# An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-
Rao Gradient Flow

Carles Domingo-Enrich<EMAIL_ADDRESS>
Courant Institute of Mathematical Sciences
New York University Aram-Alexandre Pooladian<EMAIL_ADDRESS>
Center for Data Science
New York University

###### Abstract

Let $V_{*}:\mathbb{R}^{d}\to\mathbb{R}$ be some (possibly non-convex)
potential function, and consider the probability measure $\pi\propto
e^{-V_{*}}$. When $\pi$ exhibits multiple modes, it is known that sampling
techniques based on Wasserstein gradient flows of the Kullback-Leibler (KL)
divergence (e.g. Langevin Monte Carlo) suffer poorly in the rate of
convergence, where the dynamics are unable to easily traverse between modes.
In stark contrast, the work of Lu et al. (2019; 2022) has shown that the
gradient flow of the KL with respect to the Fisher-Rao (FR) geometry exhibits
a convergence rate to $\pi$ is that independent of the potential function. In
this short note, we complement these existing results in the literature by
providing an explicit expansion of $\text{KL}(\rho_{t}^{\text{FR}}\|{\pi})$ in
terms of $e^{-t}$, where $(\rho_{t}^{\text{FR}})_{t\geq 0}$ is the FR gradient
flow of the KL divergence. In turn, we are able to provide a clean asymptotic
convergence rate, where the burn-in time is guaranteed to be finite. Our proof
is based on observing a similarity between FR gradient flows and simulated
annealing with linear scaling, and facts about cumulant generating functions.
We conclude with simple synthetic experiments that demonstrate our theoretical
findings are indeed tight. Based on our numerics, we conjecture that the
asymptotic rates of convergence for Wasserstein-Fisher-Rao gradient flows are
possibly related to this expansion in some cases.

##  1 Introduction

Sampling from a distribution with an unknown normalization constant is a
widespread task in several scientific domains. Namely, the goal is to generate
samples from a probability measure

$\displaystyle\pi(x)\propto e^{-V_{*}(x)}\,,$

where $V_{*}:\mathbb{R}^{d}\to\mathbb{R}$ is some (possibly non-convex)
potential function that is available for queries. In most cases, the target
measure $\pi$ is only known up to the normalization constant. Applications of
sampling from $\pi$ include Bayesian statistics, high-dimensional integration,
differential privacy, statistical physics and uncertainty quantification; see
Gelman et al. (1995); Robert et al. (1999); MacKay (2003); Johannes & Polson
(2010); Von Toussaint (2011); Kobyzev et al. (2020); Chewi (2022) for thorough
treatments.

Recent interest in the task of sampling stems from the following paradigm:
sampling is nothing but optimization over the space of probability measures
(Wibisono, 2018). This interpretation is due to the connection between the
celebrated work of Jordan, Kinderleher, and Otto (Jordan et al., 1998) and the
Langevin diffusion dynamics given by

$\displaystyle\,{\textnormal{d}}X_{t}=-\nabla
V_{*}(X_{t})\,{\textnormal{d}}t+\sqrt{2}\,{\textnormal{d}}B_{t}\,,$ (1)

where $\,{\textnormal{d}}B_{t}$ is Brownian motion.111This equation is to be
understood from the perspective of Itô calculus. Indeed, the work of Jordan et
al. (1998) demonstrates that the path in the space of proabability measures
given by the law of Eq. (1) is the same as the Wasserstein gradient flow (i.e.
steepest descent curve in the Wasserstein metric) of the Kullback-Leibler (KL)
divergence

$\displaystyle\text{KL}(\rho\|\pi)=\int\log\frac{\rho}{\pi}\,{\textnormal{d}}\rho\,.$

We write $(\rho_{t}^{\text{W}})_{t\geq 0}\subseteq\mathcal{P}(\mathbb{R}^{d})$
for the law of the path given by Eq. (1) (see Section 2.2.1 for a precise
definition).

A central problem in this area has been to bound the convergence rate of
$\rho_{t}^{\text{W}}$ to $\pi$ in certain similarity metrics (e.g. the KL
divergence itself, or the Wasserstein distance) under different conditions on
$\pi$. These bounds translate to convergence rates for the Langevin Monte
Carlo (LMC) sampling algorithm (Dalalyan & Tsybakov, 2012; Vempala & Wibisono,
2019; Durmus et al., 2021; Chewi et al., 2022), upon accounting for
discretization errors.

The classical result is as follows: assuming that $\pi$ satisfies a Log-
Sobolev inequality (LSI) with constant $C_{\texttt{LSI}}>0$, we obtain the
following convergence rate (Stam, 1959; Gross, 1975; Markowich & Villani,
1999)

$\displaystyle\text{KL}(\rho_{t}^{\text{W}}\|\pi)\leq\text{KL}(\rho_{0}^{\text{W}}\|\pi)e^{-\frac{2t}{C_{\texttt{LSI}}}}\,,$
(2)

which holds for all $t\geq 0$. Recall that $\pi$ satisfies a LSI if for all
smooth test functions $g$,

$\displaystyle\text{ent}_{\pi}(f^{2})\leq
2C_{\texttt{LSI}}\mathbb{E}_{\pi}\|\nabla f\|^{2}\,,$ (3)

where $\text{ent}_{\pi}(g)\coloneqq\mathbb{E}_{\pi}(g\log
g)-\mathbb{E}_{\pi}g\log\mathbb{E}_{\pi}g.$ For example, when $V_{*}$
$\alpha$-strongly convex, an LSI with $C_{\texttt{LSI}}=1/\alpha$ holds. LSI
hold more generally, but sometimes with very large constants
$C_{\texttt{LSI}}$. Indeed, for multimodal distributions such as mixtures of
Gaussians, $C_{\texttt{LSI}}$ scales exponentially in the height of the
potential barrier between modes (Holley & Stroock, 1987; Arnold et al., 2000).
This impacts convergence at the discrete-time level, and thus hinders our
ability to generate samples using LMC.

Another geometry that gives rise to gradient flows over probability measures
is the Fisher-Rao (FR) geometry; see Section 2.2.2 for definitions. Similar to
the case of Wasserstein gradient flows, we let $(\rho_{t}^{\text{FR}})_{t\geq
0}$ be the FR gradient flow of the KL divergence. Recent work by Lu and
collaborators has shown that the convergence $\rho_{t}^{\text{FR}}\to\pi$
occurs at a rate that is independent of the potential function $V_{*}$. This
is in stark contrast to the case of Wasserstein gradient flows, where the rate
of convergence is intimately related to the structure of $V_{*}$ through the
LSI constant. In their first work, Lu et al. (2019) show that for any
$\delta\in(0,\tfrac{1}{4}]$ there exists a $t_{*}\gtrsim\log(\delta^{3})$ such
that for all $t\geq t_{*}$,

$\displaystyle\text{KL}(\rho_{t}^{\text{FR}}\|\pi)\leq\text{KL}(\rho_{0}^{\text{FR}}\|\pi)e^{-(2-3\delta)(t-t_{*})}\,,$
(4)

where they require a warm-start condition
$\text{KL}(\rho_{0}^{\text{FR}}\|\pi)\leq 1$, and assumption (B) (see Section
3). In Lu et al. (2022), the authors show that the KL divergence is always
contracting under $(\rho_{t}^{\text{FR}})_{t\geq 0}$ even in the absence of a
warm-start, though with a worse rate. Combined, these two results provide the
first continuous-time convergence rates of the gradient flow of the KL
divergence under the FR geometry to $\pi$.

Merging both these geometries gives rise to the well-defined Wasserstein-
Fisher-Rao (WFR) geometry. The WFR geometry has recently been used to analyse
the convergence dynamics of parameters of neural networks Chizat (2022), mean-
field games Rotskoff et al. (2019), and has shown to be useful in statistical
tasks such as Gaussian variational inference Lambert et al. (2022), and
identifying parameters of a Gaussian mixture model Yan et al. (2023). In the
context of sampling, particle-based methods that follow dynamics governed by
WFR gradient flow of the KL, written $(\rho_{t}^{\text{WFR}})_{t\geq 0}$, are
known to escape the clutches of slow-convergence that plague the Wasserstein
geometry. A simple observation (Lu et al., 2022, Remark 2.4) gives the
following continuous-time convergence rate for $t\geq t_{*}$:

$\displaystyle\text{KL}(\rho_{t}^{\text{WFR}}\|\pi)\leq\min\\{\text{KL}(\rho_{t}^{\text{W}}\|\pi),\text{KL}(\rho_{t}^{\text{FR}}\|\pi)\\}\leq\text{KL}(\rho_{0}^{\text{WFR}}\|\pi)\min\left\\{e^{-C_{\texttt{LSI}}t},e^{-(2-3\delta)(t-t_{*})}\right\\}\,,$
(5)

where $\delta$ and $t_{*}$ are as in the FR convergence rate (4). Loosely
speaking, this “decoupled rate” is a consequence of the Wasserstein and FR
geometries being orthogonal to one another; this is made precise in Gallouët &
Monsaingeon (2017).

As elegant as this last connection may seem, the convergence rate in Eq. (4),
and consequently Eq. (5), should appear somewhat unsatisfactory to the reader.
It raises the natural question of whether or not the factor of $\delta$
appearing in the rate is avoidable, and whether the upper bound in Eq. (4) is
tight.

###  1.1 Main contributions

We close this gap for the KL divergence and any $q$-Rényi divergence. Using a
different proof technique than existing work, we prove the following
asymptotic rate of convergence for the flow $(\rho_{t}^{\text{FR}})_{t\geq
0}$, namely for $t$ sufficiently large,

$\displaystyle\text{KL}(\rho_{t}^{\text{FR}}\|\pi)=\tfrac{1}{2}\text{Var}_{\pi}\left(\log\frac{\rho_{0}^{\text{FR}}}{\pi}\right)e^{-2t}+O(e^{-3t})\,,$
(6)

and a similar result holds for all $q$-Rényi divergences. Our assumptions are
weaker to that of prior work, and given that this is a tight asymptotic
convergence rate, we conjecture that the assumptions are likely unavoidable in
the large $t$ regime. Our proof technique provides an explicit expansion of
$\text{KL}(\rho_{t}^{\text{FR}}\|\pi)$ (and $q$-Rényi) in terms of $e^{-t}$.
We supplement our finding with simulations for all three geometries,
indicating that our convergence rate is in fact tight for Fisher-Rao gradient
flows, and sheds light on possible conjectures for the convergence rate of WFR
gradient flows.

#### Notation

For a probability measure $\rho\in\mathcal{P}(\mathbb{R}^{d})$ and a function
$f:\mathbb{R}^{d}\to\mathbb{R}$, we sometimes use the shorthand $\langle
f\rangle_{\rho}\coloneqq\int f\,{\textnormal{d}}\rho$. We let $\log(\cdot)$
denote the natural logarithm, and we use the standard shorthand notation
$f=O(g)$, meaning there exists a constant $C>0$ such that $f\leq Cg$.

