator
Kν :GMMD = Kν ◦ GFR (ν),

KMMD = KFR (ν) ◦ Kν−1 .
18Approximation and Kernelization of Gradient Flows

We note that the naming convention of kernelization is consistent with the Stein-Wasserstein
relation.
Therefore, in applications such as (1.6), one considers the Fisher-Rao minimizing movement1
min F (µ) + ⟨µ − ν, GFR (ν)(µ − ν)⟩L2 ,+2τµ∈M
which is also the theoretical basis for applications such as distributionally robust optimization proposed by Ben-Tal et al. (2013). Here µ is a given non-negative measure. In suchcases, kernelization can be used to construct the MMD minimizing movement
min F (µ) +µ∈M+1⟨µ − ν, GMMD (µ − ν)⟩L2 .
2τ

This has been subsequently studied by Zhu et al. (2021) to take advantage of MMD’s manyfavorable properties for practical optimization.
3.3 Flattened Fisher-Rao, Allen-Cahn, and φ-divergences
Motivated by the relation between the MMD and the Fisher-Rao distance, we now discuss another class of divergences via an analogous construction from the dynamic formulation. This amounts to changing the state-dependent dissipation potential (??) to thestate-independent
1 δu1RFR (u) = ∥ ∥2L2ω , R∗FR (ξ) = ∥ξ∥2L2ω ,2 δω
2(3.13)

for a fixed reference measure ω. Like the MMD and the classical Allen-Cahn (i.e. L2 ), theflatten disspation potentials are state-independent. Using the dissipation potential (3.13),
We obtain the dynamic formulation
Z 1

2
2
(3.14)
FRω (µ, ν) = min∥ξt ∥L2 dtu̇ = ω · ξt , u(0) = µ, u(1) = ν .
u,ξ

0

ω

Similar to the MMD setting, the adjoint equation of the Hamiltonian dynamics simplifies toζ̇t = 0, resulting in the following static formulation in Proposition 3.9. The proof is omitted
since it is a slight modification of the proof in (Otto and Villani, 2000, Section 3) thatof the de-kernelized Wasserstein distance we show later. Similar to the linearized optimaltransport (Wang et al., 2013) and generalized geodesics (Ambrosio et al., 2005), it is natural
to consider a reference measure ω along the Fisher-Rao geodesic between µ and ν. By doing
so, we recover the following connections with the φ-divergences.
Proposition 3.9 (Static formulation of flattened Fisher-Rao distance) The flattened
Fisher-Rao distance (3.14) is equivalent to the static formulation

Z
1
2
2
FRω (µ, ν) = supζ d(µ − ν) − ∥ζ∥L2ω .
(3.15)
4
ζIf the reference measure is the Lebesgue measure ω = Λ, then the flattened Fisher-Rao
2
distance coincides with the L2 norm FRω (µ, ν) = ∥µ − ν∥2L2 . The resulting graient flow is
the L2 Hilbert space gradient flow (classical Allen-Cahn, Example 2.1).
19

Zhu and Mielke

Furthermore, Suppose the reference measure ω is chosen along the
Fisher-Rao geodesic√ 2√
between µ and ν given in (2.9), i.e., ω(s) = (1 − s) µ + s ν , s ∈ [0, 1]. Then,
2

FRω(s) (µ, ν) conincides with,2
1. if s = 0, the χ2 -divergence Dχ2 (µ|ν) = ∥ dµdν − 1∥L2 ;νdν
2. if s = 1, the reverse χ2 -divergence Dχ2 (ν|µ) = ∥ dµ
− 1∥2L2 ;µ√√3. if s = 12 , the squared Fisher-Rao (Hellinger) distance itself FR2 (µ, ν) = 4∥ µ − ν∥2L2 .

Remark 3.10 (Fisher-Rao geodesic and flatness) The third case demonstrates a particular nice property of the Fisher-Rao geometry. The Fisher-Rao geometry is not flat andpossesses a geodesic structure (Geod-FR). Yet its geodesic distance can be computed by aflat distance FRω( 1 ) (µ, ν) characterized in Proposition 3.9.
2

4 Wasserstein setting
We now apply some of our results in the Fisher-Rao type gradient flows from the previous
section to the Wasserstein type flows, exploiting the similarity in their dissipation geometry.
First, we revisit the Stein gradient flow in Section 4.1, where we establish the gradient flow
structure and dissipation potentials for Stein and its regularized version. In Section 4.2,
we derive a new Wasserstein-type gradient-flow geometry by drawing the parallel to therelation between Fisher-Rao and MMD. We further show a related Cahn-Hilliard dissipationgeometry.
4.1 Gradient structure for the (regularized) Stein gradient flow
The Stein geometry (Liu and Wang, 2019; Duncan et al., 2019) has been studied in thecontext of sampling for statistical inference. We first write down explicitly the gradientstructure for the Stein gradient system by providing the dissipation potentials for the Steingeometry, implied in the dynamic formulation of (Duncan et al., 2019)1
RStein (ρ, u) = ∥u∥2Stein,ρ ,2∥u∥2Stein,ρ := inf ∥Kρ v∥2H : u = − div(ρ · Kρ v) .

(4.1)

We refer to the ∥u∥Stein,ρ as the primal Stein norm dual dissipation potential. By Fenchelduality, we find the velocity-kernelized dual dissipation potential of the Stein gradient flow
1
1∗
RStein(ρ, ξ) = ⟨∇ξ, Kρ ∇ξ⟩L2ρ = ∥Kρ ∇ξ∥2H .
2
2

(4.2)

Using this gradient structure, we obtain Stein (variational) gradient-flow equation∂t µ = div(µ · Kµ ∇

δF
[µ]).
δµ

(4.3)

Regularized Stein gradient flow as approximation to Wasserstein gradient flow
Following a similar route as in the kernelized Fisher-Rao setting, we consider the regularized20

Approximation and Kernelization of Gradient Flows

primal and dual dissipation potential
1
λRλ-Stein (ρ, u) := RW2 (ρ, u) + λRStein (ρ, u) = ∥u∥2H −1 (ρ) . + ∥u∥2Stein,ρ for u = div(ρv)
2
2
 1
1
−1
−1
⟨v, v⟩L2ρ + ⟨v, Kρ v⟩L2ρ = ⟨v, Kρ (Kρ + λ Id)v⟩L2ρ ,
(4.4)
=
2
2
1
R∗λ-Stein (ρ, ξ) = ⟨∇ξ, (Kρ + λ Id)−1 Kρ ∇ξ⟩L2ρ ,(4.5)
2
resulting in the following approximate Wasserstein gradient system.
Proposition 4.1 (Kernel-approximate Wasserstein gradient flow with KRR velocity field)
The generalized gradient system (P, F, λ-Stein) generates the gradient flow equation where
the approximate growth field r is given by the KRR solution


δF
2
2
µ̇t = div(µt · vt ), vt = argmin ∥f − ∇[µt ] ∥L2µ + λ∥f ∥H .
(4.6)
t
δµ
f
Specifically, the closed-form solution for the velocity field is
vt = (Kµ + λ Id)−1 Kµ ∇

δF
[µ] .
δµ

(4.7)

It is worth noting that the above approximation fits the setting of learning the diffusion
model (1.7), where the velocity is approximated using KRR. Later, we rigorously justify
this approximation.
4.2 De-Stein: de-kernelized Wasserstein geometry
As we have witnessed in Section 3.2, the MMD bears the intuition of the de-kernelized andflat Fisher-Rao distance. What then is the de-kernelized (flat) Wasserstein geometry, as
MMD is to Fisher-Rao? Our starting point is the following de-kernelized H −1 -type norm∥u∥2De-Stein := inf ∥v∥2H : u = − div(ρ · Kρ−1 v) = inf ∥v∥2H : u = − div(· K−1 v) .
Similar to the gradient structure of MMD, this quantity no longer depends on the measure
ρ. As a consequence of the above formulation, the primal and dual dissipation potential forthe approximation system are state-independent
1
1
1
1
∗
RDe-Stein (u) = ∥u∥2De-Stein , RDe-Stein(ξ) = ∥K− 2 ∇ξ∥2L2 = ∥∇ξ∥2H .
2
2
2
Therefore, like the MMD, this geometry is a flat geometry and its gradient flow equation is∂t µ = − div( K−1 ∇

δF
[µ]).
δµ

We can now write down the dynamic formulation of a new distance, which we term thede-kernelized Wasserstein (De-Stein) distance4 .
Z 1

−1
2
2
De-Stein (µ, ν) = inf∥∇ξt ∥H dt ∂t u = − div(K ∇ξt ), u(0) = µ, u(1) = ν . (4.8)
0

4. Due to the presence of static dual representation in Proposition 4.2, this distance should be more appropriately termed kernel-Sobolev distance. However, similar terms have already been used in the machinelearning literature to denote a heuristically regularized H −1 geometry.