##  2 Background

###  2.1 Definitions

The study of gradient flows has a rich history in both pure and applied
mathematics. The development of the relevant calculus to understand gradient
flows is not the purpose of this note, and we instead provide a barebones
introduction. However, we strongly recommend the interested reader consult
standard textbooks on the topic, namely Ambrosio et al. (2005), and the first
chapter of Chewi (2022).

Let $\mathcal{P}(\mathbb{R}^{d})$ be the space of probability measures over
$\mathbb{R}^{d}$. A functional
$\mathcal{F}:\mathcal{P}(\mathbb{R}^{d})\to\mathbb{R}$ is defined on the space
of probability measures, with $\rho\mapsto\mathcal{F}(\rho)\in\mathbb{R}$. We
call ${\delta\mathcal{F}}(\rho)$ the first variation of $\mathcal{F}$ at
$\rho$ if for a signed measure $\eta$ such that
$\int\,{\textnormal{d}}\eta=0$, it holds that

$\displaystyle\lim_{\varepsilon\to
0}\frac{\mathcal{F}(\rho+\varepsilon\eta)-\mathcal{F}(\rho)}{\varepsilon}=\int{\delta\mathcal{F}}(\rho)\,{\textnormal{d}}\eta\,.$
(7)

The Kullback-Leibler (KL) divergence of a measure $\rho$ with respect to some
fixed target measure $\pi$ is defined as
$\text{KL}(\rho\|\pi)=\int\log\frac{\rho}{\pi}\,{\textnormal{d}}\rho$ for
$\rho$ absolutely continuous with respect to $\pi$. For $\pi\propto
e^{-V_{*}}$, the first variation of the KL divergence is given by

$\displaystyle{\delta\text{KL}(\cdot\|\pi)}(\rho)(x)=\log\frac{\rho(x)}{\pi(x)}=\log\rho(x)+V_{*}(x)+\log
Z_{1}\,,$ (8)

where $Z_{1}$ is the normalizing constant for $\pi$.

A more general notion of dissimilarity between probability measures is the
$q$-Rényi divergence: for $q\in[1,\infty]$, we define
$\mathcal{R}_{q}(\rho\|\pi)$ to be the $q$-Rényi divergence with respect to
$\pi$, given by

$\displaystyle\mathcal{R}_{q}(\rho\|\pi)\coloneqq\frac{1}{q-1}\log\int\left(\frac{\rho}{\pi}\right)^{q}\,{\textnormal{d}}\pi\,,$
(9)

for measures $\rho$ that are absolutely continuous with respect to $\pi$.
$\mathcal{R}_{q}$ recovers the KL divergence in the limit $q\to 1$, and when
$q=2$, $\mathcal{R}_{2}(\rho\|\pi)=\log(\chi^{2}(\rho\|\pi)+1)$, where
$\chi^{2}$ is the chi-squared divergence, written explicitly as

$\displaystyle\chi^{2}(\rho\|\pi)=\text{Var}_{\pi}\left(\frac{\rho}{\pi}\right)=\int\left(\frac{\rho}{\pi}\right)^{2}\,{\textnormal{d}}\pi-1\,.$

###  2.2 Gradient flows of the Kullback-Leibler divergence

####  2.2.1 Wasserstein gradient flow

In its dynamic formulation, the 2-Wasserstein distance between two probability
measures $\rho_{0},\rho_{1}$ with bounded second moments can be written as
(Villani, 2008; Benamou & Brenier, 2000)

$\displaystyle\mathrm{W}_{2}^{2}(\rho_{0},\rho_{1})\coloneqq\inf_{(\rho_{t},v_{t})}\int_{0}^{1}\int\|v_{t}(x)\|^{2}\rho_{t}(x)\,{\textnormal{d}}x\,{\textnormal{d}}t\quad\text{s.t.}\quad\partial_{t}\rho_{t}+\nabla\cdot(\rho_{t}v_{t})=0\,,$
(10)

where $(\rho_{t})_{t\in[0,1]}$ is a curve of probability densities over
$\mathbb{R}^{d}$, and $(v_{t})_{t\in[0,1]}$ is a curve of
$L^{2}(\mathbb{R}^{d})^{d}$ vector fields. The constraint is known as the
continuity equation, with endpoints $\rho_{0}$ and $\rho_{1}$. For a
functional $\mathcal{F}:\mathcal{P}(\mathbb{R}^{d})\to\mathbb{R}$, the
Wasserstein gradient flow is the curve of measures
$(\rho_{t}^{\text{W}})_{t\geq 0}$ that satisfies the continuity equation with
the vector field replaced by the steepest descent under the Wasserstein
geometry,

$\displaystyle
v_{t}=-\nabla_{W_{2}}\mathcal{F}(\rho_{t}^{\text{W}})\coloneqq\nabla{\delta\mathcal{F}}(\rho_{t}^{\text{W}})\,,$

where the last equation is simply the (standard) spatial gradient of the first
variation of $\mathcal{F}$. Plugging in the expression for the first variation
of the KL divergence (8), we see that the law of the Langevin diffusion is
given by $\rho_{t}^{\text{W}}$ which satisfies

$\displaystyle\partial_{t}\rho_{t}^{\text{W}}=\nabla\cdot\left(\rho_{t}^{\text{W}}(\nabla\log\rho_{t}^{\text{W}}+\nabla
V_{*})\right)\,.$ (11)

This equation may be rewritten as
$\partial_{t}\rho_{t}^{\text{W}}=\nabla\cdot(\nabla
V_{*}\rho_{t})+\Delta\rho_{t}$, which one readily identifies as the Fokker-
Planck equation for the potential $V_{*}$. The equation describes the
evolution of the distribution of a particle that moves according to the
stochastic differential equation 1. At the particle level, the key aspect of
Wasserstein gradient flows is that they model particle transport, and that
makes them useful for high-dimensional applications such as LMC. In what
follows, we will sometimes abbreviate Wasserstein gradient flow to W-GF.

####  2.2.2 Fisher-Rao gradient flow

The Fisher-Rao distance, or Hellinger-Kakutani distance, between probability
measures has a long history in statistics and information theory (Hellinger,
1909; Kakutani, 1948). It can be defined as (Bogachev, 2007; Gallouët &
Monsaingeon, 2017)

$\displaystyle\mathrm{FR}^{2}(\rho_{0},\rho_{1})\coloneqq\inf_{(\rho_{t},r_{t})}\int_{0}^{1}\int
r_{t}(x)^{2}\rho_{t}(x)\,{\textnormal{d}}x\,{\textnormal{d}}t\quad\text{s.t.}\quad\partial_{t}\rho_{t}=r_{t}\rho_{t}\,,$

where $(\rho_{t})_{t\in[0,1]}$ is again a curve of probability measures, and
$(r_{t})_{t\in[0,1]}$ is a curve of $L^{2}(\mathbb{R}^{d})$ functions.
Together, they satisfy the prescribed equation, with endpoints equal to
$\rho_{0}$ and $\rho_{1}$. The Fisher-Rao gradient flow of the KL divergence,
also known as Birth-Death dynamics, is the curve of measures
$(\rho_{t}^{\text{FR}})_{t\geq 0}$ that satisfies (Gallouët & Monsaingeon,
2017; Lu et al., 2019)

$\displaystyle\partial_{t}\rho^{\text{FR}}_{t}=-\rho^{\text{FR}}_{t}\alpha_{t}\,,\quad\alpha_{t}\coloneqq\log\frac{\rho^{\text{FR}}_{t}}{\pi}-\text{KL}(\rho^{\text{FR}}_{t}\|\pi)\,.$

The first term adjusts mass (i.e. gives birth to or kills mass) according to
the log-ratio of $\rho_{t}^{\text{FR}}$ and the target measure $\pi$. The last
term preserves the total mass, so that
$\rho_{t}^{\text{FR}}\in\mathcal{P}(\mathbb{R}^{d})$ for all time.

Expanding this equation, we have

$\displaystyle\partial_{t}\rho_{t}^{\text{FR}}(x)=-\big{(}\log(\rho_{t}^{\text{FR}}(x))+V_{*}(x)-\big{\langle}\log(\rho_{t}^{\text{FR}})+V_{*}\big{\rangle}_{\rho_{t}^{\text{FR}}}\big{)}\rho_{t}^{\text{FR}}(x).$
(12)

We henceforth omit the superscript FR for the Fisher-Rao gradient flow of the
KL divergence unless the notation becomes ambiguous. For short-hand, we make
use of the abbreviation FR-GF for Fisher-Rao gradient flows.

The FR-GF may be simulated using a system of weighted particles (see Appendix
B). Unlike for the W-GF, in this case the positions of the particles are
fixed; only the weights change over time. Hence, to simulate the FR-GF one is
forced to grid the underlying space $\mathbb{R}^{d}$. This is feasible only
for small dimensions $d$. Consequently, FR-GFs cannot be simulated in high
dimensions, which makes them impractical for sampling applications.

####  2.2.3 Wasserstein-Fisher-Rao geometry gradient flow

The Wasserstein-Fisher-Rao distance between probability measures arises as a
combination of the Wasserstein and the Fisher-Rao distances (Chizat et al.,
2018; 2015; Kondratyev et al., 2016; Liero et al., 2016; 2018). It is defined
as

$\displaystyle\mathrm{WFR}^{2}(\rho_{1},\rho_{1})\coloneqq\inf_{(\rho_{t},v_{t},r_{t})}\int_{0}^{1}\int(\|v_{t}(x)\|^{2}+r_{t}(x)^{2})\rho_{t}(x)\,{\textnormal{d}}x\,{\textnormal{d}}t\quad\text{s.t.}\quad\partial_{t}\rho_{t}+\nabla\cdot(\rho_{t}v_{t})=r_{t}\rho_{t}\,,$

where, for each $t\in[0,1]$, the triple $(\rho_{t},v_{t},r_{t})$ lives in
$\mathcal{P}(\mathbb{R}^{d})\times L^{2}(\mathbb{R}^{d})^{d}\times
L^{2}(\mathbb{R}^{d}),$ and they simultaneously satisfy the constraint
equation, which has endpoints $\rho_{0}$ and $\rho_{1}$, as well. Similarly,
the Wasserstein-Fisher-Rao gradient flow of the KL divergence is the solution
of PDE that incorporates the terms in the Wasserstein and Fisher-Rao gradient
flows (Eq. 11 and Eq. 12):

$\displaystyle\partial_{t}\rho_{t}^{\text{WFR}}=\nabla\cdot\left(\rho_{t}^{\text{WFR}}(\nabla\log\rho_{t}^{\text{WFR}}+\nabla
V_{*})\right)-\big{(}\log(\rho_{t}^{\text{WFR}})+V_{*}-\big{\langle}\log(\rho_{t}^{\text{WFR}})+V_{*}\big{\rangle}_{\rho_{t}^{\text{WFR}}}\big{)}\rho_{t}^{\text{WFR}}$
(13)

Similar to the other geometries, we write WFR-GF as shorthand for Wasserstein-
Fisher-Rao gradient flow At the particle level, WFR-GFs are able to capture
both _transport_ and _weight updates_ , which is why they enjoy a convergence
rate that at least matches the better rate between W- and FR-GFs (recall Eq.
5), and is clearly superior in practice in some instances. Hence, any
improvement in the convergence analysis of either W- or FR-GFs translates to
improving our understanding of WFR-GFs.

###  2.3 Simulated annealing dynamics

Simulated annealing is a technique seen in several works when attempting to
either optimize a function or sample from a multimodal probability
distribution, and has a long history (Pincus, 1970; Kirkpatrick et al., 1983),
and plays a crucial role in our analysis. In what follows, we introduce the
annealing path with linear scaling, and conclude with a proposition.

Consider the time-dependent measure $(\mu_{\tau})_{\tau\in[0,1]}$
corresponding to the annealing path, with linear scaling, initialized at the
measure $\mu_{0}=\rho_{0}\propto e^{-V_{0}}$. By definition, $\mu_{\tau}$
admits the density

$\displaystyle\mu_{\tau}(x)=\frac{e^{-\tau(V_{*}(x)-V_{0}(x))-V_{0}(x)}}{Z_{\tau}},\quad
Z_{\tau}=\int_{\mathbb{R}^{d}}e^{-\tau(V_{*}(x)-V_{0}(x))-V_{0}(x)}\,dx,$ (14)

for $\tau\in[0,1]$. Note that indeed, $\mu_{1}=\pi$. To this end, it will be
convenient to rewrite Eq. 14 in terms of the log-density of $\mu_{\tau}$.
Remark that

$\displaystyle\log(\mu_{\tau}(x))=-\tau(V_{*}(x)-V_{0}(x))-V_{0}(x)-\log
Z_{\tau}\,.$ (15)

One can check that the pointwise derivative of the density $\mu_{\tau}$ (with
respect to $\tau$) is

$\displaystyle\partial_{\tau}\mu_{\tau}(x)=-(V_{*}(x)-V_{0}(x)-\langle
V_{*}-V_{0}\rangle_{\mu_{\tau}})\mu_{\tau}(x)\,.$ (16)

From this, we obtain that

$\displaystyle\begin{split}&\log(\mu_{\tau}(x))+E(x)-\big{\langle}\log(\mu_{\tau})+V_{*}\big{\rangle}_{\mu_{\tau}}\\\
&=-\tau(V_{*}(x)-V_{0}(x))-V_{0}(x)-\langle-\tau(V_{*}-V_{0})-V_{0}+V_{*}\rangle_{\mu_{\tau}}+V_{*}(x)-\langle
V_{*}\rangle_{\mu_{\tau}}\\\
&=-\tau(V_{*}(x)-V_{0}(x))-V_{0}(x)+V_{*}(x)-\langle-\tau(V_{*}-V_{0})-V_{0}+V_{*}\rangle_{\mu_{\tau}}\\\
&=(1-\tau)\big{(}V_{*}(x)-V_{0}(x)\big{)}-(1-\tau)\langle
V_{*}-V_{0}\rangle_{\mu_{\tau}}\\\ &=(1-\tau)\big{(}V_{*}(x)-V_{0}(x)-\langle
V_{*}-V_{0}\rangle_{\mu_{\tau}}\big{)}\,.\end{split}$ (17)

Note that in the first equality we used that the log-partition is a constant
and gets cancelled. Consequently, Eq. 16 can be rewritten, for $\tau\in(0,1)$,
as

$\displaystyle\partial_{\tau}\mu_{\tau}(x)=-\frac{1}{1-\tau}\big{(}\log(\mu_{\tau}(x))+V_{*}(x)-\big{\langle}\log(\mu_{\tau})+V_{*}\big{\rangle}_{\mu_{\tau}}\big{)}\mu_{\tau}(x).$
(18)

A first observation is that that the linear schedule $\tau$ in the exponent of
Eq. 14 results in dynamics that resemble the Fisher-Rao gradient flow of the
KL divergence, up to a reparameterization that can be made explicit. Indeed,
if one compares Eq. 18 with Eq. 12, the only difference is the factor
$\frac{1}{1-\tau}$ in the right-hand side of Eq. 18. Since the solution of the
Fisher-Rao gradient flow of the KL divergence is unique (see Proposition 4 in
Appendix A), an appropriate time reparameterization of the annealed dynamics
(14) will yield the solution (12). We summarize this observation in the
following proposition, which we were unable to find a citation for in the
literature.

######  Proposition 1.

Let $(\mu_{\tau})_{\tau\in[0,1]}$ be as defined in Eq. 14. The Fisher-Rao
gradient flow $(\rho_{t})_{t\geq 0}$ of $\text{KL}(\rho\|\pi)$ (i.e. solving
Eq. 12) is given by $\rho_{t}=\mu_{1-e^{-t}}$.

###### Proof.

If we write $t$ as a function of $\tau$, we have that

$\displaystyle\partial_{\tau}\rho_{t(\tau)}=\partial_{t}\rho_{t(\tau)}\frac{dt}{d\tau}(\tau)=-\frac{dt}{d\tau}(\tau)\big{(}\log(\rho_{t(\tau)}(x))+E(x)-\big{\langle}\log(\rho_{t(\tau)})+E\big{\rangle}_{\rho_{t(\tau)}}\big{)}\rho_{t(\tau)}(x).$
(19)

Identifying $\rho_{t(\tau)}$ with $\rho_{\tau}$, and establishing a direct
comparison with Eq. 18, we obtain that for Eq. 19 to hold, $t(\tau)$ must
fulfill $\frac{dt}{d\tau}(\tau)=\frac{1}{1-\tau}$. With the initial condition
that $\tau(0)=0$, this differential equation has the following unique
solution:

$\displaystyle t(\tau)=\int_{0}^{\tau}\frac{1}{1-s}\,ds=-\log(1-\tau)\,.$ (20)

That is, we have that $t(\tau)=-\log(1-\tau)$, or equivalently,
$\tau(t)=1-e^{-t}$. ∎

###  2.4 Cumulants and their power series

Our core argument hinges on observing a relation between the above gradient
flows and their connection to cumulants of a random variable. Recall that for
a random variable $Y$, its cumulant-generating function to be
$K_{Y}(z)=\log\mathbb{E}[e^{Yz}]$. The $n^{\text{th}}$ cumulant $\kappa_{n}$
of the random variable $Y$ is defined as the $n^{\text{th}}$ derivative of
$K_{Y}$ evaluated at $z=0$, that is, $\kappa_{n}=K^{(n)}_{Y}(0)$. Similar to
moment-generating functions, if $K_{Y}(z)$ is finite in some neighborhood of
$z\in(-\epsilon_{0},\epsilon_{0})$, then it holds that $K_{Y}$ is smooth (in
fact, holomorphic) (see e.g. (Shiryaev, 1984, Section II.12.8). Moreover,
$K_{Y}(z)$ admits the following infinite series expansion

$\displaystyle K_{Y}(z)=\sum_{n\geq 1}\frac{\kappa_{n}}{n!}z^{n}\,.$

In particular, one can easily check that $\kappa_{1}=\mathbb{E}[Y]$ and
$\kappa_{2}=\text{Var}(Y)$.

##  3 Main result

The goal of this section is to prove our main result, which is an explicit
expansion of the KL divergence in terms of log-cumulants of the random
variable $\log\frac{\rho_{0}(X)}{\pi(X)}$ where $X\sim\pi$. We make the
following assumptions throughout, and we will make their uses explicit when
necessary.

(A1) $V_{*}\in L_{1}(\pi)$,

(A2) There exists $\alpha\in\mathbb{R}_{+}$, such that
$\inf_{x}\frac{\rho_{0}(x)}{\pi(x)^{1+\alpha}}>0$.

Assumption (A1) ensures that $\pi$ has finite differential entropy, and is a
relatively weak condition. (A2) asks that at least some mass is initially
placed along the support of $\pi$. (A2) is, however, a much weaker assumption
that what is currently used in the literature. To be precise, Lu et al. (2019;
2022) assume a particular case of (A2), namely

(B) There exists $M>0$ such that $\inf_{x}\frac{\rho_{0}(x)}{\pi(x)}\geq
e^{-M}$.

This is essentially the same as (A2), though $\alpha$ is constrained to be 0,
and a precise lower bound on the infimum is needed. Note that (A2) is weaker
the larger $\alpha$ is, as $\pi(x)^{1+\alpha}$ decreases faster. As a
comparison, if $\rho_{0}$ and $\pi$ are Gaussians, (A2) covers the setting
where both have arbitrary means and covariances, while constraining $\alpha=0$
only covers the cases in which the covariance matrix of $\rho_{0}$ is strictly
larger than the one of $\pi$ in the positive definite order.

The following theorem is our main contribution. While here we have stated an
asymptotic expression, in fact a more general expression is available as an
infinite power series under the same assumptions, and appears explicitly in
the proof.

######  Theorem 1.

Suppose (A1) and (A2) hold. Then for $t$ large enough and any
$q\in(1,\infty)$,

$\displaystyle\text{KL}(\rho_{t}\|\pi)=\frac{\kappa_{2}}{2}e^{-2t}+O(e^{-3t})\,,\quad\text{
and
}\quad\mathcal{R}_{q}(\rho_{t}\|\pi)=\frac{q\kappa_{2}}{2}e^{-2t}+O_{q}(e^{-3t})\,,$
(21)

where $\kappa_{2}=\text{Var}_{\pi}\left(\log\frac{\rho_{0}}{\pi}\right)$.

######  Remark 1.

The coefficient $\kappa_{2}$ is nothing more than the variance under $\pi$ of
the first-variation of the KL divergence at $\rho_{0}$ (recall Eq. 8).

###  3.1 Proof

Henceforth, we will always write

$\displaystyle Y\coloneqq\log\frac{\rho_{0}(X)}{\pi(X)}\text{ where
}X\sim\pi\,.$ (22)

The first step in our proof is to represent these divergences as a function of
the cumulants of $Y$, which is possible due to the aforementioned time-
reparameterization of the FR flow.