21

Zhu and Mielke

The Hamiltonian formulation for De-Stein is(µ̇ = − div(K−1 ∇ξ),ξ˙ = 0.
where the Hamilton-Jacobi equation in the Wasserstein geometry is replaced by the “static”
adjoint variable ξ due to the state-independent dissipation potential (4.8). More concretely,
following the derivation of the relation between the static Kantorovich dual formulationand dynamic Benamou–Brenier formulation (see, e.g., (Otto and Villani, 2000)), we now
derive the static defition of the metric. Different from the Wasserstein and Stein setting, weonly need one static (i.e., time-independent) test function, because of the adjoint (geodesic)
equation in the Hamiltonian dynamics ∂t ζ = 0. Consequently, we obtain a simple static
formulation as a weak norm, similar to an IPM.
Proposition 4.2 (Static dual formulation of the De-Stein distance) The dynamic formulation of the De-Stein distance (4.8) is equivalent to the static dual formulation
Z

1
2
2
De-Stein (µ, ν) = supζ d(µ − ν) − ∥∇ζ∥H .
(4.9)
4
ζ

The intuition is self-evident — compared with the static dual of MMD, the regularization by
the RKHS norm ∥ζ∥H is replaced by the RKHS norm of its gradient.
The Euler-Lagrangeequation of the optimization problem in (4.9) is 12 div −K−1 ∇ζ = µ − ν. Plugging it backinto (4.9) and integrating by parts, we find
De-Stein2 (µ, ν) = inf ∥v∥2H s. t. div −K−1 v = 2(µ − ν) ,v

which is simply a kernel-weighted H −1 norm.
4.3 Flattened Wasserstein, Cahn-Hilliard, and Sobolev discrepancy
Mirroring the development of the flattened Fisher-Rao setting in Section 3.3, we now focus
on the a similar construction in the Wasserstein setting. Similar to the De-Stein setting,
we obtain the state-independent dissipation potentials
1
RW 2 (u) = ∥u∥2H −1 ,
ω
2

1
∗
RW
(ξ) = ∥∇ξ∥2L2ω ,2
2

where the reference measure ω is fixed. Its dynamic formulation is given by
Z 1

2
2
W ω (µ, ν) = min∥∇ξt ∥L2ω dt ∂t u = − div(ω · ∇ξt ), u(0) = µ, u(1) = ν .

(4.10)

0

The adjoint equation ∂ζt = 0 implies the static dual formulation
Z

1
2
2
W ω (µ, ν) = supζ d(µ − ν) − ∥∇ζ∥L2ω .
4
ζ
22(4.11)

Approximation and Kernelization of Gradient Flows

Similar to the setting after Proposition 4.2, we find static dual formulation is equivalent to


1
2
2
W ω (µ, ν) = inf ∥ξ∥L2ω s. t. div(ξ · ω) = µ − ν ,ξ2which concincides with the weighted Hω−1 norm. If the reference measure ω is chosen as theLebesgue measure, then 21 ∆ζ = (µ − ν). Consequently, we obtain the static formulationZ2(4.12)W (µ, ν) = ∥µ − ν∥2H −1 = − (µ − ν) ∆−1 (µ − ν) dx,
which is equivalent to the classical Cahn-Hilliard H −1 Hilbert space in Example 2.1. If thereference measure ω is chosen as ω = ν, the flattened Wasserstein distance is the Hν−1 norm,
which is equivalent to the Sobolev discrepancy proposed by Mroueh et al. (2019).

5 Wasserstein-Fisher-Rao settingThis section applies our framework to the Wasserstein-Fisher-Rao gradient flow, also referred to as Hellinger-Kantorovich by Liero et al. (2018) for a better accounting of thehistorical developments. We first recall an elementary fact regarding duality of the infconvolution of functionals.
Lemma 5.1 (Dissipation potential with inf-convolution) Suppose the effective primal dissipation potential R is given by the inf-convolution of two dissipation potentials
R1 , R2 , i.e., R(µ, ·) = R1 (µ, ·)□R2 (µ, ·), where □ denotes inf-convolution. The resulting
gradient-flow equation generated by the gradient system (X, F, R) is given byµ̇ = ∂R∗1 (µ, −DX F ) + ∂R∗2 (µ, −DX F ),(5.1)

Note that the two differentials DX F in (5.1) must be taken w.r.t. the same space X.
Example 5.1 (Hellinger-Kantorovich) In the setting of the Wasserstein-Fisher-Rao(Hellinger-Kantorovich) distance (Liero et al., 2018; Chizat et al., 2019), the gradient-flow
equation (5.1) corresponds to the reaction-diffusion PDEµ̇ = R∗W2 (µ, −

δF
δF
δF
δF
[µ]) + ∂R∗DHellinger (µ, −[µ]) = div(µ∇[µ]) − µ
[µ] .
δµ
δµ
δµ
δµ

(5.2)

5.1 Kernelization and approximation of the WFR gradient flow
To produce a Stein-type geometry for the WFR distance, we consider the primal and dual
dissipation potentials as in Lemma 5.1, where two dissipation potentials are obtained fromthe Stein and kernelized Fisher-Rao
RK-WFR = RStein □RFRk ,R∗K-WFR = RStein + RFRk .

Our starting point is therefore the kernelized reaction-diffusion equation


δF
δF
µ̇ − div µ · Kµ ∇[µ] = −µ · Sµ
[µ] .
δµ
δµ
23

(5.3)

Zhu and Mielke

where Sµ is the integral operator associated with the another kernel s(·, ·) that may bedifferent from k. We find the following dynamic formulation of the kernelized WassersteinFisher-Rao distance with the kernerlized reaction-diffusion equation
Z 1
k-WFR (µ, ν) = min∥Kut ∇ξt ∥2H + ∥Sut ζt ∥2H dt
20u̇t − div (ut · Kut ∇ξt ) = −ut · Sut ζt , u(0) = µ, u(1) = ν . (5.4)
Going beyond kernelization, we now construct the kernel-approximate WFR geometryby considering both inf-convolution and additive regularization
RRK-WFR (ρ, ·) := Rλ-Stein (ρ, ·)□Rλ-FRk (ρ, ·),(5.5)

R∗RK-WFR (ρ, ·) = R∗λ-Stein (ρ, ·) + R∗λ-FRk (ρ, ·),(5.6)

where Rλ-Stein (ρ, ·) is defined in (4.4) and Rλ-FRk (ρ, ·) is defined in (3.6). We summarize
the result below by performing the calculation using the inf-convolution rules for boththe kernelized version, i.e., Stein-type metric for the Wasserstein-Fisher-Rao case, and theregularized kernel-Wasserstein-Fisher-Rao gradient flow.
Corollary 5.2 (Kernel-approximate WFR gradient flow) The generalized gradient system (M+ , F, WFR) generates the reaction-diffusion equation where the velocity and growthfield v, r are given by the nonparametric regression solutionsµ̇t = div(µt · vt ) − µt · rt ,




2
δF
δF
2
2
2
rt = argmin f −[µt ]+ λ∥f ∥H , vt = argmin ∥f − ∇[µt ] ∥L2µ + λ∥f ∥H .
t
δµ
δµ
f
f
L2µ
t

(5.7)
5.2 De-kernelized Wasserstein-Fisher-Rao and Kernel Sobolev-Fisher
Similar to the De-Stein and MMD, we now consider the de-kernelized Wasserstein-FisherRao (D-WFR) distance by de-kernelizing the reaction-diffusion equation (5.2)


δF
−1 δF
[µ] = −µ · Kµ−1[µ] .
µ̇ − div µ · Kµ ∇
δµ
δµ

(5.8)

We find the dynamic formulation and the regularized version
2

Z 1

D-WFR (µ, ν) = inf∥∇ξt ∥2H + ∥ζt ∥2H dt

0u̇t − div

ut · Ku−1∇ξtt

The important insight here is:24= −ut · Su−1
ζt , u(0) = µ, u(1) = νt, (5.9)Approximation and Kernelization of Gradient Flows

Proposition 5.3 (De-kernelized Wasserstein-Fisher-Rao) The De-kernelized WassersteinFisher-Rao distance (5.9) coincides with the inf-convolution of the MMD and the De-Stein
distance, i.e., D-WFR2 (µ, ν) := inf π∈M+ De-Stein2 (µ, π)□ MMD2 (π, ν).
Furthermore, it admits the static dual formulation
Z

1
1
2
2
2
D-WFR (µ, ν) = supζ d(µ − ν) − ∥∇ζ∥H − ∥ζ∥H .
(5.10)
44ζCompared with the MMD (3.12), the test function in the de-kernelized WFR is additioanlly
reguarlized by the RKHS norm of the gradient, ∥∇ζ∥2H .
5.3 Flattened Wasserstein-Fisher-Rao and kernel-Sobolev-Fisher
Using the WFR-type inf-convolution with the flattened Fisher-Rao (Section 3.3) and theflattened Wasserstein (Section 4.3), we obtain the following flattened Wasserstein-FisherRao formulation
Z 1

WFR2ω (µ, ν) := min0∥∇ξt ∥2L2ω + ∥ζt ∥2L2 dt1ω2u̇t − div (ω1 ∇ξt ) = −ω2 ζt , u(0) = µ, u(1) = ν , (5.11)
The flattened geometry is the inf-convotion of the weighted L2ω and Hω−1 norms, with thestatic formulation

Z
1
1
2
2
2
WFRω (µ, ν) = supζ d(µ − ν) − ∥∇ζ∥L2ω − ∥ζ∥L2ω .
(5.12)
44
ζIn the simplest case that ω1 , ω2 are both the Lebesgue measure, we recover the inf-convolutionof the Allen-Cahn (L2 ) and Cahn-Hilliard (H −1 ). Choosing ω1 = ω2 = ν, this distance becomes the Sobolev-Fisher discrepancy (Mroueh et al., 2019). Furthermore, those authors
proposed a regularized version with a heuristic RKHS norm regularization, of which we givethe precise characterization.
Example 5.2 (Regularized Kernel-Sobolev-Fisher discrepancy) The regularized kernel Sobolev-Fisher discrepancy proposed by Mroueh and Rigotti (2020)
Z

1
1
a
2
2
2
2
KSF (µ, ν) = supζ d(µ − ν) − ∥∇ζ∥L2ν − ∥ζ∥L2ν − ∥ζ∥H ,(5.13)
442
ζ
for a > 0, is equivalent to the following dynamic formulation
Z 1

1
2
2
2
−1
min
∥∇ξt ∥L2ν + ∥ζt ∥L2ν + ∥κt ∥H dt u̇t − div (∇ξt ) = −ζt − K κt , u(0) = µ, u(1) = ν .
2a0
It is the inf-convolution of three dissipaton geometries, the MMD, Hν−1 norm, and L2ν norm.
25

Zhu and Mielke

6 Analysis
6.1 Entropy dissipation and functional inequalities
In machine learning applications, functional inequalities are building blocks for analysis
of many algorithms, such as sampling and optimization via particle schemes. The aim of
this section is to develop an intuition for the Lojasiewicz type inequalities for the various
gradient-flow geometries studied in this paper.
Historically, the celebrated Bakry-Émery theorem (Bakry and Émery, 1985) gives asufficient condition for the logarithmic Sobolev inequality (LSI) to hold along the solutionof the Fokker-Planck equations: the target probability measure π satisfies the Bakry-Émery
condition if π ∝ exp (−V ) for the potential function V that satisfies∇2 V ≥ C · Id, C > 0.