######  Proposition 2.

Let $\pi\propto e^{-V_{*}}$ and $\rho_{0}\propto e^{-V_{0}}$ be probability
measures on $\mathbb{R}^{d}$, and let $Y$ be as in Eq. 22. Let
$(\mu_{\tau})_{\tau\in[0,1]}$ be follow the simulated annealing dynamics from
Eq. 14. It holds that

$\displaystyle\text{KL}(\mu_{\tau}\|\pi)=(1-\tau)K_{Y}^{\prime}(1-\tau)-K_{Y}(1-\tau)\,,$
(23)
$\displaystyle\mathcal{R}_{q}(\mu_{\tau}\|\pi)=\frac{1}{q-1}K_{Y}(q(1-\tau))-\frac{q}{q-1}K_{Y}(1-\tau)\,.$
(24)

###### Proof.

We first identify the following relationship, which arises from a simple
manipulation of Eq. 14

$\displaystyle\log Z_{\tau}=K_{Y}(1-\tau)+\log Z_{1}\,.$ (25)

Using this expression, we can expand the KL divergence between $\mu_{\tau}$
and $\pi$ as follows:

$\displaystyle\text{KL}(\mu_{\tau}\|\pi)$
$\displaystyle=\int\log\frac{\mu_{\tau}}{\pi}\mu_{\tau}=\int\log\left(\frac{e^{-\tau(V_{*}-V_{0})-V_{0}}Z_{\tau}^{-1}}{e^{-V_{*}}Z_{1}^{-1}}\right)\,{\textnormal{d}}\mu_{\tau}$
$\displaystyle=\log Z_{1}-\log Z_{\tau}+(1-\tau)\langle
V_{*}-V_{0}\rangle_{\mu_{\tau}}$ $\displaystyle=(1-\tau)\langle
V_{*}-V_{0}\rangle_{\mu_{\tau}}-K_{Y}(1-\tau)\,.$

Another fact about cumulant generating functions that we can exploit is the
following differential relationship

$\displaystyle-\langle
V_{*}-V_{0}\rangle_{\mu_{\tau}}=\frac{\,{\textnormal{d}}}{\,{\textnormal{d}}\tau}Z_{\tau}=-K_{Y}^{\prime}(1-\tau)\,.$
(26)

Altogether, this gives

$\displaystyle\text{KL}(\mu_{\tau}\|\pi)=(1-\tau)K_{Y}^{\prime}(1-\tau)-K_{Y}(1-\tau)\,.$
(27)

The general $q$-Rényi case is deferred to the appendix, where the computation
is similar. ∎

The following lemma uses both (A1) and (A2) to establish that $K_{Y}(z)$ is
finite in some neighborhood of $z\in B_{\epsilon_{0}}(0)$, which implies that
$K_{Y}$ admits the series expansion we will require in the sequel. The proof
is deferred to the appendix.

######  Proposition 3.

Suppose (A1) and (A2) are satisfied. Then there exists some constant
$\epsilon_{0}>0$ such that the cumulant generating function of $Y$,
$K_{Y}(z)=\log\mathbb{E}[e^{Yz}]$ is finite on some neighborhood of $z\in
B_{\epsilon_{0}}(0)$. Moreover, inside this neighborhood, $K_{Y}(z)$ is
holomorphic and we have the series expansion

$\displaystyle K_{Y}(z)=\sum_{n\geq 1}\frac{\kappa_{n}}{n!}z^{n}\,.$ (28)

We conclude with the proof of our main result.

###### Proof of Theorem 1.

We begin with the expression of the KL divergence. Note that since $K_{Y}(z)$
is smooth for $z$ sufficiently close to the origin, it holds that

$\displaystyle K_{Y}^{\prime}(z)=\sum_{n\geq
1}\frac{\kappa_{n}}{(n-1)!}z^{n-1}\,.$

Using the parameterization of Eq. 27 and the series expansion for
$K_{Y}^{\prime}(1-\tau)$, our expression for $\text{KL}(\mu_{\tau}\|\pi)$
reads

$\displaystyle\text{KL}(\mu_{\tau}\|\pi)$ $\displaystyle=(1-\tau)\sum_{n\geq
1}\frac{\kappa_{n}}{(n-1)!}(1-\tau)^{n-1}-\sum_{n\geq
1}\frac{\kappa_{n}}{n!}(1-\tau)^{n}$ $\displaystyle=\sum_{n\geq
1}\kappa_{n}\left(\frac{n}{n!}-\frac{1}{n!}\right)(1-\tau)^{n}$
$\displaystyle=\sum_{n\geq 2}\frac{\kappa_{n}}{n(n-2)!}(1-\tau)^{n}\,.$

Expanding the relation and replacing $\tau(t)=1-e^{-t}$ gives

$\displaystyle\text{KL}(\rho_{t}\|\pi)=\frac{\kappa_{2}}{2}e^{-2t}+\sum_{n\geq
3}\frac{\kappa_{n}}{n(n-2)!}e^{-nt}\,.$

We now do the same manipulations for $\mathcal{R}_{q}(\mu_{\tau}\|\pi)$.

$\displaystyle\mathcal{R}_{q}(\mu_{\tau}\|\pi)$
$\displaystyle=\frac{1}{q-1}\sum_{n\geq
1}\frac{\kappa_{n}}{n!}(q(1-\tau))^{n}-\frac{q}{q-1}\sum_{n\geq
1}\frac{\kappa_{n}}{n!}(1-\tau)^{n}$
$\displaystyle=\frac{1}{q-1}\left(\frac{\kappa_{1}}{q}(1-\tau)+\sum_{n\geq
2}q^{n}\frac{\kappa_{n}}{n!}(1-\tau)^{n}\right)-\frac{q}{q-1}\left(\kappa_{1}(1-\tau)+\sum_{n\geq
2}\frac{\kappa_{n}}{n!}(1-\tau)^{n}\right)$ $\displaystyle=\sum_{n\geq
2}\frac{q^{n}-q}{q-1}\frac{\kappa_{n}}{n!}(1-\tau)^{n}\,.$

Substituting $\tau(t)=1-e^{-t}$ and expanding out the first term yields

$\displaystyle\mathcal{R}_{q}(\rho_{t}\|\pi)=q\frac{\kappa_{2}}{2}e^{-2t}+\sum_{n\geq
3}\frac{q^{n}-q}{q-1}\frac{\kappa_{n}}{n!}e^{-nt}\,.$

Our proof concludes by taking the limit $t\to\infty$, which we fully justify
in the appendix (Lemma 2). ∎

##  4 Numerical simulations

We present simple numerical simulations that demonstrates our asymptotic
convergence rate of the KL divergence the FR gradient flows, as well as a
comparison with the WFR- and W-GFs. We consider two target distributions over
the set $[-\pi,\pi)$, each with two initializations:

  1. 1.

Target distribution $\pi_{1}$: We set $\pi_{1}\propto e^{-V_{1}}$ with
$V_{1}(x)=2.5\cos(2x)+0.5\sin(x)$. This distribution has two modes with
different weights and has been studied previously by Lu et al. (2019). We
consider two initial distributions:

    1. (a)

$\pi_{a}\propto e^{-V_{a}}$ with $V_{a}=-V_{1}$, which has two modes in
locations where $\pi$ has little mass.

    2. (b)

$\pi_{b}\propto e^{-V_{b}}$ with $V_{b}=2.5\cos(2x)$, which has two modes in
almost the same positions as $\pi$, but with equal weight.

  2. 2.

Target distribution $\pi_{2}$: We set $\pi_{2}\propto e^{-V_{2}}$ with
$V_{2}(x)=-6\cos(x)$. This distribution has one mode. We consider two initial
distributions:

    1. (c)

$\pi_{c}\propto e^{-V_{c}}$ with $V_{c}=-V_{2}$, which has one mode in a
location where $\pi$ has little mass.

    2. (d)

$\pi_{d}\propto e^{-V_{d}}$ with $V_{d}=0$, which is the uniform distribution.

|
---|---
Figure 1: Energies of the target and initial distributions.

Fig. 1 shows the target energies $V_{1}$, $V_{2}$ and the initial energies
$V_{a}$, $V_{b}$, $V_{c}$, $V_{d}$ introduced above. Fig. 2 shows the
evolution of the KL divergence along the FR, WFR and W gradient flows. It also
contains plots of the dominant term $\frac{\kappa_{2}}{2}e^{-2t}$ of the
approximation of the KL divergence decay for FR flows (see Theorem 1),
displayed as dotted lines. Table 1 shows the slopes of each curve from Fig. 2,
at large times (see Appendix B for details on the computation of slopes).

|
---|---
Figure 2: Evolution of the KL divergence with respect to $\pi_{1}$ (_left_)
and $\pi_{2}$ along their respective FR (_solid lines_), WFR (_dash-dotted
lines_) and W (_dashed lines_) gradient flows. Each plot contains flows
initialized at two probability measures: in the left plot these are $\pi_{a}$
(_blue_ , top curves at $t=0$) and $\pi_{b}$ (_orange_); in the right plot,
$\pi_{c}$ (_blue_ , top curves at $t=0$) and $\pi_{d}$ (_orange_). The
_dotted_ lines show the curves $\frac{\kappa_{2}}{2}e^{-2t}$ (for the
appropriate values $\kappa_{2}$), introduced in Theorem 1.

Some observations are in order:

  * •

As predicted by Theorem 1, the curves $\text{KL}(\rho_{t}^{\text{FR}}\|\pi)$
approach the curves $\frac{\kappa_{2}}{2}e^{-2t}$ as $t$ grows.

  * •

For $\pi_{1}$, the curves $\text{KL}(\rho_{t}^{\text{FR}}\|\pi)$ and
$\text{KL}(\rho_{t}^{\text{WFR}}\|\pi)$ initialized at $\pi_{b}$ are very
close for small times. The reason is that $\nabla V_{1}$ and $\nabla V_{b}$
are very close in the regions where $\pi_{1}$ and $\pi_{b}$ have most of the
mass. Consequently, the term
$\nabla\cdot\left(\rho_{t}^{\text{WFR}}(\nabla\log\rho_{t}^{\text{WFR}}+\nabla
V_{1})\right)$, which is the difference between the FR and the WFR PDEs, is
small at initialization.

  * •

The curves $\text{KL}(\rho_{t}^{\text{W}}\|\pi)$ behave very differently for
$\pi_{1}$ and $\pi_{2}$ (see Table 1). Indeed, since $\pi_{1}$ is bimodal
$C_{\texttt{LSI}}(\pi_{1})$ is quite large (thus convergence is slow), whereas
$\pi_{2}$ is unimodal, with a much smaller log-Sobolev constant.

  * •

The curves $\text{KL}(\rho_{t}^{\text{WFR}}\|\pi)$ also behave differently for
both target distributions. For $\pi_{1}$, it decays only slightly faster than
$\text{KL}(\rho_{t}^{\text{FR}}\|\pi)$, while for $\pi_{2}$ it goes down much
faster than both $\text{KL}(\rho_{t}^{\text{FR}}\|\pi)$ and
$\text{KL}(\rho_{t}^{\text{WFR}}\|\pi)$. Interestingly, looking at Table 1 we
observe that the asymptotic slopes of the WFR are very close to the sum of the
slopes for FR and W. This seems to indicate that at large times, the KL
divergence decays like $e^{-2t-\frac{2t}{C_{\texttt{LSI}}}}$, i.e. that the W
and FR terms act more or less independently.

| Target $\pi_{1}$ | Target $\pi_{2}$
---|---|---
| Init. $\pi_{a}$ | Init. $\pi_{b}$ | Init. $\pi_{c}$ | Init. $\pi_{d}$
FR | -2.0016 | -2.0002 | -2.0028 | -2.0014
WFR | -2.0771 | -2.0759 | -12.8190 | -12.8632
W | -0.0811 | -0.0811 | -10.7784 | -10.8538
Table 1: Large-time slopes of the KL divergence vs. time curves in a semi-
logarithmic plot (Fig. 2), for the three flows. See Appendix B for details on
the computation of the slopes.

##  5 Conclusion

In this work, using a relatively simple proof technique, we showed that the
Kullback-Leibler divergence along its Fisher-Rao gradient flow
$(\rho_{t}^{\text{FR}})_{t\geq 0}$ can be written as a power-series expansion,
resulting in a tight asymptotic convergence rate for large times. A similar
expansion holds for $\mathcal{R}_{q}(\rho_{t}^{\text{FR}}\|\pi)$, where
$\mathcal{R}_{q}$ is any $q$-Rényi divergence. Our findings were verified with
simple numerical experiments, where we also simulated Wasserstein and
Wasserstein-Fisher-Rao gradient flows. Our simulations indicated that, in some
cases, the convergence rate of the WFR gradient flow scales like
$e^{-(2+(2/C_{\texttt{LSI}}))t}$, an observation that we hope can be made
precise in future work. A second direction is to extend our proof technique
from the KL divergence to general Bregman divergences.

#### Acknowledgments

The authors thank Joan Bruna, Jonathan Niles-Weed, Sinho Chewi, and Andre
Wibisono for helpful discussions. CD acknowledges Meta AI Research as a
funding source. AAP acknowledges NSF Award 1922658 and and Meta AI Research.

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##  Appendix A Remaining proofs

######  Proposition 4 (Uniqueness of the Fisher-Rao gradient flow of the KL
divergence).

Given a target potential $V_{*}$ and an initial measure $\rho_{0}$, the
solution of Eq. 12 is unique.

###### Proof.

Consider the PDE

$\displaystyle\partial_{t}\mu_{t}(x)=-\big{(}\log(\mu_{t}(x))+V_{*}(x)\big{)}\mu_{t}(x),\qquad\mu_{0}=\rho_{0}$
(29)

Note that this is in fact an ODE for each point $x$, that we can rewrite as
$\partial_{t}\log\mu_{t}(x)=-\big{(}\log(\mu_{t}(x))+V_{*}(x)\big{)}$. The
unique solution of this ODE with initial condition $\log\mu_{0}(x)$ is
$\log\mu_{t}(x)=(\log\mu_{0}(x)-V_{*}(x))e^{-t}+V_{*}(x)$. Thus, we conclude
that Eq. 29 has a unique solution.

Now, given a solution $\rho_{t}$ of Eq. 12 with initial condition $\rho_{0}$,
define $\tilde{\rho}_{t}$ as

$\displaystyle\log\tilde{\rho}_{t}(x)=\log\rho_{t}(x)-\int_{0}^{t}\big{\langle}\log(\rho_{s})+V_{*}\big{\rangle}_{\rho_{s}}\,{\textnormal{d}}s.$
(30)

Remark that $\tilde{\rho}_{t}$ is a solution of Eq. 29, since

$\displaystyle\partial_{t}\log\tilde{\rho}_{t}(x)$
$\displaystyle=\partial_{t}\log\rho_{t}(x)-\big{\langle}\log(\rho_{t})+V_{*}\big{\rangle}_{\rho_{t}}=-\big{(}\log(\rho_{t}(x))+V_{*}(x)-\big{\langle}\log(\rho_{t})+V_{*}\big{\rangle}_{\rho_{t}}\big{)}-\big{\langle}\log(\rho_{t})+V_{*}\big{\rangle}_{\rho_{t}}$
$\displaystyle=-\big{(}\log(\rho_{t}(x))+V_{*}(x)\big{)}.$

Also, note that the map $(\rho_{t})_{t\geq 0}\to(\tilde{\rho}_{t})_{t\geq 0}$
defined by Eq. 30 is invertible, as
$\rho_{t}(x)=\tilde{\rho_{t}}(x)/\int\tilde{\rho}_{t}(y)\,{\textnormal{d}}y$.
This follows from the fact that $\rho_{t}$ and $\tilde{\rho}_{t}$ are
proportional to each other, and that $\rho_{t}$ integrates to 1.

Finally, suppose that $\rho_{t}^{a}$ and $\rho_{t}^{b}$ are two solutions of
Eq. 12 with initial condition $\rho_{0}$. Via the construction Eq. 30, they
yield solutions $\tilde{\rho}_{t}^{a}$ and $\tilde{\rho}_{t}^{b}$ of Eq. 29
with initial condition $\rho_{0}$. The uniqueness of the solution of Eq. 29
implies that $\tilde{\rho}_{t}^{a}=\tilde{\rho}_{t}^{b}$. Since the map
$(\rho_{t})_{t\geq 0}\to(\tilde{\rho}_{t})_{t\geq 0}$ is invertible, we obtain
that $\rho_{t}^{a}=\rho_{t}^{b}$, which concludes the proof ∎

###### Proof of Proposition 2 (Continued).

We perform similar manipulations as in the case with the KL divergence:

$\displaystyle\mathcal{R}_{q}(\mu_{\tau}\|\pi)$
$\displaystyle=\frac{1}{q-1}\log\int\frac{e^{-q\tau(V_{*}-V_{0})-qV_{0}}(Z_{\tau})^{-q}}{e^{-qV_{*}}(Z_{1})^{-q}}\,{\textnormal{d}}\pi$
$\displaystyle=\frac{1}{q-1}\log\int
e^{q(1-\tau)(V_{*}-V_{0})}\left(\frac{Z_{\tau}}{Z_{1}}\right)^{q}\,{\textnormal{d}}\pi$
$\displaystyle=\frac{1}{q-1}K_{Y}(1-\tau)-\frac{q}{q-1}(\log Z_{\tau}-\log
Z_{1})$
$\displaystyle=\frac{1}{q-1}K_{Y}(1-\tau)-\frac{q}{q-1}K_{Y}(1-\tau)\,,$

where in the last line we again used Eq. 25. This completes the proof. ∎

###### Proof of Proposition 3.

By (A1), the partition function $F(t)=\int_{\mathbb{R}^{d}}e^{-tV_{*}(x)}\,dx$
is differentiable at $t=1$. This is because $F^{\prime}(t)=-\int
V_{*}(x)\,d\pi(x)$. Hence, $F(t)$ is finite on an interval
$(1-2\epsilon_{1},1]$ for some $\epsilon_{1}$.

Note that the assumption (A2) can be written equivalently as
$\xi:=\inf_{x}\alpha V_{*}(x)-V_{0}(x)>-\infty$. We obtain that for all
$\epsilon\in[0,\epsilon_{1}/\alpha)$,

$\displaystyle\begin{split}-\epsilon(V_{*}(x)-V_{0}(x))-V_{*}(x)&=-\epsilon((1+\alpha)V_{*}(x)-V_{0}(x))+(\epsilon\alpha-1)V_{*}(x)\\\
&\leq-\epsilon\xi+(\epsilon\alpha-1)V_{*}(x)\leq-\epsilon\xi+(\epsilon_{1}-1)V_{*}(x)\end{split}$
(31)

Equivalently,

$\displaystyle\exp(K_{Y}(-\epsilon))=\int_{\mathbb{R}^{d}}e^{-\epsilon(V_{*}(x)-V_{0}(x))-V_{*}(x)}\,dx\leq
e^{-\epsilon\xi}\int_{\mathbb{R}^{d}}e^{-(1-\epsilon_{1})V_{*}(x)}\,dx=e^{-\epsilon\xi}F(1-\epsilon_{1})<+\infty.$
(32)

Also, for all $\epsilon\in[0,1)$, using the convexity of the exponential
function we have that

$\displaystyle\exp(K_{Y}(\epsilon))$
$\displaystyle=\int_{\mathbb{R}^{d}}e^{\epsilon(V_{*}(x)-V_{0}(x))-V_{*}(x)}\,dx=\int_{\mathbb{R}^{d}}e^{-(1-\epsilon)V_{*}(x)-\epsilon
V_{0}(x)}\,dx$ (33)
$\displaystyle\leq\int_{\mathbb{R}^{d}}(1-\epsilon)e^{-V_{*}(x)}+\epsilon
e^{-V_{0}(x)}\,dx=(1-\epsilon)Z_{1}+\epsilon Z_{0}<+\infty.$ (34)

Hence, the cumulant-generating function $K_{Y}(t)=\log\mathbb{E}e^{tY}$ is
finite on a neighborhood $(-\epsilon_{0},\epsilon_{0})$ with
$\epsilon_{0}=\min\\{1,\epsilon_{1}/\alpha\\}$. Applying Lemma 1, we conclude
that there exists $\epsilon>0$ such that for $z\in B_{\epsilon}(0)$, we have
that $K_{Y}(z)=\sum_{n=1}^{+\infty}\frac{\kappa_{n}}{n!}z^{n}$. ∎

The following lemma, which we make explicit, is a well-known fact in
probability theory. In short, since the moment-generating function is analytic
in some neighborhood, and is non-negative, taking the logarithm is safe as
everything is analytic. The interested reader can consult e.g. (Shiryaev,
1984, Section II.12.8) which dissects this in detail.

######  Lemma 1.

Assume that the cumulant-generating function $K_{Y}(t)=\log\mathbb{E}e^{tY}$
is finite on a neighborhood $(-\epsilon_{0},\epsilon_{0})$ of zero. Then,
$K_{Y}(z)=\log\mathbb{E}e^{zY}$ as a function on the complex plane is
holomorphic on the open ball $B_{\epsilon}(0)$ of radius $\epsilon$ centered
at zero, for some $\epsilon>0$. Moreover, for $z\in B_{\epsilon}(0)$, we have
that

$\displaystyle K_{Y}(z)=\sum_{n=1}^{+\infty}\frac{\kappa_{n}}{n!}z^{n}.$ (35)

######  Lemma 2 (End of the proof of Theorem 1).

We have that

$\displaystyle|\text{KL}(\rho_{t}\|\pi)-\frac{\kappa_{2}}{2}e^{-2t}|=O(e^{-3t}),\qquad|\mathcal{R}_{q}(\rho_{t}\|\pi)-\frac{q\kappa_{2}}{2}e^{-2t}|=O(e^{-3t}).$
(36)

###### Proof.

Lemma 1 implies that the series for $K_{Y}$ centered at zero has convergence
radius $\epsilon$, for some $\epsilon>0$. Since the derivative of a series has
the same radius of convergence, we obtain that

$\displaystyle H(z):=zK_{Y}^{\prime}(z)-K_{Y}(z)=\sum_{n\geq
2}\frac{\kappa_{n}}{n(n-2)!}z^{n}.$

has convergence radius $\epsilon$ as well. Hence, by the Cauchy-Hadamard
theorem, $\frac{1}{\epsilon}\geq\limsup_{n\to\infty}(|c_{n}|^{1/n})$, where
$c_{n}:=\frac{\kappa_{n}}{n(n-2)!}$.