(BE)

Our starting point here is the differential energy dissipation balance (EDB) relation ofgeneralized gradient flow systems,d∗
F (µ(t)) = ⟨DF, µ̇⟩ = − R(µ, µ̇) + R (µ, −DF ) =: −I(µ(t)).
(6.1)
dt
where the R, R∗ quantities are the dissipation potentials discussed in the previous sections.
We refer to the quantity I as the energy dissipation. From this, we impose the following
version of the Lojasiewicz condition.
Definition 6.1 (Lojasiewicz inequality for generalized gradient systems) We saythat the Lojasiewicz inequality holds ifα∗R(µ, µ̇) + R (µ, −DF ) ≥ c · F (µ(t)) − inf F (µ) .
(L)µholds for some c > 0, α > 0.
For conciseness, this paper only focuses on the case of c > 0, α = 1, i.e., the PolyakLojasiewicz inequality due to its relevance to machine learning and optimization, simply
referred to as the Lojasiewicz inequality in the rest of the paper. We refer to articles suchas (Otto and Villani, 2000; Blanchet and Bolte) for a wider scope of related inequalities.
An immediate consequence of (L) is that the energy of the generalized gradient system
converges exponentially via Grönwall’s lemma, i.e.,−c·t
F (µ(t)) − inf F (µ) ≤ e
F (µ(0)) − inf F (µ) .
µµTherefore, on the formal level, the intuition of the analysis is to produce the Lojasiewicz
type relations in the form ofI ≥ c · (Ft − F ∗ ) .

(6.2)

Concretely, in the Wasserstein gradient flows and the Fokker-Planck PDEs, entropy
dissipation can be easily calculatedZ2dδF(along WGF)
W
IF (µ(t)) = − F (µ(t))
=
µ∇
[µ] .
(6.3)
dt
δµ
26

Approximation and Kernelization of Gradient Flows

Carrying out the similar derivation, we find the entropy dissipation for the Fisher-Rao
gradient flow
IFFR (µ(t)) = −d(along FRGF)
F (µ(t))
=
dt

Z
µ

2
δF
[µ] .
δµ

(6.4)

As a toy example, we now formally check the inequality (L) for the the Wasserstein gradientsystemwith the KL entropy energy function (P(Ω̄), DKL (·∥π), W22 ), where DKL (µ|π) =R
dµlog( dπ )µ. We calculate the entropy dissipation− I(µ(t)) =dµµ2
DKL (µ|π) = ⟨DL DKL (µ|π), µ̇⟩L2 = ⟨log , − div(µ∇ log )⟩L2dt
ππµ(IBP )= −∥∇ log ∥2L2µ . (6.5)π

Specializing the Lojasiewicz inequality (L) and form (6.2), we find the LSI for some c > 0∥∇ logµ(t) 2∥ 2≥ c · DKL (µ(t)∥π).
π L (µ(t))(LSI)

By Grönwall’s lemma, the entropy decays exponentially DKL (µ(t)∥π) ≤ e−c·t DKL (µ(0)∥π).
As discussed previously, the Bakry-Émery Theorem states that (BE) =⇒ (LSI) for the
Wassrstein gradient flow. However, the Lojasiewicz inequality (L), of which the (LSI) is a
special case, cannot be expected to hold globally for arbitrary geometry in general, e.g., theFisher-Rao geometry. Consider the Fisher-Rao gradient flow with the KL entropy energy,i.e., F (µ) = DKL (µ|π). Then, the specialized Lojasiewicz condition reads, for some c > 0,
∥ log

dµ 2
∥ 2 ≥ c · DKL (µ(t)∥π).
dπ Lµ

(6.6)

Lemma 6.2 (No global Lojasiewicz condition in Fisher-Rao flows of KL) Thereexists no 0 < c < ∞ such that (6.6) holds along the Fisher-Rao flow of the KL-entropy, i.e.,
the gradient system (M+ , DKL (·|π), FR) does not satisfy the global Lojasiewicz condition for
any constant.
See Figure 2 and the caption for an illustration. Despite this lack of the global Lojasiewicz
condition in general, a local condition can be satisfied trivially around the equilibrium
measure µ = π. However, from this paper’s perspective, we are not interested in the local
version for the reason stated below.
Example 6.1 (Birth escaping zero: focus on global instead of local) Suppose we wish
to minimize the energy F (µ) = Dφ (µ|π) starting from the initial distribution µ0 . It is very
common that the distribution µ0 does not have the full support of the target distribution π
as in Figure 3 (left), i.e., supp(µ0 ) ⊊ supp(π). Many variational inference methods, e.g.,(Khan and Nielsen, 2018; Lambert et al., 2022; Chen et al., 2023) use Gaussian densities
to approximate the target distribution. In such cases, the distributions share the support
as in Figure 3 (right), i.e., supp(µ0 ) = supp(π), but the density ratio is nearly zero at27

Zhu and Mielke

Valueslogs s + 1
s(logs)2

0.0

0.2

0.4

0.6

s

0.8

1.0

1.2

1.4

Figure 2: The plot illustrates the lack of global Lojasiewicz inequality as in Lemma 6.2. We
plot the KL-entropy generator function φ(s) = s log s − s + 1. The blue dottedcurve represents the KL-entropy generator φ(s). The function s| log s|2 is plottedin solid black. The Lojasiewicz inequality condition is satisfied locally around theequilibrium s = 1 (red dot). However, it can never be satisfied in a neighborhood
around s = 0.

many points. For example, Figure 3 (right) depicts a Gaussian mixture distribution as theinitial µ0 that has very little mass near x = 2. This can be quite likely in practice withhigh-dimensional data. The most difficult part of the minimization is to escape the nearzero region with enough slopes provided by the energy. For example, the reaction dynamicsµ̇ = −µ δFδµ [µ] implies that a significant growth field is needed to escape when µ is nearzero, i.e., the birth process. Our theory precisely characterizes this escape threshold via the
global Lojasiewicz condition, e.g., in Corollary 6.5. In contrast, the local convergence behavior near the equilibrium is much easier to capture, see Figure 2, Figure 4. Therefore, weplace our current scope on the global Lojasiewicz condition without delving into the detailed
equilibrium behavior.

While Lemma 6.2 shows that the Fisher-Rao flow of the KL-entropy cannot satisfy the
global Lojasiewicz, we now show a positive result for the case when the energy functional
is a nicer one: the squared Fisher-Rao distance F (µ) = 12 FR2 (µ, π). First, note that the
s
δ 1 2
dπ
′
first variation of the squared Fisher-Rao distance is
FR (µ, π) = f (µ) = 2 − 2
.
δµ 2
dµ
28

Approximation and Kernelization of Gradient Flows

4.0

0.5

0

0

3.5
3.0
2.5

0.3

DensityDensity / Probability Mass

0.4

2.0
1.5

0.2

1.0
0.1

0.0

0.5
0.0
4

2

0
x

2

4

4

2

0
x

2

4

Figure 3: Illustration of Example 6.1: birth escaping zero.
Specializing the Lojasiewicz inequality to this setting,sdπ 2
1∥ 2 ≥ c · FR2 (µ|π).
4 · ∥1 −
dµ Lµ
2

(6.7)

It can be easily checked by definition that we have the unconditional satisfaction of the
global Lojasiewicz inequality in this case.
Lemma 6.3 (Global Lojasiewicz with Fisher-Rao energy) The Lojasiewicz inequality (6.7) holds for the Fisher-Rao gradient system (M+ , 12 FR2 (·, π), FR) globally for anyc ∈ (−∞, 2].
Going beyond the Fisher-Rao energy, we are now ready to extract some general principles.
Historically, following Bakry and Émery (1985), Arnold et al. (2001) provided an elementary
proof of the Bakry-Émery theorem, i.e., for the convex entropy function φ satisfying
2 1
φ(1) = φ′ (1) = 0, φ′′ (1) > 0 andφ′′′ (s) ≤ φ′′ (s)φ(4) (s),(6.8)
2
the Wasserstein gradient flow with the corresponding φ-divergence energy converges exponentially. That is, the following sufficient relation holds(BE) + (6.8) =⇒ Lojasiewicz for Wasserstein =⇒ exp. convergence.
The natural question is whether such relation exists for the Fisher-Rao geometry. To answer
that, we first establish the condition for global Lojasiewicz condition for a class of entropies.
Proposition 6.4 (Global Lojasiewicz for FR flow of φ-divergence energy) Given the
Fisher-Rao gradient system with φ-divergence energy, i.e., (M+ , Dφ (·∥π), FR). If φ : (0, ∞) →[0, ∞) is a convex entropy generator function satisfying
2
φ(1) = φ′ (1) = 0, φ′′ (1) > 0 and ∃ c∗ > 0 such that ∀ s > 0 : s φ′ (s) ≥ c∗ φ(s), (6.9)
then the Lojasiewicz inequality holds along the Fisher-Rao gradient flow , i.e.,
  2
dµ
′
φ
≥ c∗ Dφ (µ|π).
dπ L2µ29(L-FR)

Zhu and Mielke

Proof of Proposition 6.4. Theof the φ-divergence is given by (Ambrosio
 first-variations
dµδ
Dφ (µ|π) = φ′
. Thus, using the Fisher-Rao metric, we obtain the
et al., 2005)
δµdπddissipation relation dt
Dφ (µ|π) = −IφFR (µ) with
Z 
Z 
 2
 2
2 dµ
′ dµ
FR
′ dµ
′ dµ
dµ
=
φ
Iφ (µ) = φ dπ
=
φ
dπ
dπ
dπ dπ.
2LµX

X

Now exploiting the assumption (6.9) for estimating the integrand, we immediately obtain(L-FR).
In short, for the φ-divergence energy,(6.9) ⇐⇒ (L-FR) =⇒ exp. convergence
Because of the simple point-wise estimate in the above proof, it is also clear that condition(6.9) is necessary and sufficient for the Lojasiewicz estimate (L-FR).
Corollary 6.5 (Lojasiewicz: necessary sufficient condition for power-like entropy)
The Lojasiewicz inequality (L-FR) for the Fisher-Rao gradient system with the power-likeentropy φp (1.15) energy, (M+ , Dφp , FR), holds globally if and only if p ≤ 12 . Furthermore,
the constant is c∗ = 1/(1−p) in that case.
In practice, it is common to use entropy with p ≥ 0. In such cases, the above condition canalso be stated with p ∈ [0, 21 ] to avoid the technicality when p = −∞; see (Ambrosio et al.,2005). In particular, Corollary 6.5 shows that the energies for which the globally Lojasiewicz
estimate holds include the squared Fisher-Rao (p = 21 ), the reverse KL (p = 0), and thepower-like entropies between those. On the negative side, it states that the Lojasiewicz
estimate does not hold globally for many commonly used entropy energy functionals suchas the KL (p = 1) and χ2 (p = 2).
Remark 6.6 (Entropy power threshold p = 12 ) The relevance of the threshold p = 1/2
can be seen from two perspectives. First, we observe that µ = 0 is a steady state solution for
the gradient systems M+ , Dϕp (·|π), FR for p > 1/2. However, if µ(t) = 0 is a solution,then it cannot converge exponentially to the equilibrium measure π. The point is that the
metric slopeDφp (0) − Dφp (µ) +|∂Dφp |FR (0) = lim sup
FR(0, µ)µ→0
can be calculated explicitly and satisfies the relation
for p > 1/2,
0|∂Dφp |FR (0) = 1
for p = 1/2,

∞ for p < 1/2.
In the case p > 0 where Dφp (0) < ∞ the curve t 7→ µ(t) = 0 can still be considered asolution of the gradient-flow equation, however, the exponential decay only applies to thecurves of maximal slopes satisfyingd1
1
Dφp (µ(t) = − |µ′ |FR (t)2 − |∂Dφp |FR (µ(t))2 .
dt
2
2
30

Approximation and Kernelization of Gradient Flows5

p=0
p=0.25
p=0.5
p=1
p=2

4

Value

3

2

1

0
0.0

0.2

0.4

0.6

s

0.8

1.0

1.2

1.4

Figure 4: The plot illustrates the left-hand side Lojasiewicz inequality (L-FR) for the
Fisher-Rao geometry. The purple curve represents p = 0 (reverse KL), the green
curve represents p = 0.25, the blue curve represents p = 0.5 (FR), the red curve
represents p = 1 (KL), and the orange curve represents p = 2 (χ2 ). The red dot
represents the equilibrium at s = 1. This plot provides insights into the slopes ofthe power-like entropies in the Fisher-Rao gradient flow. We observe the threshold p = 0.5 (FR; green) where the behavior near s = 0 jumps. See the main text,
especially Remark 6.6, for analysis.

We refer to (Laschos and Mielke, 2023, Section 2) for a more detailed discussion.
A second way to see the importance of the threshold p ≤ 21 involves the results in(Otto and Villani, 2000), showing that geodesic Λ-convexity of a functional implies the
Lojasiewicz inequality with cLoj = 2Λ. For the condition of geodesic Λ-convexity for funcRdµtionals Dφ (µ|π) = X φ( dπ) dπ in the FR geometry, it can be shown thatno1w φ′′ (w) + φ′ (w) .
w≥02

Λ := inf

This gives the same result when considering the p-power family φp . But for general φ, we
may have 2Λ ≨ cLoj .
For the Wasserstein distance, the McCann condition (see, e.g., (Ambrosio et al., 2005))
shows that Dφp (·| dx) (i.e., the reference measure is Lebesgue) is geodesically convex only
for p ≥ (d−1)/d where d is the dimension. In (Liero et al., 2023), necessary and sufficientconditions for the geodesic convexity of entropy functionals with respect to the WassersteinFisher-Rao distance were derived. The upper threshold p = 1/2 was also observed in the31Zhu and Mielke

sense that densities with p ∈ [p∗ , 1/2] ∪ (1, ∞) lead to geodesically convex p-divergences,where p∗ = 1/3 for space dimension d = 1 and p∗ = 1/2 for d = 2. For d ≥ 3 only the rangep > 1 is admitted. However, only the convexity constant Λ equals 0 has been shown for allp > 1. We now provide our result on the WFR Lojasiewicz in the following corollary.
Corollary 6.7 For power-like entropy energy with p ∈ [−∞, 21 ], the global Lojasiewicz in1.
equality holds for the Wasserstein-Fisher-Rao gradient system with a constant c∗ ≥1−p
In relating those results to the aforementioned literature, we first note that geodesic convexity implies Lojasiewicz inequality but only with a non-negative constant c ≥ 0. As thedimension increases, Liero et al. (2023)’s result and the McCann condition have an increasing power threshold for the value of p. For dimension d ≥ 3, their intervals no longer overlap
with our threshold of p ≤ 12 for the global Lojasiewicz in the Fisher-Rao geometry. Yet, weare able to provide a further Lojasiewicz result that is weaker than (Liero et al., 2023)’s
geodesic convexity condition. We further compare our results with the literature in Table 2
for greater clarity. (Liu et al., 2023, Section 2.3.2) suggests that the Lojasiewicz inequality
for the Wasserstein-Fisher-Rao geometry holds whenever the Lojasiewicz inequalities for
the Wasserstein (LSI) and Fisher-Rao both hold. Improving on that, we do not require the
LSI type condition to hold for the Wasserstein dissipation at all. This is an improvementin the understanding of dissipation mechanisms in the Wasserstein-Fisher-Rao geometry.
See Corollary 6.7. In fact, a close examination of the proof of Corollary 6.7 shows that thecondition can hold for more general flows than WFR, e.g., non-gradient flows, since theWasserstein part of the dissipation only needs to be non-negative.
6.2 Kernel discrepancies and entropy dissipation in kernelized geometries
We now consider entropy dissipation in the kernelized Fisher-Rao gradient flow. First, as in
the pure Fisher-Rao geometry in Lemma 6.2, the KL-entropy energy does not have “enoughslope” near the zero.
Lemma 6.8 Suppose the kernel k is bounded, i.e., ∥k∥∞ < ∞. Then, there exists noconstant 0 < c < ∞ such that
Z
dµ
dµ
dµ
log
· Kµ logdπ ≥ c · DKL (µ(t)∥π).
(6.10)dπdπdπTherefore, there is no global Lojasiewicz inequality for the kernelized Fisher-Rao gradient
flow of the KL-entropy.
By Corollary 6.5, we findCorollary 6.9 (Necessary condition for Lojasiewicz in kernelized FR flow of entropy)
Suppose the global Lojasiewicz inequality holds for kernelized-Fisher flow of power-like entropy φp (1.15) energy. If the kernel is bounded, then p ≤ 12 .
A sufficient condition for the Lojasiewicz type results in kernelized geometries is unclear
at the moment. However, one contribution of this section is to show that the entropy
32

Approximation and Kernelization of Gradient Flows

dissipation in the kernelized gradient flows generates a class of discrepancies meaningful
for machine learning applications, in the form of interaction energy. Similar to the pure
Fisher-Rao and Wasserstein dissipation in (6.4), (6.3), we obtain the following.
Proposition 6.10 (Discrepancies via entropy dissipation in kernelized FR) The
dissipation of energy F in the kernelized Fisher-Rao gradient flow (FRk ) is an interactionenergy and a kernel discrepancy between measures:
Z Z
δF
δF
FRk
IF (µ) =[µ] (x) k(x, y)[µ] (y) dµ(x) dµ(y).
(6.11)
δµδµIf the energy is the φ-divergence energy, i.e., F (µ) = Dφ (µ|π), then, the dissipation is




Z Z
dµ
dµ
dµ
dµ
′
′
FRk
(x) · φ(x) k(x, y) (y) · φ(y) dπ(x) dπ(y). (6.12)
Iφ (µ|π) =
dπ
dπ
dπ
dπ
Specifically, for the power-like entropy (1.15), we find
!
!
p−1

p−1
Z Z 
1
dµ
dµ
FRk
Ip (µ|π) =(x)− 1 k(x, y)(y)− 1 dµ(x) dµ(y).
(p − 1)2dπ
dπ(6.13)
Specifically, we find the commonly used entropy dissipation as kernel discrepancies:
1. p = 0, the reverse KL energy, we obtain MMD2 (µ, π) as its dissipationZ Z
FRk
I0 (µ|π) =(µ(x) − π(x)) k(x, y) (µ(y) − π(y)) dx dy,(6.14)
2. p = 21 , the squared Fisher-Rao energy,
Z Z p

p
 √
p
p
√
k
I FR
(µ|π)
=
4
µ(x)
−
π(x)
k(x,
y)
µ(y)
−
π(y)
d µ(x) d µ(y),12(6.15)
3. p = 1, the KL-entropy energy,
Z Z
dµ
dµ
FRk
(x)k(x, y) log(y) dµ(x) dµ(y),I1 (µ|π) =
log
dπ
dπ