This implies that for all $0<\epsilon^{\prime}<\epsilon$, there exists a
constant $C_{\epsilon^{\prime}}>0$ such that for all $n\geq 0$, $|c_{n}|\leq
C_{\epsilon^{\prime}}/(\epsilon^{\prime})^{n}$. Consequently, for all
$z\in\mathbb{C}$ with $|z|<1/\epsilon^{\prime}$,

$\displaystyle|H(z)-\frac{\kappa_{2}}{2}z^{2}|=\big{|}\sum_{n=3}^{+\infty}\frac{\kappa_{n}}{n(n-2)!}z^{n}\big{|}\leq
C_{\epsilon^{\prime}}\sum_{n=3}^{+\infty}\big{(}\frac{|z|}{\epsilon^{\prime}}\big{)}^{n}=C_{\epsilon^{\prime}}\frac{\big{(}\frac{|z|}{\epsilon^{\prime}}\big{)}^{3}}{1-\frac{|z|}{\epsilon^{\prime}}}$
(37)

Using Eq. 23, we get that for any constant $\gamma>0$, if
$t\geq-\log\epsilon^{\prime}+\gamma$ (or equivalently,
$e^{-t}\leq\epsilon^{\prime}e^{-\gamma}$),

$\displaystyle|\text{KL}(\rho_{t}\|\pi)-\frac{\kappa_{2}}{2}e^{-2t}|\leq
C_{\epsilon^{\prime}}\frac{\big{(}\frac{e^{-t}}{\epsilon^{\prime}}\big{)}^{3}}{1-\frac{e^{-t}}{\epsilon^{\prime}}}=C_{\epsilon^{\prime}}\frac{e^{-3t}}{(\epsilon^{\prime})^{3}(1-e^{-\gamma})}=O(e^{-3t}),$
(38)

which concludes the proof for the KL divergence. For the Rényi divergence, the
proof is analogous (note that in that case the series
$\frac{1}{q-1}K_{Y}(qz)-\frac{q}{q-1}K_{Y}(z)$ has convergence radius
$\epsilon/q$). ∎

##  Appendix B Details on the numerical simulations

To run the simulations in Section 4, we discretized the interval $[-\pi,\pi)$
in $n=2000$ equispaced points. Let $h=2\pi/n$. For each algorithm and
initialization, we construct sequences ${(x_{k})}_{k\geq 0}$, where
$x_{k}\in\mathbb{R}^{n}$ represents the normalized log-density at each point.
We let $v_{*}\in\mathbb{R}^{n}$ be the (non-normalized) energy of the target
distribution, obtained by evaluating $V_{*}$ at the discretization points.
Similarly, $\nabla v_{*},\Delta v_{*}\in\mathbb{R}^{n}$ are the evaluations of
$\nabla V_{*}$ and $\Delta V_{*}$ at the $n$ points (note that $\nabla V_{*}$
is a scalar because the distributions are one-dimensional).

We used the following discretizations for the Fisher-Rao, Wasserstein and
Wasserstein-Fisher-Rao gradient flows:

  1. (i)

Fisher-Rao GF: We use mirror descent in log-space. The update reads:

$\displaystyle\tilde{x}_{k+1}$ $\displaystyle\leftarrow
x_{k}+\epsilon(-v_{*}-x_{k}),$ $\displaystyle x_{k+1}$
$\displaystyle\leftarrow\tilde{x}_{k+1}-\log\bigg{(}\sum_{i=1}^{n}e^{-\tilde{x}_{k+1}^{i}}\bigg{)}.$

  2. (ii)

Wasserstein GF: We approximate numerically the gradient and the laplacian of
the log-density:

$\displaystyle\begin{split}\forall i\in[n],\qquad(\nabla
x_{k})^{i}&\leftarrow(x_{k}^{i+1}-x_{k}^{i-1})/(2h),\\\ \forall
i\in[n],\qquad(\Delta
x_{k})^{i}&\leftarrow(x_{k}^{i+1}+x_{k}^{i-1}-2x_{k}^{i})/h^{2},\\\
x_{k+1}&\leftarrow x_{k}+\epsilon(\Delta v_{*}+\Delta x_{k}+(\nabla
v_{*}+\nabla x_{k})\nabla x_{k}).\end{split}$ (39)

We use periodic boundary conditions, so that the first discretization point is
adjacent to the last one for the purposes of computing derivatives.

  3. (iii)

Wasserstein-Fisher-Rao GF: We combine the two previous updates. Letting
$\nabla x_{k}$ and $\Delta x_{k}$ be as in Eq. 39, we have

$\displaystyle\tilde{x}_{k+1}$ $\displaystyle\leftarrow
x_{k}+\epsilon(-v_{*}-x_{k}+\Delta v_{*}+\Delta x_{k}+(\nabla v_{*}+\nabla
x_{k})\nabla x_{k}),$ $\displaystyle x_{k+1}$
$\displaystyle\leftarrow\tilde{x}_{k+1}-\log\bigg{(}\sum_{i=1}^{n}e^{-\tilde{x}_{k+1}^{i}}\bigg{)}.$

We used stepsizes $\epsilon=$2.5\text{\times}{10}^{-6}$$ and
$\epsilon=$1\text{\times}{10}^{-6}$$ for the experiments on target
distributions (1) and (2), respectively. The slopes in Table 1 are obtain by
taking $0<t_{1}<t_{2}$ and computing

$\displaystyle\frac{\log(\text{KL}(\rho_{t_{2}}\|\pi))-\log(\text{KL}(\rho_{t_{1}}\|\pi))}{t_{2}-t_{1}}.$

We use different values for $t_{1}$ and $t_{2}$ for each target distribution;
$t_{1}$ and $t_{2}$ must be large enough to capture the asymptotic slope of
the curve, but not too large to avoid numerical errors. For all the curves
corresponding to target $\pi_{1}$, we take $t_{1}=7.0$ and $t_{2}=7.5$. For
target $\pi_{2}$, we take: for FR, $t_{1}=6.875$ and $t_{2}=7.0$; for WFR,
$t_{1}=1.875$ and $t_{2}=2.0$; for W, $t_{1}=2.75$ and $t_{2}=2.875$.


# Single and Multi-Speaker Cloned Voice Detection: From Perceptual to Learned
Features ††thanks: This work was partially funded by a grant from the UC
Berkeley Center For Long-Term Cybersecurity (CLTC), an award for open-source
innovation from the Digital Public Goods Alliance and United Nations
Development Program, and from an unrestricted gift from Meta. The public
codebase can be found at https://github.com/audio-df-ucb/ClonedVoiceDetection.

Sarah Barrington1, Romit Barua1, Gautham Koorma1, Hany Farid1,2 School of
Information1, Electrical Engineering and Computer Sciences2, University of
California, Berkeley
Berkeley, CA USA
{sbarrington, romit_barua, gautham.koorma<EMAIL_ADDRESS>
###### Abstract

Synthetic-voice cloning technologies have seen significant advances in recent
years, giving rise to a range of potential harms. From small- and large-scale
financial fraud to disinformation campaigns, the need for reliable methods to
differentiate real and synthesized voices is imperative. We describe three
techniques for differentiating a real from a cloned voice designed to
impersonate a specific person. These three approaches differ in their feature
extraction stage with low-dimensional perceptual features offering high
interpretability but lower accuracy, to generic spectral features, and end-to-
end learned features offering less interpretability but higher accuracy. We
show the efficacy of these approaches when trained on a single speaker’s voice
and when trained on multiple voices. The learned features consistently yield
an equal error rate between $0\%$ and $4\%$, and are reasonably robust to
adversarial laundering.

###### Index Terms:

deepfakes, generative AI, audio forensics

##  I Introduction

Computational techniques for modifying a recorded voice to sound like another
person while preserving the original semantic meaning–voice
conversion–predates today’s deepfakes and generative AI by some $65$ years
[23]. The semiannual voice conversion challenge111http://vc-challenge.org
evaluates voice cloning submissions on naturalness (rated from 1 = completely
unnatural to 5 = completely natural) and speaker identity (rated on a scale of
“same, absolutely sure,” “same, not sure,” “different, not sure,” or
“different, absolutely sure”). In the first challenge of 2016, the best-
performing system received an average of $3.0$ on the five-point naturalness
scale and $70\%$ of the samples were judged on identity to be “same.” In 2018,
the best-performing system received an average $4.1$ naturalness score, and
$80\%$ of the samples were judged on identity to be “same.” In 2020, the best
naturalness scores continued to hover around $4.0$, but identity ratings were
nearly perfect.

Over the past few years, AI-powered voice synthesis has continued to improve
(in terms of naturalness and identity), culminating this year in dramatic
breakthroughs. Perhaps most striking is zero-shot, multi-speaker text-to-
speech (ZS-TTS)222https://edresson.github.io/YourTTS for cloning a voice
identity not seen during training from a few seconds to minutes of reference
audio [6]. Also striking is the easy access to these voice-cloning
technologies through low-cost commercial
services333http://https://beta.elevenlabs.io.

While these advances are a major success of the research community, they have
also come at a price. Reports of phone scams have emerged in which a call
purportedly from a family member claims they were in an accident, arrested, or
kidnapped after which the scammer takes over in an attempt to extort money
[13, 15]. Similar reports have emerged that financial institutions using voice
identification can now be spoofed with voice cloning [7]. And, fake audio is
adding to already existing problems of disinformation [16].

From these disinformation campaigns to small- and large-scale fraud and to the
continued erosion in trust of all digital media, it is critical that we
develop techniques to distinguish the real from the fake.

Detection strategies fall into two general categories: (1) active techniques
which, at the point of synthesis, embed a perceptible or imperceptible
watermark into [22], or extract a perceptual fingerprint [22] from,
synthetically-generated content. These watermarks/fingerprints can then be
used to identify content once it is released into the wild; and (2) in the
absence of watermarks/fingerprints, passive techniques detect a range of
statistical to semantic inconsistencies in synthetically-generated content
(see Section I-A).

Our efforts fall into the second category where we describe three related
passive approaches for distinguishing real from cloned voices using
handcrafted perceptual, generic spectral, or learned features. The benefit of
the perceptual features is that they afford a low-dimensional, explainable
classifier, while the learned features generally afford better classification
performance, with the spectral features affording a compromise between these.
These different approaches (Section III) are evaluated (Section IV) against
two different real audio datasets and three cloned audio datasets (Section
II).

We consider two basic scenarios in which the three feature sets are trained to
distinguish real from cloned voices of a single speaker (Section IV-A) and
trained simultaneously from multiple speakers (Section IV-C).

###  I-A Related Work

By way of background, Almutairi and Elgibreen [2] provide a review and a
quantitative comparison of various audio deepfake detection methods, and the
First International Workshop on Deepfake Detection for Audio Multimedia
focused on synthetic audio detection [27]. In this section, we highlight a few
of these approaches and those most closely related to ours.

Classical approaches for detecting synthetic speech typically exploit
statistical differences between synthetic and human speech. Ogihara et al.
[20], for example, proposed a technique that exploits differences in pitch
between synthetic and human speech. De Leon et al. [8] extended this work by
exploiting additional pitch features including stability and jitter. In
addition to these pitch differences, they also observed that the transition
between phonemes occurs more rapidly in synthetic speech. AlBadawy et al. [1]
showed that synthetic speech contains specific and unusual higher-order
spectral correlations that are not typically found in human speech.