(6.16)

4. p = 2, the χ2 -divergence energy,
I2FRk (µ|π) =

Z Z 

2

2
dµ
dµ
(x) − 1 k(x, y)(y) − 1 dµ(x) dµ(y).
dπdπ(6.17)The MMD enjoys a computational advantage as it allows Monte Carlo estimation and does
not require the measures µ, π to have common support. (6.14) reveals the insight that this isdue to the reverse KL-entropy dissipation structure. Furthermore, we obtain an interesting
insight regarding optimization
33

Zhu and Mielke

Lemma 6.11 The gradient flow equation generated by
 the pure Fisher-Rao gradient systemwith the squared MMD energy M+ , MMD2 (·, π), FR conincides with that of the kernelized
Fisher-Rao gradient system with the reverse KL-entropy energy (M+ , Dφ0 (·|π), FRk ), i.e.,µ̇ = −µ · K (µ − π) .
This means that solving the optimization problem min MMD2 (µ, π) in the pure Fisher-Raoµgeometry is equivalent to min Dφ0 (µ|π) in the kernelized Fisher-Rao geometry.
µ

It is already established that the dissipation of the KL-divergence in the Stein geometry
results in the Stein-Fisher information (Duncan et al., 2019), also referred to as the squared
kernel Stein discrepancy (KSD) (Liu et al.). First, we generalize this result.
Proposition 6.12 (Kernel discrepancies via entropy dissipation in Stein) Energy dissipation in the Stein gradient flow follows
Z Z
δF
δF
Stein
IF (µ(t)) =∇[µ] (x) k(x, y) ∇[µ] (y) dµ(x) dµ(y).
(6.18)
δµδµFor the φ-divergence energy, i.e., F (µ) = Dφ (µ|π),




Z Z
dµ
dµ
dµ
dµ
′
′
Stein
(x)∇φ
(x) k(x, y) (y)∇φ(y) dπ(x) dπ(y). (6.19)
Iφ (µ|π) =
dπ
dπ
dπ
dπ
In the case of the power-like entropy (1.15),
p−1

p−1
Z Z 
dµ
dµ
dµ
dµ
Stein
Ip (µ|π) = −(x)∇ (x)k(x, y)∇ (y)(y)dπ(x) dπ(y).
dπ
dπ
dπ
dπ
(6.20)
Specifically, we find
1. p = 0, the reverse KL energy, we obtain KSD2 (π|µ), i.e., squared reverse KSD
Z Z
dµ
dµ
(x)k(x, y)∇ log(y) dπ(x) dπ(y).
(6.21)
I0Stein (µ|π) = −
∇ log
dπ
dπ
2. p = 21 , the squared Fisher-Rao energy,
r
Z Z r
dµ
dµ
Stein
I 1 (µ|π) = −4∇(x)k(x, y)∇(y) dπ(x) dπ(y).
2dπdπ(6.22)

3. p = 1, the KL-entropy energy, we obtain KSD2 (µ|π), a.k.a. the Stein-Fisher information
Z Z
dµ
dµ
Stein
I1 (µ|π) = −∇ log(x)k(x, y)∇ log(y) dµ(x) dµ(y).
(6.23)dπ
dπ
4. p = 2, the χ2 -divergence energy,
Z Z
dµ
dµ
Stein
I2 (µ|π) = −∇ (x)k(x, y)∇ (y) dµ(x) dµ(y).
dπdπ(6.24)

Furthermore, we observe a new connection between the MMD and the KSD – they are bothgenerated by the reverse KL (0-th order power entropy) dissipation in kernelized geometries.
34

Approximation and Kernelization of Gradient Flows

6.3 Explicit connection between nonparametric regression and RayleighPrinciple
In this section, we uncover a connection between the entropy dissipation in gradient flows
and the nonparametric regression
Z
22
argmin(f (x) − y(x)) dµ(x) + λ∥f ∥F ,(6.25)
f ∈F

where y is some target functions such as the generalized force δFδµ [µ] for Fisher-Rao andδFvelocity ∇ δµ [µ] for Wasserstein. This can be further unified under the general MLE formulation with negative log-likelihoodargmin NLL(f ; y, µ),(6.26)

f ∈F

where NLL is the negative-log-likelihood, e.g., 2σ1 2 ∥f − y∥2L2 + λ∥f ∥2L2 with variance σ 2 .
µµWe now give an explicit characterization that connects gradient flow dissipation geometry
with nonparametric regression. Since Fisher-Rao and Wasserstein type flows are based on
L2 type geometries, we only focus on the least-squares type losses (6.25). Our main resultin this subsection is a connection between the nonparametric regression (6.25) and theHelmholtz-Rayleigh Principle (Rayleigh, 1873).
Proposition 6.13 (Nonparametric regression as Helmholtz-Rayleigh Principle)
In approximate Fisher-Rao, Wasserstein, and Wasserstein-Fisher-Rao flows, the nonparametric regression (6.25), where the target y is the velocity field of the corresponding (pseudo)Riemannian manifold, is equivalent to the optimization problem


d
λ
1
f
2
2
argminF (µt ) + ∥f ∥L2µ + ∥f ∥F .
(6.27)
dt
2
2
f ∈F
Furhermore, solving the nonparametric regression (6.25) is equivalent to the minimization
problem in the form of the Rayleigh Principle (Rayleigh, 1873)d1
λargmin ⟨DF, µft ⟩ + ∥f ∥2L2µ + ∥f ∥2F .
f ∈F |{zdt } |2{z 2}
energydissipation potential

Therefore, the approximation of the gradient flow via nonparametric regression (6.25) isequivalent to searching for the approximate growth or velocity field f ∈ F that maximallydissipates the energy F , while regularized by its function class F, L2 norm, and optionallya regularization term λ2 ∥f ∥2F .
Remark 6.14 (Helmholtz-Rayleigh Principle) While our discussion around Proposition 6.13 is formal, it can be made mathematically rigorous in the form of the (Helmholtz)Rayleigh Principle (Rayleigh, 1873), also known as the maximum dissipation principle; see(Mielke, 2015, Proposition 5.2.1) for the rigorous statement. Our notation for the dissipation geometry R, R∗ is also due to the Rayleigh dissipation function.
35

Zhu and Mielke

As discussed in the previous subsection, the reverse KL entropy dissipation generates
the MMD. Using Proposition 6.13, we show further connection to tools in machine learningin the following examples. First, Proposition 6.13 specialized to the Wasserstein setting
gives a connection to generative models.
Example 6.2 (Wasserstein case as implicit score-matching) A standard technique forsolving a score-matching problem is the implicit score-matching (ISM) for solving regression
argmin ∥f − ∇ log
f ∈F

µ 2
∥ 2.
π Lµ

(6.28)

Apply Proposition 6.13 (with λ = 0) in the Wasserstein setting with the KL entropy energy,
we recover the ISM result (Hyvarinen; Vincent, 2011)

Z
µ
(IBP)
argmin −⟨f, ∇ log ⟩2L2µ + ∥f ∥2L2µ =
π
f ∈F



f 2 + div f + f · ∇ log π µ .

(6.29)

In practice, the estimator fˆ in (6.29) can be fitted using, e.g., deep neural networks, and usedas the velocity field to update the particle locations, i.e., performing Langevin update Xt+1 ←Xt + τ · fˆ(Xt ). This technique has also been applied with neural network approximation forsampling by Dong et al. (2022).
Going beyond Wasserstein, specializing Proposition 6.13 to the Fisher-Rao setting results in a commonly used tool in machine learning applications.
Example 6.3 (Reverse KL dissipation in approximate FR2 and two-sample test)
Let the energy functional be the reverse KL (0-th order power) entropy dissipation F (µ) =D0 (µ|π). Energy-dissipation formulation of nonparametric regression (6.27) is equivalent to
the weak-norm formulation
Z

1
λ
2
2
sup
f d (µ − π) − ∥f ∥L2µ − ∥f ∥F
2
2
f ∈F
We have already seen this type of weak-norms in the de-kernelized and flattened geometries.
Specifically, the choice F as the RKHS gives a regularized version of the MMD. Weak-norm
formulations, often termed the integral probability metric (IPM), are also commonly used inmachine learning applications such as generative models (Nowozin et al., 2016), two-sample
testing (Gretton et al., 2012), and robust learning under distributional shifts (Zhu et al.,2021).
Another implication of Proposition 6.13 is that the Stein gradient flow can not be castin the form (6.27) since kernel smoothing is not an M-estimator, i.e., it cannot be written
as the solution of an optimization problem like (6.25). However, it is possible to cast it asa local regression; cf. texts in nonparametric statistics (Spokoiny, 2016; Tsybakov, 2009).
Example 6.4 (Approximation in the Stein setting as local regression and MLE)
The particle approximation scheme for the Stein PDE (1.4) was first proposed as the SVGD
36

Approximation and Kernelization of Gradient Flows

algorithm by Liu and Wang (2019). We now cast the approximation in the Stein setting as
local regression and thus local MLE
Z

2
δF
′
′
′
′
µ̇t = div(µt · vt ), vt (x) = argminµ(x )k(x − x) θ − ∇[µt ] (x ) dx , (6.30)
δµθ∈Rdwhere the regression is now locally weighted using the shift-invariant kernel k(x′ − x). This
problem admits a closed-form solution as a kernelized velocityZδFk(x′ − x)∇
vt (x) = µ(x′ ) R[µt ] (x′ ) dx′ .
(6.31)′′′µ(x )k(x − x) dx δµGiven data samples {xi }N
i=1 , we obtain the kernel smoothing method known as the NadarayaN
1 X
δFk(xi − x)·∇[µt ] (xi ), which is the SVGD velocity
Watson estimator v̂(x) =PNNδµi=1 k(xi − x)i=1with a normalized kernel.
Example 6.5 (Nadaraya-Watson estimator in the kernelized Fisher-Rao setting)
Similar to Stein, in the kernelized Fisher-Rao setting, we have

Z
2
δF
µ̇t = −µt · rt , rt (x) = argmin[µt ] (x′ ) dx′ ,(6.32)µ(x′ )k(x′ − x) θ −δµθ∈Rdwith the closed-form solution as a kernelized growth fieldZk(x′ − x)δFrt (x) = µ(x′ ) R[µt ] (x′ ) dx′ ,′′′µ(x )k(x − x) dx δµ(6.33)NδF
1 X
k(xi − x)·[µt ] (xi ).
PN
N
δµ
k(x
−
x)
i
i=1
i=1
In particular, setting the energy as the 0-th order power entropy dissipation F (µ) =
R
′ −x)
′
′
′
D0 (µ|π) in (6.33), we obtain growth field R µ(xk(x′ )k(x′ −x) dx′ (µ(x ) − π(x )) dx . Given two
and a Nadaraya-Watson estimator r̂(x) =′ M
samples {xi }N
i=1 ∼ µ and {xi }i=1 ∼ π, A sample based estimator of the growth field is thedifference between two kernel density estimators
!
N
M
1
1 X
1 X
′
r̂(x) = PN
k(xi − x) −k(xi − x) .
·NM
i=1 k(xi − x)
i=1

i=1

6.4 Energy dissipation in kernel-approximate flows: formal arguments
To set the stage for the evolutionary Γ-convergence results in Section 6.5, we now analyze
the energy dissipatioin in the kernel-approximate gradientn flows. First, we examine thekernel-regularized Fisher-Rao setting studied in Section 3.1. For the RKHS approximationin this paper, we may use the standard approximation theory characterization, e.g., (Cucker
and Zhou, 2007, Chapter 8), which was applied to the Stein setting by He et al. (2022).
s
One difference is that we do not make the assumption that δFδµ [µ] ∈ range(Kµ ). In the37

Zhu and Mielke

δF
Wasserstein or the Fisher-Rao setting, the regularity of δFδµ [µ] or ∇ δµ [µ] is determined by
the gradient system and geodesic structure introduced in Section 2. It is not clear whether
the range condition ξ ∈ Range(Kµs ) can be satisfied in any meaningful gradient systems.
−s

Furthermore, since Kµ is a compact operator, hence the quantity ∥Kµ 2 ξ∥2L2 is unbounded.
µInstead, we rely on regularization to avoid unbounded estimate. We emphasize that thefollowing arguments are merely formal and we provide a rigorous justification in Section 6.5.
Proposition 6.15 (Energy dissipation of kernel-approximate Fisher-Rao flows) For
0 ≤ s ≤ 1, the energy dissipation satisfies
s δF
d
δF
F (µ(t)) ≤ −∥[µt ]∥2L2µ + λs ∥ (Kµ + λ Id)− 2[µt ] ∥2L2µ .
dt
δµ
δµ
|
{z
} |
{z
}

approximation error

FR dissipations

Note that the estimate ∥ (Kµ + λ Id)− 2 ξ∥2L2 is finite for any fixed λ.
µ
The implication of the above estimate can be seen in the following scenario. Suppose
the Lojasiewicz condition in the pure Fisher-Rao geoemtry holds, i.e.,


δF
2
∥
[µ] ∥L2µ ≥ c · F (µ) − inf F (µ) .
(6.34)µ
δµNote that we have already justified the global Lojasiewicz inequality holds for the power-likeentropy energy satisfying our threshold condition. By the generalized Gronwall’s lemma,
the energy decay of the approximate Fisher-Rao gradient system satisfies the following
estimate. For 0 ≤ s ≤ 1,−c·T

F (µ(T )) − inf F (µ) ≤ e
µ
|




· F (µ(0)) − inf F (µ)µ{z}
FR flow

Z T
+
|0

s δF
e−c·(T −t) · λs ∥ (Kµt + λ Id)− 2[µt ] ∥2L2µ dt . (6.35)
t
δµ
{z
}

approximation error

As mentioned, those are merely formal arguments and do not justify the asymptotic
convergence behavior. However, based on our characterization of the entropy dissipationin the pure Fisher-Rao geometry, we expect the approximation such as in Proposition 6.15
to be close. Next, we give a rigorous justification, at the rigor level of applied analysis, ofconvergence as λ → 0 for the kernel-approximate Fisher-Rao flows.
6.5 Evolutionary Γ-convergence at the approximation limit
As the regularization parameter λ → 0, techniques from approximation theory and statistics
can be used to show consistency for fixed time t point-wise. That is, the regression problems
such as (3.7) (4.6) yield the velocity vt or growth rt that converge to the target counterpart,
λ→0+e.g., vt −→ ∇ δF
δµ [µt ] for fixed t. However, this point-wise convergence does not directly38Approximation and Kernelization of Gradient Flows

imply the convergence or the existence of the approximating gradient systems. For example,
it has not been shown that the approximate system generating µ̇t = −µt ·rt in (3.7) converges
to the pure Fisher-Rao gradient system.
Different from the point-wise convergence of regression problems, we now rigorously
justify this approximation limit of the gradient systems using evolutionary Γ-convergencein this section. We focus on the kernel approximation to the Fisher-Rao gradient system
in Section 3.1; see (3.7), (3.6). Note that the rigor level of this subsection’s analysis iselevated above the rest of the paper, i.e., not merely formal arguments. A rigorous result
for the approximate Wasserstein(-Fisher-Rao) setting is beyond our current scope due tothe technicality involved.
For general curves u : [0, T ] → X, where X denotes the state space, we define thedissipation functionals Dλ as
Z T(Rλ (u(t), u̇(t)) + R∗λ (u(t), −DF (u(t)))) dt,
Dλ (u) :=0
Z T(R(u(t), u̇(t)) + R∗ (u(t), −DF (u(t)))) dt.
D0 (u) =0

The energy-dissipation principle (EDP) states that, under suitable technical assumptions(cf. (Mielke, 2023, Theorem 3.9)), u : [0, T ] → X is a solution to the gradient-flow equation(2.2) if and only if it satisfies the following energy-dissipation inequality:
F (u(t)) + Dλ (u) ≤ F (u(0)).

(6.36)

Thus, Dλ intrinsically encodes the gradient-flow dynamics.
Definition 6.16 (EDP-convergence) A sequence of gradient systems (X, F, Rλ ) is said
to converge in the sense of the energy-dissipation principle (EDP-converge) to (X, F, R),
EDP

shortly written as (X, F, Rλ ) −→ (X, F, R), if Dλ Γ-converges to D0 with bounded energies
for all T > 0, i.e.,(Γinf)

uλ → u andsupF (uλ (t)) < ∞ =⇒ lim inf Dλ (uλ ) ≥ D0 (u),λ→0+

λ>0,0≤t≤T(Γsup)∀ub ∈ L2 ([0, T ]; X) ∃ u
bλ withsup(6.37)F (buλ (t)) < ∞, :λ>0, 0≤t≤Tubλ → ub and lim sup Dλ (buλ ) ≤ D0 (bu).

(6.38)

λ→0Recall that the dissipation geometry of the regularized-approximate Fisher-Rao gradientsystem in Section 3.1.
 δu
1 δu
1
Rλ-FRk (ρ, u) = ⟨ , Id +λKρ−1⟩L2ρ , R∗λ-FRk (ρ, ξ) = ⟨ξ, (Kρ +λ Id)−1 Kρ ξ⟩L2ρ .
2 δρ
δρ
2
A useful observation is that Rλ-FRk (ρ, u) is decreasing with decreasing λ. In fact, it is evenaffine. Since the Legrendre transform is anti-monotone, Rλ-FRk (ρ, u) is increasing for λdecreasing to 0. To avoid technicalities, we do not show full EDP-convergence, but onlythe Γ-liminf estimate (6.37), this means we stay in the Γ-convergence framework of Serfaty
39

Zhu and Mielke(2011). This will be enough to conclude that solutions µλ of the regularized gradient-flow
equation of (M+ , F, Rλ-FRk ) converge to solutions µ of the pure Fisher-Rao gradient-flow
equation of (M+ , F, RFR ), see Corollary 6.18.
Theorem 6.17 (Γ-convergence of the kernel-approx. FR gradient systems)
Assume that the functional F : M+ → R satisfies the following assumptions:the λ-FRk -dissipation µ 7→ Rλ-FRk µ, δF(6.39)δµ (µ) is weakly lower semicontinuous.
Then, the dissipation functional Dλ for the regularized approximate Fisher-Rao gradientsystem (M+ , F, Rλ-FRk ) satisfies the Γ-liminf estimate (6.37).
slope
Proof. As in Serfaty (2011) we decompose Dλ = Drate
into a rate part and a slope
λ + Dλ
part:
Z T
Z T

slope
rate
Rλ-FRk (µ, µ̇) dt and Dλ (µ) =R∗λ-FRk µ, δFDλ (µ) =δµ (µ) dt.
00(A) Extraction of a converging subsequence. By standard arguments, it is sufficient to consider a family (µλ )λ>0 of curves with Dλ (µλ ) ≤ C < ∞. Using Rλ-FRk ≥ RFR we obtain
RT ′ 20 |µλ |FR (t) dt ≤ C. This implies that there exists a subsequence (not relabeled) and a limit
curve µ0 with
Z T
(6.40)|µ′0 |2FR (t) dt ≤ C and ∀ t ∈ [0, T ] : µλ (t) ⇀ µ0 (t),0where weak convergence is meant in the sense of testing with continuous functions.
1 ′ 2(B) Γ-liminf estimate for Drate
λ . For this we exploit Rλ-FRk ≥ RFR = 2 |µ |FR :rate
lim inf Drateλ (µλ ) ≥ lim inf D0 (µλ ) = lim inf
λ↓0
λ↓0
λ↓0