Moving beyond these statistical approaches, more recent approaches have
incorporated explicit vocal and perceptual models. Blue et al. [4] employed
fluid-dynamic models to estimate the arrangement of the vocal tract during
speech generation, and argued that synthetic speech yields unlikely anatomical
structures. Li et al. [18] compared $16$ physical and perceptual features for
synthetic audio detection and highlighted the importance of perceptual
features. They found that in noisier conditions where the quality of the
synthetic audio is low, the perceptual linear prediction technique [12], which
combines spectral analysis with linear prediction analysis, outperforms other
features. They also analyzed the distribution of these features for real and
synthetic speech, providing useful benchmarks for selecting discriminative
features.

Variations in prosody have also been used to detect synthetic audio. For
example, Attorresi et al. [3] combined a speaker embedding representing
distinct voice features (e.g., timbre and pitch contour) with a prosodic
embedding representing variational style (e.g., rhythm and intonation). Their
experiments on the ASVspoof19 dataset show that a combination of these two
embeddings yields a $3-15$ percentage point improvement in equal error rate
(EER) over baseline models (RawNet2, MFCC-ResNet, Spec-ResNet).

End-to-end deep learning has also been deployed to identify synthetically-
generated speech. Muller et al. [19], for example, evaluated the
generalizability of various deepfake detection algorithms of $12$ end-to-end
architectures, and tested them on a novel in-the-wild (IWA) dataset of public
figures collected from social networks and video-streaming
platforms444https://deepfake-demo.aisec.fraunhofer.de/in_the_wild. They
observed that the raw audio-based end-to-end models outperformed the feature-
based models, with the RawNet2 model proposed by Tak et al. [24] achieving the
lowest equal error rate (EER) of $3.2\%$ on the ASVspoof19 dataset and an EER
of $33.9\%$ on the IWA dataset (with chance performance at $50\%$).

Lastly, Pianese et al. [21] evaluated the use of various off-the-shelf speaker
verification tools for synthetic voice detection and found them effective and
robust to intentional and unintentional laundering (e.g., transcoding,
resampling, etc.). This approach yielded an average EER of $15.0\%$ on the
ASVspoof19, FakeAVCeleb, and IWA datasets.

Most forensic approaches seek to distinguish real from synthetic voices
regardless of identity. A more personalized biometric approach can also be
taken in which a person’s distinct voice characteristics are used to
distinguish the real from the fake [21].

Beyond classifying speech as synthetic or real, recent efforts have also
focused on identifying fingerprints that can identify specific synthesis
architectures [26]. And, although somewhat outside of the scope of our work,
there has also been an effort to detect audio spoofing in the form of a
rebroadcast attack in which a person’s voice is recorded and replayed [25,
24].

We take a hybrid approach in terms of the audio features–leveraging learned,
spectral, and perceptual features–and in terms of considering both single-
speaker (personalized) detectors and multi-speaker (non-personalized)
detectors. We evaluate our detectors on a number of real and cloned voices and
evaluate the vulnerability to standard laundering attacks.

SINGLE-SPEAKER
---
Type | Name | Clips (#) | Duration (sec)
Real | LJSpeech | $13{\small,}100$ | $86{\small,}117$
Synthetic | WaveFake | $91{\small,}700$ | $603{\small,}081$
| ElevenLabs | $13{\small,}077$ | $78{\small,}441$
| Uberduck | $13{\small,}094$ | $83{\small,}322$
MULTI-SPEAKER
Type | Name | Clips (#) | Duration (sec)
Real | TIMIT | $4{\small,}620$ | $14{\small,}192$
Synthetic | ElevenLabs | $5{\small,}499$ | $15{\small,}413$

TABLE I: An overview of the real and synthetic datasets used in our single-
speaker (top) and multi-speaker (bottom) evaluations. The $91{\small,}700$
WaveFake samples correspond to $13{\small,}100$ samples per each of seven
different vocoder architectures, hence the larger number of clips and
duration. REAL
---
SYNTHETIC
Figure 1: Example real audio (top) and synthetic audio (bottom) temporal
waveforms (each normalized into the amplitude range $[-1,1]$) for the same
utterance. Note the difference in the length of the silences and the
differences in overall amplitude and amplitude modulation over time.

##  II Datasets

A selection of publicly available datasets was used to develop and test our
models (see Table I). For the evaluation of single-speaker detection, the
LJSpeech [14] and WaveFake datasets [10] were used. The LJSpeech dataset is a
publicly available dataset consisting of $13{\small,}100$ short audio clips of
a single female speaker, Linda Johnson, reading passages from seven non-
fiction books. The WaveFake dataset555https://github.com/RUB-SysSec/WaveFake
comprises $117{\small,}985$ audio clips generated from the LJSpeech dataset
using seven different vocoder architectures. Linda Johnson’s voice was cloned
from the LJSpeech dataset using the leading commercial text-to-speech (TTS)
platforms ElevenLabs and Uberduck. Each transcript from the LJSpeech corpus
was re-generated in the cloned voice.

For the evaluation of our perceptual features (Section III) and multi-speaker
detection (Section IV-C), we used the TIMIT dataset[11], consisting of $462$
real male and female American-English speakers, uttering a total of
$1{\small,}718$ different phonetically-rich sentences [11]. Each of these
phrases was fed to ElevenLabs with one of $11$ distinct voices: nine of the
voices were built into ElevenLabs, and we cloned the remaining two voices to
mimic Presidents Biden and Obama using $1{\small,}038$ and $1{\small,}192$
seconds of audio recordings. The resulting dataset provided a diverse range of
real and synthesized voices with a one-to-one correspondence of the underlying
utterances. To ensure balanced representation, utterances with only one human
speaker were removed from the dataset, and the remaining audio clips were
randomly sampled to select clips with the greater count of the real or
synthetic voice per utterance. This process yielded a total of $763$ real and
$763$ synthesized audio clips. Lastly, each real and synthesized audio was
normalized into the amplitude range $[-1,1]$.

All audio files were downsampled to 16khz and the seven WaveFake architectures
were randomly sampled such that the total number of WaveFake clips were equal
to that of Uberduck and ElevenLabs.

##  III Methods

We describe three approaches for classifying speech as synthetic or real
(single class), and for identifying the underlying synthesis architecture
(multi class). These approaches range from low-dimensional (and interpretable)
handcrafted features to higher-dimensional generic spectral audio features, to
even higher-dimensional (and less interpretable) learned neural features. The
next three sections describe these features followed by a description of a
simple classifier that ingests these features for the purpose of single- and
multi-class classification.

###  III-A Perceptual

Shown in Fig. 1 is a pair of real (top) and synthetic (bottom) waveforms (each
normalized into the amplitude range $[-1,1]$) for the same utterance (“nuclear
rockets can destroy airfields with ease”) from which we can see some
qualitative differences. For the same utterance, the real human voice shows a
lower average normalized amplitude and higher amplitude variability. And, we
observe that real voices exhibit more frequent and noticeable pauses between
certain words. Using the real and fake TIMIT dataset, and as described next,
we designed a set of handcrafted features to determine if these simple
temporal-domain observations would yield reliable classification between real
and synthetic audio.

Pause: A pause in the temporal waveform is identified as a segment of audio
with $100$ consecutive samples with a rolling average amplitude less than
$0.5\%$ of the maximum normalized amplitude (all audios are normalized into
the range $[-1,1]$).

The mean/standard deviation of pause length (as a percentage of the audio
length) for real and synthetic audio contained within the TIMIT dataset is
$27.27/8.49$ and $13.57/6.56$. A two-sided t-test reveals a strong statistical
difference in these distributions ($p\ll 10^{-10}$).

We quantify these differences by extracting four summary statistics from the
identified pauses: the pause ratio (the ratio of pauses relative to the length
of the audio), the mean pause length (specified as the number of samples), the
standard deviation of pause length, and the number of pauses (the number of
pauses, of course, depends on the number of words per utterance, but our
training dataset consisted of the same utterances for both real and synthetic
audio).

Amplitude: Two amplitude features are extracted capturing the consistency and
variation in voices. To begin, the absolute values of each waveform are
temporally smoothed with a fifth-order Butterworth low-pass filter. From this
smoothed waveform, we compute the overall mean amplitude and mean amplitude of
the temporal derivative. The mean/standard deviation of mean amplitude for
real and synthetic audio contained within the TIMIT dataset is $0.06/0.02$ and
$0.10/0.02$ ($p\ll 10^{-10}$), again showing a significant difference.

###  III-B Spectral

For generic spectral features, we employed the openSMILE library (speech &
music interpretation by large-space extraction) [9]. For an arbitrary-length
audio clip, openSMILE generates $6{\small,}373$ scalar-valued features such as
summary statistics (mean, standard deviation, etc.), regression coefficients,
linear predictive coding coefficients, and various peak-related functionals. A
simple dimensionality reduction (SelectFromModel666https://scikit-
learn.org/stable/modules/generated/sklearn.feature_selection.SelectFromModel.html)
was used to reduce the number of features to $20$.

###  III-C Learned

For the end-to-end learned audio features, we employed Nvidia’s open-source
TitaNet model [17]. TitaNet was initially trained for speaker identification
using end-to-end additive margin angular loss, which enhances the separation
of speaker identity in the latent space. Using an encoder-decoder
architecture, TitaNet converts 16KHz sampled raw audio files into $192$-D
embeddings. We treat these embeddings as features for the downstream
classification task.

###  III-D Classification

For each of the three feature sets described above, we employed a linear
logistic regression and a non-linear random forest classifier for a single-
class (real vs. synthetic) or multi-class (real vs. specific synthesis
architecture) task. In each case, the full data set was split into a $60\%$
training, $20\%$ validation (for hyper-parameter tuning), and $20\%$ testing.
All results below are for the testing portion of the dataset.

##  IV Results

We describe classification accuracy for a personalized, single-speaker task in
which a classifier is trained on learned, spectral, or perceptual features for
a single-speaker identity. We next describe the generalization of these
classifiers to a multi-speaker task in which a classifier is trained across
multiple speakers. The classifiers are evaluated against the generated voices,
and laundered voices. Lastly, we compare our results to a ElevenLabs’
detector.