Z T
0

1 ′
|µ |(t)2 dt ≥
2 λ

Z T
0

1 ′
|µ |(t)2 dt = Drate0 (µ0 ).
2 0(C) Γ-liminf estimate for Dslope. To treat the slope term we use the monotonicity R∗λ-FRk ≥λR∗δ-FRk for 0 < λ ≤ δ and the weak lower semi-continuity assumed in (6.39). For fixed δ > 0
we have∗∗δFδF
lim inf R∗λ-FRk (µλ , δFδµ (µλ )) ≥ lim inf Rδ-FRk (µλ , δµ (µλ )) ≥ Rδ-FRk (µ0 , δµ (µ0 )),
λ↓0

λ↓0

where the last estimate follows because of (6.39). Integration over t ∈ [0, T ], Fatou’s lemmayields
Z T
δ↓0
slope
slope
lim inf Dλ (µλ ) ≥lim inf R∗λ-FRk (µλ , δF(µ0 ) −→ Dslope(µ0 ),0δµ (µλ )) dt ≥ Dδ
λ↓0

0

λ↓0

where the last convergence follows by the monotone convergence principle.
Hence, the desired Γ-liminf estimate for Dλ = Drate+ Dslope
is established.
λλThe above result allows us to conclude that the solution of the approximate Fisher-Rao
gradient flows converges to that of the pure Fisher-Rao.
40

Approximation and Kernelization of Gradient Flows

Corollary 6.18 (Convergence of Gradient flow solutions) Let µλ be a sequence ofsolutions to the regularized Fisher-Rao gradient system (M+ , F, Rλ-FRk ) in the sense ofenergy-dissipation balance. Assume that the assumptions of Theorem 6.17 are satisfied.
Suppose that for all t ∈ [0, T ] we have µλ (t) ⇀ µ(t) and that F (µλ (0)) → F (µ(0)) < ∞.
Then, µ : [0, T ] → (M+ , FR) is absolutely continuous and a solution to the Fisher-Rao
gradient system (M+ , F, FR).
Proof. By Theorem 6.17 we know that µ satisfies
Z T|µ̇|FR (t)2 dt < ∞ and F (µ(T )) + D0 (µ) ≤ F (µ(0)).

0The last relation follows by the EDP for µλ , namely F (µλ (T )) + Dλ (µλ ) ≤ F (µλ (0)) (see(6.36)) and the limit passage λ ↓ 0. Now exploiting the EDP for λ = 0 we see that µ is asolution for (M+ , F, RFR ).
Thus far, we have answered the question we posed earlier: the approximate flow, using a
regression formulation such as (1.12), is indeed a gradient flow that converges to the target
gradient-flow system such as the Fisher-Rao gradient flow, in the sense of evolutionary
Γ-convergence. This provides the mathematical basis for “learning” the flow for machinelearning applications.

7 Other related works
There is a well-studied line of works using regularized energies of the Wasserstein gradient
flow (Carrillo et al., 2019) and Fisher-Rao gradient flow (Lu et al., 2023). While closely
related, our work differs significantly in 1) the focus on modifying the dissipation geometry
and the gradient structure instead of the energy – our flows use the same energy as thetarget flow. 2) the focus on regression type approximation (3.7) and (4.6) instead of relying
on convolution and letting the kernel bandwidth go to zero. Point 2) is important as itlets us use straightforward replacement of the kernel machines with deep neural networks.
Furthermore, Section 6.3 shows a direct connection between our regression formulation andthe Rayleigh Principle for gradient flows. Related to kernel methods and Wasserstein flows,
Arbel et al. (2019) studied the Wasserstein gradient flow of the MMD. In comparison, we
have investigated the dynamics of generalized gradient flows (not necessarily Wasserstein)
and their dissipation geometries, which should not be confused with the WGF with different
energy objectives. There also exist works such as (Divol et al., 2022) that characterizes thestatistical error in the setting of approximating the optimal transport map using the static
formulation when computing the Wasserstein distance. Marzouk et al. (2023) also took
a nonparametric regression perspective for ODE learning. Although their results do notconcern gradient flows. Compared with works such as (Liu et al., 2023; Lu et al., 2023), ourdiscussion in Example 6.1 distinguishes our results regarding the pure FR dynamics fromtheirs. Furthermore, our analysis covers a spectrum of power-like entropies with an explicit
convergence threshold of p = 12 . Lastly, there exists a large body of works that use gradient
flows such as Langevin or birth-death dynamics to analyze neural network training, which
we do not exhaust here.
41

Zhu and Mielke

8 Further proofspp
Proof of Lemma 3.6.
R By definition, MMD (µ, ν) = ∥Kµ − Kν∥H . We introduce theauxiliary variable f = k(x, ·) dµ(x), then apply the Lagrange duality to the constrained
optimization problemZp
inf ∥f − Kν∥H s. t. f = k(x, ·) dµ(x).
f ∈HFinally, we associate the equality constraint f =h ∈ H. By the first order optimality condition,Rk(x, ·) dµ(x) with the dual variable1
2 · (f − Kν) = −h, f = Kν − h.
2

(8.1)

Hence,2Z1hh d(ν − µ) + ∥ ∥2H − ∥h∥2H
2
2
h∈H
Z
1
= sup h d(ν − µ) − ∥h∥2H .
4h∈H

MMD (µ, ν) = sup

(8.2)
(8.3)

The optimizing h∗ can be further obtained by directly solving the quadratic program.
For aesthetic reasons, we need the following lemma whose derivation is an exercise in convex
analysis.
Lemma 8.1 For a scaling parameter τ > 0, we findZ
1τ· MMD2 (µ, ν) = sup h d(µ − ν) − ∥h∥2H .
2τ
2
h∈F

Proof of Theorem 3.7.

(8.4)

We derive the force-kernelized MMD gradient flow, i.e., we1

replace the differential ξt with its kernelization Kµ2t ξt by Definition 3.1. The boundary
conditions of (3.11) are trivially equivalent to µ(0) = ν, µ(1) = µ. Following the dynamic
definition of the Stein distance, the cost of the trajectory optimization problem in the MMD
formulation 21 ∥ξt ∥2H = 21 ⟨ξt , K−1 ξt ⟩L2µ , as well as the dual dissipation potential, should be
replaced with an additional (weighted) Kµt operation by1
1
1 − 1 121
⟨K 2 Kµt ξt , K− 2 Kµ2t ξt ⟩L2 = ∥ξt ∥2L2 (µt ) .
2
2

Using this gradient structure, the MMD gradient-flow equation is obtained as
Kµ̇ = −Kµt ξt .
Since the convolutional operator K is positive definite, this gradient-flow equation is equivalent to the reaction equation in (1.11), i.e., µ̇ = −µt · ξt . Therefore, we have recovered theFisher-Rao geodesics.
42

Approximation and Kernelization of Gradient Flows

Proof of Proposition 3.9. If s = 0, then ω(0) = ν. By Fenchel-duality, the linearized
FR can be written as

Z
Z 
dµ
1
1 2
2
sup ζ d(µ − ν) − ∥ζ∥L2ν = supζ ·(dν− 1) − ζ
4
dν
4
ζ
ζ
2
Z 
dµ
=
− 1 dν = Dχ2 (µ|ν),dν
hence the equivalence to the χ2 -divergence. The case of s = 1, ω(1) = µ is similar.
√√
If s = 12 , then ω( 12 ) = 14 ( µ + ν)2 . We find
Z 
sup
ζ

√

√
√ 2
µ− ν
1 2
√
√ − ζ ·
ζ·√µ + ν dx
16
µ+ ν
√ 2
Z √
√ 2
√
µ− ν
√
√
√
µ + ν dx = 4∥ µ − ν∥2L2 = FR2 (µ, ν). (8.5)
·
= 4 √
µ+ ν

Proof of Proposition 4.2. Let ∇ζt ∈ ∇C0∞ be the test function with zero boundary
condition. Taking time derivative along the flow solution,
Z
Z
Z
Z
Z
d
(product)(dynamics)
ζt µt=∂t ζt µt + ζt ∂t µt=∂t ζt µt − ζt div(K−1 ∇ξ)
dt
Z
Z Z
1
1
(IBP)
=
∂ t ζt µt +K− 2 ∇ζt K− 2 ∇ξ dx dt. (8.6)
Completing the squares for the last term
Z
Z Z
Z
1
1
− 12
− 21
K ∇ζt K ∇ξ dx dt =dt
K− 2 ∇ζt K− 2 ∇ξ dx
 11  −1
2
− 21
2
− 12
2
− 12
2
2
−
∥K ∇ζt ∥L2 + ∥K ∇ξt ∥L2 +∥K ∇ζt ∥L2 + ∥K ∇ξt ∥L2
2
2
Z


1
=
dt −∥∇ζ − ∇ξ∥2H + ∥∇ζt ∥2H + ∥∇ξt ∥2H .
2Integrating (8.6) w.r.t. time t and rearranging the terms
Z1∥∇ξt ∥2H dt
2
Z
Z Z
Z
Z
1
1
2
= ζ1 dµ1 − ζ0 dµ0 −∂t ζt · µt dx dt −∥∇ζt ∥H dt +∥∇ζ − ∇ξ∥2H dt. (8.7)
2
2
We now consider the duality in the optimization problem of the BB-formula, i.e.,Z1
inf∥∇ξt ∥2H dt = sup inf (RHS of (8.7))
ξ,µ 2
ζ ξ,µ
43

Zhu and Mielke

R
The crucial feature of (8.7) is that the term 21 ∥∇ζt ∥2H dt is independent of the measure µt .
Therefore, in order for the infimum w.r.t. µ to be finite, we require the condition ∂t ζt ≤ 0
to hold. At optimality, we recover the adjoint equation in the Hamiltonian dynamics∂t ζ = 0,i.e., ζ is a time-independent (static) function. Hence, the optimization problem is greatly
simplified to a static setting, which is in contrast to the Wasserstein and Stein settings. We
find
Z

Z
1
1
2
2
· inf ∥∇ξt ∥H dt = supζ d(µ1 − µ0 ) − ∥∇ζ∥H
2 ξ,µ
2
ζ
Noting the scaling in the dual formulation from Lemma 8.1, we find that the transport costcoincides with the static Kantorovich dual formulation.
Proof of Lemma 6.2. This can be seen easily by calculating the KL-entropy dissipation
I(µ(t)) = −ddµdµDKL (µ(t)∥π) = ⟨log
, Kµ log
⟩ 2
dt
dπ
dπ Lµ


Z
Z
dµ
dµ
dµ
dµ
dµ 2
=
log
· Kµ logdπ ≤ ∥k∥∞ ·log
dπ. (8.8)
dπ
dπ
dπ
dπ
dπ

Similar to the pure Fisher-Rao geometry in Figure 2, this last quantity decays to zerodµ
as dπ
→ 0+ while the KL-entropy itself does not. Hence, by the analogous argument inLemma 6.2, no global Lojasiewicz condition can hold.
Proof of Corollary 6.5. According to the previous result we need to find c∗ = cp which
is given via
φp (s)
1= sup Φp (s) with Φp (s) =.
cp 0<s̸=1s(φ′p (s))2√Obeserve that φ1/2 (s) = 2( s − 1)2 implies Φ1/2 ≡ 1/2, and hence c1/2 = 2.
The derivative of the power-like entropy generator (1.15) is
φ′p (s) =11· sp−1 − 1 for p ∈ R \ {0, 1}, φ′0 (s) = 1 − , φ′1 (s) = log s.
p−1s

For general p ∈ R, explicityly calculatingΦp (s) =p − 1 sp − p(s − 1) − 1·,ps(sp−1 − 1)2

we easily verify that Φp is continuous at the s = 1 and hence continuous on (0, ∞). Moreover,
we have Φp (s) → max{0, 1−p} for s → ∞. For s → 0 we obtain Φp (s) → ∞ for p > 1/2
and Φ(s) → 0 for p < 1/2.
Thus, we conclude sup Φp = ∞ for p > 1/2. For p ≤ 1/2 a closer inspection shows thatsup Φp = 1−p. and hence cp = 1/(1−p) as stated.
44

Approximation and Kernelization of Gradient Flows

Proof of Corollary 6.7. Lojasiewicz inequality in the WFR geometry reads∥δFδF[µ] ∥2L2µ + ∥∇[µ] ∥2L2µ ≥ c · (F (µ) − F (π)) .
δµ
δµ

(8.9)

1
By our power threhold condition, for p ∈ [−∞, 12 ], the constant cFR = 1−psatisfies∥δF[µ] ∥2L2µ ≥ cFR · (F (µ) − F (π)) .
δµSince the dissipation of the Wasserstein part is always non-negative, we trivially have∥δFδF[µ] ∥2L2µ + ∥∇[µ] ∥2L2µ ≥ cFR · (F (µ) − F (π)) + 0,
δµ
δµwhich is the desired statement.
Proof of Proposition 6.10 and Proposition 6.12.
For the φ-divergence energy
dissipation in kernelized Fisher-Rao gradient flow,
 
 
dµ
dµ
d
FRk
′
′
, Kµ φ⟩ 2
Iφ (µ|π) = − Dφ (µ∥π) = −⟨φ
dt
dπ
dπ Lµ




Z Z
dµ
dµ
dµ
dµ
(IBP)
′
′
=
(x) · φ(x) k(x, y) (y) · φ(y) dπ(x) dπ(y).
dπ
dπ
dπ
dπ
For the φ-divergence energy dissipation in Stein gradient flow,
 

 
d
dµ
dµ
Stein
′
′
Iφ (µ(t)|π) = − Dφ (µ(t)∥π) = −⟨φ, div µKµ ∇φ
⟩L2
dt
dπ
dπ




Z Z
dµ
dµ
dµ
dµ
(IBP)
=
(x)∇φ′(x) k(x, y) (y)∇φ′(y) dπ(x) dπ(y).
dπ
dπ
dπ
dπ
Other calculation is straightforward.
Proof of Proposition 6.13. The proof is formal. Consider the Fisher-Rao type approximate flows µ̇ft = −µft · ft where the approximate growth field f is obtained by solving the
nonparametric regression (6.25). Here, the target of the regression is DF , the derivative ofenergy F in the respective geoemtry, e.g., in the sense of Fréchet. Then,∥f − DF ∥2L2 + λ∥f ∥2F = ∥f ∥2L2 − 2⟨f, DF ⟩L2 + ∥DF ∥2L2 + λ∥f ∥2F
µf

µf

µf

µf

d
= 2 · F (µft ) + ∥f ∥2L2 + λ∥f ∥2F + ∥DF ∥2L2 .
dtµfµf
In the Wasserstein type approximate flows µ̇t = div(µt · ft ), a similar derivation with IBP
yields∥f − ∇DF ∥2L2 + λ∥f ∥2F = ∥f ∥2L2 − 2⟨f, ∇DF ⟩L2 + ∥DF ∥2L2 + λ∥f ∥2F
µf

µf

µf

(IBP)

= 2·

µf

d
F (µft ) + ∥f ∥2L2 + λ∥f ∥2F + ∥∇DF ∥2L2 .
dt
µf
µf
45

Zhu and Mielke

Combining those two cases, we obtain the similar formulation for the Wasserstein-FisherRao type flows,∥g − DF ∥2L2

µg,h

=2·

+ ∥h − ∇DF ∥2L2µg,h+ λ(∥g∥2F + ∥h∥2F )

d
2
2
2
2
2
2
F (µg,ht ) + ∥g∥L2 g,h + ∥h∥L2 g,h + λ(∥g∥F + ∥h∥F ) + ∥DF ∥L2 f + ∥∇DF ∥L2 g,h .
dt
µ
µ
µ
µ

By setting the product f = g ⊗ h, we observe that the formal calculation in different geometries above results in the same optimization objectives on the right-hand sides independentof the dissipation geometries.
Proof of Proposition 6.15. For constant 0 ≤ s ≤ 1 and a set of orthonormal bases {ej }of L2µ ,
d
dt



δF
δF
δF
F (µ(t)) − inf F (µ) = −⟨[µ], u̇⟩L2µ = −⟨[µ] , (Kµ , +λ Id)−1 Kµ ,[µ]⟩L2µ
t
µ
δµ
δµ
δµ
2
X λ
δF
δF
δF
= −∥
[µ]∥2L2µ +· ⟨[µ] , ej ⟩ = −∥[µt ]∥2L2µ
δµσj + λ
δµ
δµ
j

+ λs

X
j

λ
σj + λ

1−s
·

⟨ δF
δµ [µt ] , ej ⟩s

(σj + λ)

2

≤ −∥

s δF
δF
[µt ]∥2L2µ + λs ∥ (Kµ + λ Id)− 2[µt ] ∥2L2µ .
δµδµ(8.10)

9 Discussion
Historically, geometries over probability measures such as optimal transport and information geometry had a tremendous impact on computational algorithms in optimization andstatistical inference. It is our hope that the new geometric structure studied in this paper
can motivate downstream applications and further investigations.
For example, in this paper, we used kernel approximation due to its simplicity andwell-established approximation theory. In modern machine learning, practitioners often
use nonlinear models such as deep neural networks and scalable approximations of kernel
machines. With our nonparametric regression formulation, it is straightforward to use thoselearning models.
In terms of generative models, we observe a direct correspondence of our nonparametric regression formulation in Proposition 6.13 to generative models such as the flowmatching (Lipman et al., 2022) and score-matching (Song et al., 2020) algorithms. The
advantage of our characterization using the Rayleigh Principle in Proposition 6.13 is thatit does not designate a specific geometry, i.e., matching of quantities such as score, velocity,
force, and growth can be expressed in the same formalism of the Rayleigh Principle.
46

Approximation and Kernelization of Gradient FlowsAs a final remark, we wish to comment on the asymptotic behavior of the functional
inequality for the approximate systems, e.g., in Table 1. Although we have provided statements such as in Proposition 6.15 with an upper estimate for a finite regularization parameter λ, we have not yet proved that there always exists a finite λ such that the flow in
the approximate geometry satisfies the functional inequality. However, motivated by the
Γ-convergence in Section 6.5, our hope is that the approximate Fisher-Rao gradient flow
with nice energy, e.g., power-like entropy with p ≤ 21 , can satisfy a Lojasiewicz inequality
for some small λ > 0. As a consequence, we expect the resulting approximate flows witha small regularization parameter λ to exhibit a similar convergence behavior as the target
flows.
Acknowledgement

We thank Anna Korba for the helpful discussion on Stein geometry.

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