SINGLE-SPEAKER
---
Dataset | Model | Synthetic Accuracy (%) | Real Accuracy (%) | EER (%)
|  | Learned | Spectral | Perceptual | Learned | Spectral | Perceptual | Learned | Spectral | Perceptual
EL | single (L) | 100.0 | 99.2 | 78.2 | 100.0 | 99.9 | 72.5 | 0.0 | 0.5 | 24.9
| single (NL) | 100.0 | 99.9 | 82.2 | 100.0 | 100.0 | 80.4 | 0.0 | 0.1 | 18.6
UD | single (L) | 99.8 | 98.9 | 51.9 | 99.9 | 98.9 | 54.0 | 0.1 | 1.1 | 47.2
| single (NL) | 99.7 | 99.2 | 54.4 | 99.9 | 99.0 | 56.5 | 0.2 | 0.9 | 44.5
WF | single (L) | 96.5 | 78.4 | 57.8 | 97.1 | 82.3 | 45.6 | 3.3 | 19.7 | 48.5
| single (NL) | 94.5 | 87.6 | 50.3 | 96.7 | 90.2 | 52.7 | 4.4 | 11.2 | 48.6
EL+UD | single (L) | 99.7 | 94.8 | 63.4 | 99.9 | 97.1 | 60.3 | 0.2 | 4.2 | 37.9
| single (NL) | 99.7 | 99.2 | 57.3 | 99.9 | 99.6 | 69.0 | 0.2 | 0.8 | 37.6
EL+UD+WF | single (L) | 93.2 | 79.7 | 58.4 | 98.7 | 93.0 | 57.6 | 3.6 | 15.9 | 42.1
| single (NL) | 91.2 | 90.6 | 53.1 | 99.0 | 94.1 | 64.7 | 4.1 | 7.9 | 41.6
EL+UD | multi (L) | 99.9 | 96.6 | 61.0 | 100.0 | 94.6 | 35.7 | - | - | -
| multi (NL) | 99.7 | 98.3 | 65.6 | 100.0 | 97.2 | 43.2 | - | - | -
EL+UD+WF | multi (L) | 98.8 | 80.2 | 45.1 | 97.3 | 64.3 | 22.9 | - | - | -
| multi (NL) | 98.1 | 94.2 | 48.6 | 96.3 | 84.4 | 27.6 | - | - | -
SINGLE-SPEAKER: ADVERSARIAL LAUNDERING
Dataset | Model | Synthetic Accuracy (%) | Real Accuracy (%) | EER (%)
|  | Learned | Spectral | Perceptual | Learned | Spectral | Perceptual | Learned | Spectral | Perceptual
EL | single (L) | 95.5 | 94.3 | 61.1 | 94.5 | 92.6 | 65.2 | 4.9 | 6.7 | 36.6
| single (NL) | 96.0 | 96.2 | 70.4 | 95.4 | 95.6 | 69.6 | 4.1 | 4.1 | 30.1
UD | single (L) | 95.4 | 81.1 | 61.4 | 91.8 | 84.3 | 44.7 | 6.3 | 17.3 | 46.7
| single (NL) | 95.4 | 86.8 | 52.9 | 93.3 | 86.1 | 55.9 | 5.5 | 13.6 | 45.6
WF | single (L) | 87.6 | 60.7 | 59.6 | 85.0 | 70.4 | 42.5 | 13.9 | 34.4 | 49.4
| single (NL) | 83.6 | 77.1 | 51.4 | 85.6 | 76.7 | 53.9 | 15.3 | 23.1 | 47.3
EL+UD | single (L) | 95.2 | 79.1 | 54.0 | 91.7 | 78.4 | 59.8 | 6.2 | 21.3 | 43.1
| single (NL) | 94.8 | 86.1 | 55.2 | 93.3 | 90.0 | 62.4 | 6.0 | 12.0 | 41.4
EL+UD+WF | single (L) | 83.7 | 70.9 | 50.6 | 88.6 | 72.9 | 59.7 | 13.2 | 28.2 | 44.8
| single (NL) | 83.4 | 79.2 | 53.0 | 90.7 | 85.1 | 60.7 | 12.5 | 17.9 | 43.6
EL+UD | multi (L) | 94.2 | 85.6 | 50.9 | 91.0 | 77.1 | 29.1 | - | - | -
| multi (NL) | 94.5 | 91.7 | 53.2 | 90.3 | 82.9 | 41.3 | - | - | -
EL+UD+WF | multi (L) | 89.8 | 65.4 | 35.3 | 83.1 | 44.3 | 26.2 | - | - | -
| multi (NL) | 88.8 | 78.8 | 39.8 | 82.1 | 63.0 | 28.6 | - | - | -
MULTI-SPEAKER
Dataset | Model | Synthetic Accuracy (%) | Real Accuracy (%) | EER (%)
|  | Learned | Spectral | Perceptual | Learned | Spectral | Perceptual | Learned | Spectral | Perceptual
EL | single (L) | 100.0 | 94.2 | 83.9 | 99.9 | 98.3 | 86.9 | 0.0 | 3.0 | 13.1
| single (NL) | 92.3 | 96.3 | 82.2 | 100.0 | 99.7 | 87.7 | 0.1 | 1.6 | 13.7

TABLE II: Accuracy for a personalized, single-speaker classification of
unlaundered audio (top) and audio subject to adversarial laundering in the
form of additive noise and transcoding (middle). Shown in the bottom table is
the non-personalized, multi-speaker accuracy. Dataset corresponds to
ElevenLabs (EL), Uberduck (UD), and WaveFake (WF); Model corresponds to a
linear (L) or non-linear (NL) classifier, and for a single-classifier (real v.
synthetic) or multi-classifier (real vs. specific synthethis architecture;
accuracy (%) is reported for synthetic audio, real audio, and (for the single-
classifiers) equal error rate (EER).

###  IV-A Single Speaker

Shown in Table II (top panel) is the accuracy for distinguishing real from
synthetic audio (model: single) and real from specific synthetic audio
architecture (model: multi) using a linear (model: L) and non-linear (model:
NL) classifier, evaluated against single or multiple datasets (ElevenLabs
[EL], Uberduck [UD], WaveForm [WF]). Each column corresponds to the accuracy
for correctly classifying real and synthetic audio using the learned,
spectral, or perceptual features. The far-right columns report the equal error
rate (the EER is the point on the receiver operating curve (ROC) where the
false acceptance rate (incorrectly classifying a synthetic voice as real) and
false rejection rate (incorrectly classifying a real voice as synthetic) are
equal).

As expected, the non-linear classifier generally affords better accuracy. For
the spectral features, for example, across all dataset combinations the non-
linear classifiers afford an average $4.1$ percentage point reduction in EER.

Accuracy on the learned features outperforms the spectral and perceptual
features, with an average EER on single datasets (and linear classifier) of
$0.0\%$, $0.1\%$, and $3.3\%$ for the learned features as compared to $0.5\%$,
$1.1\%$, and $19.7\%$ for the spectral features, and $24.9\%$, $47.2\%$, and
$48.5\%$ for the perceptual features.

Generally speaking, classifiers trained and tested on a single dataset (EL,
UD, or WF) perform better than those trained on two or more datasets. And,
accuracy on the single-class task is higher than on multi-class.

###  IV-B Laundering

To test the robustness of our methods against unintentional or intentional
adversarial laundering attacks, we split our real and synthetic datasets into
four equal classes consisting of the unlaundered audio, the unlaundered audio
corrupted with additive Gaussian noise with an SNR sampled uniformly between
$10$ and $80$dB, the unlaundered audio transcoded (AAC) at a bitrate of 64K,
127K, or 196K, and the unlaundered audio transcoded and corrupted with noise.
Shown in Table II (middle panel) are the resulting classification accuracies
in the same format as described above.

As expected, laundering degrades classification accuracy. The spectral
features were particularly impacted which is perhaps not surprising since the
additive noise and transcoding introduce broad-band spectral distortions.

As compared to the unlaundered voices, the EER for the learned features jumps
by $7.5$ percentage points for the linear classifier and $6.9$ percentage
points for the non-linear classifier.

###  IV-C Multi Speaker

The above results are based on personalized classifiers trained to distinguish
real from synthetic audio for a specific individual. Shown in the lower panel
of Table II is the accuracy for a multi-speaker classifier trained and tested
on the TIMIT-ElevenLabs dataset. This classifier is trained to detect
synthetic voices regardless of the underlying identity. The learned features
yield similar EER as compared to single speaker and the spectral EER is only
slightly higher. The perceptual features, on the other hand, yield a lower EER
dropping from $18.6\%$ to $13.7\%$ (for the nonlinear classifier). We
hypothesize that this improvement is because the cadence for the single
speaker (LJ) as she is reading is highly structured, as compared to a more
conversational style. Regardless, these results imply that our features are
not speaker specific, but seem to capture synthesis artifacts regardless of
identity.

###  IV-D Comparison

ElevenLabs recently released a classifier designed to determine if an audio
sample was generated by their synthesis
engine777https://beta.elevenlabs.io/blog/ai-speech-classifier. With a reported
accuracy of ${\small>}99\%$ accuracy for unlaundered samples and
${\small>}90\%$ accuracy for laundered samples, this classifier is on par with
our classifier based on learned features (Table II, top and middle panels, row
EL). We tested the ElevenLabs classifier on a random sample of $50$ real and
$50$ ElevenLabs synthesized audio samples, each laundered with additive
Gaussian noise and transcoded at varying compression levels (see Section
IV-B). Classification accuracy was perfect, as compared to our average
accuracy of $95.8\%$ using the learned features and non-linear classifier.
Despite this slightly lower performance, our classifier, unlike the ElevenLabs
classifier, can detect samples from other synthesis engines: we verified that
ElevenLabs mis-classifies synthetically-generated audio from Uberduck and
WaveFake.

Although comparison to other published techniques is difficult due to
differences in the underlying training and tresting datasets, generally
speaking we achieve lower or equal EERs to the techniques described in Section
I-A.

##  V Discussion

In the field of digital forensics, image- and video-based techniques have
outpaced those of audio forensics. And for good reason. Until fairly recently
synthetic voices were not particularly natural or identity-preserving. This,
however, is no longer the case and it is now possible to create highly natural
and realistic voices from only a few minutes of a person’s voice. When coupled
with increasingly high-quality deepfake videos, it is quickly becoming
possible to create highly realistic deepfake videos of anyone saying anything.

Combining video and audio analyses (e.g., [5]) offers the advantage of a
richer data source and more chances to detect statistical or semantic
inconsistencies. Purely audio-based techniques, however, are needed to contend
with phone-based scams and fake leaked audios of world or corporate leaders.

While low-dimensional, interpretable features are attractive, it is clear that
the end-to-end learned features afford better discrimination. We did not
combine all three features because the learned features significantly
outperformed the others.

The advantage of a single-speaker approach is that it can learn highly
specific and distinct speaking styles that are difficult for a synthetic voice
to perfectly mimic. The drawback is that, unlike multi-speaker techniques, it
does not scale well to protect a large number of possible victims of voice
cloning. We see the need for both single- and multi-speaker approaches. Our
results suggest that the same underlying feature selection and classification
can be adapted for both tasks.

As new voice synthesis architectures emerge, it will be important for forensic
techniques to generalize across new architectures. Our results suggest that
this type of generalization is possible, but that performance generally
degrades as the classifier is tasked with categorizing voices from
increasingly more diverse synthesis architectures. To the extent that the goal
is to distinguish real from synthetic voices, a single-class approach can be
taken. It may be informative, however, to also refine multi-class approaches
in which the classifier is able to specify which synthesis architecture was
used to generate a fake voice; such information could be useful in tracking
down the source of disinformation campaigns or illegal activities.

As our field continues to develop techniques for distinguishing real from fake
content, we encourage those on the synthesis side to help mitigate potential
abuse from deepfakes by embedding imperceptible watermarks into synthetically
generated content (see, for example, Adobe’s Content Authenticity
Initiative888https://contentauthenticity.org). While this is not a panacea,
it, along with the types of forensic techniques described here, will take us a
long way to mitigating abuse from AI-generated content.

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