Gravity as an ensemble and the moment problem
Oliver Janssen⋆,† and Mehrdad Mirbabayi⋆

⋆International Centre for Theoretical Physics
†Institute for Fundamental Physics of the Universe
Trieste, Italy
including an appendix with

Peter Zograf

St. Petersburg Department of the Steklov Mathematical Institute
St. Petersburg State University
St. Petersburg, Russia

Abstract: If a bulk gravitational path integral can be identified with an
average of partition functions over an ensemble of boundary quantum theories,
then a corresponding moment problem can be solved. We review existence and
uniqueness criteria for the Stieltjes moment problem, which include an
infinite set of positivity conditions. The existence criteria are useful to
rule out an ensemble interpretation of a theory of gravity, or to indicate
incompleteness of the gravitational data. We illustrate this in a particular
class of 2D gravities including variants of the CGHS model and JT
supergravity. The uniqueness criterium is relevant for an unambiguous
determination of quantities such as $\overline{\log Z(\beta)}$ or the quenched
free energy. We prove in JT gravity that perturbation theory, both in the
coupling which suppresses higher-genus surfaces and in the temperature, fails
when the number of boundaries is taken to infinity. Since this asymptotic data
is necessary for the uniqueness problem, the question cannot be settled
without a nonperturbative completion of the theory.

##  1 Introduction

Long ago Coleman argued that microscopic Euclidean wormholes result in an
ensemble of low-energy effective field theories [1]. A variant of this idea
could explain the factorization puzzle in the context of holography [2]:
consider a bulk path integral with two identical boundaries, which we denote
by $B^{2}\equiv B\sqcup B$ where $\sqcup$ means disjoint union. Under the
conventional holographic dictionary this must be computing the partition
function of two copies of the dual quantum theory living on $B$, i.e.
$Z(B)^{2}$. However, if there are wormhole geometries that connect the two
boundaries, the answer differs from the square of the gravity result for
$Z(B)$.

There would be no puzzle if gravitational path integrals were to compute an
ensemble average over boundary duals:

$\mathcal{P}(B)\equiv\sum_{M}\int_{\partial
M=B}\mathcal{D}\Phi~{}e^{-S_{E}}=\overline{Z(B)}\,,$ (1)

where $M$ are bulk manifolds and $\Phi$ the collection of all bulk fields. Now
it is natural to expect

$\mathcal{P}(B^{2})=\overline{Z(B)Z(B)}\neq\mathcal{P}(B)^{2}\,.$ (2)

Perhaps it is more natural to attribute such puzzles to our poor understanding
of gravity, rather than a need to modify the dual quantum description. For
instance, it is not clear if additional stringy ingredients in the
conventional examples of holography could not cancel the undesired effect of
wormholes.

However, in recent years Jackiw-Teitelboim (JT) gravity [3, 4], a two-
dimensional dilaton-gravity model, has emerged as an example of such an
ensemble duality. First, as describing a particular low-energy sector of the
Sachdev-Ye-Kitaev (SYK) model. SYK is a system of fermions with random
couplings [5, 6, 7], and several computations in JT gravity coupled to
additional matter fields have been shown to agree with disorder averaged
quantities in SYK [8, 9].

Pure JT gravity was later shown to be exactly dual, to all orders in
perturbation theory, to a matrix ensemble [10]. That is, the JT path integral
with $n$ boundaries of lengths $\beta_{1},\beta_{2},\cdots,\beta_{n}$ can be
thought of as a particular limit, a so-called double scaling (DS) limit, of
the following correlation functions in a model of $L\times L$ random Hermitian
matrices:

$\mathcal{P}_{\textsf{JT}}(\beta_{1},\beta_{2},\cdots)=\overline{Z(\beta_{1})Z(\beta_{2})\cdots}=\underset{\textsf{DS}}{\lim}\int\mathrm{d}H~{}e^{-L\,\text{Tr}\,V(H)}\,\text{Tr}\exp\left(-\beta_{1}H\right)\text{Tr}\exp\left(-\beta_{2}H\right)\cdots\,.$
(3)

In the double scaling limit one takes $L\rightarrow\infty$ and focusses on the
edge of the spectrum where the density of eigenvalues is controlled by a
parameter $e^{S_{0}}$, which appears in the JT path integral as a coupling
constant. The correspondence (3) holds then to all orders in $e^{-S_{0}}$.
These results motivated exploring further the possibility of ensemble
holography. Indeed several other explicit 2D [11, 12, 13, 14, 15] and 3D [16,
17] examples have been proposed, as well as hints of such a duality for richer
bulk theories [18].

In this context, an interesting question was recently raised in [19]. Assuming
an ensemble interpretation of gravity, can we use the knowledge of
$\overline{Z(\beta)^{n}}$ for integer $n$, which we know how to compute in
gravity, to extract quantities for which there is no direct gravitational
recipe? An important example is the quenched free energy

$F_{q}(\beta)=-\frac{1}{\beta}\,\overline{\log Z(\beta)}\,.$ (4)

At low temperature $\beta\to\infty$, $F_{q}(\beta)$ significantly deviates
from its simpler counterpart, the annealed free energy
$F_{a}(\beta)=-(1/\beta)\log\overline{Z(\beta)}$. Hence computing (4) is
necessary in order to determine low-temperature thermodynamic quantities. In
this case the standard approach is the replica method, i.e. analytic
continuation in $n$:

$\overline{\log Z(\beta)}=\lim_{n\to
0}\frac{1}{n}\left(\overline{Z(\beta)^{n}}-1\right).$ (5)

The authors of [19] studied two explicit 2D models of gravity
($\widehat{\text{CGHS}}$ , see §3.1, and JT, see §3.2) and emphasized that
finding the “correct” analytic continuation is not straightforward, involving
perhaps the phenomenon of replica symmetry breaking. Indeed, as noted in [19],
without further input the analytic continuation is ambiguous: if
$f_{\beta}(z)$ is an analytic continuation of $\overline{Z(\beta)^{n}}$ away
from the integers, then $F_{\beta}(z)=f_{\beta}(z)+g_{\beta}(z)\sin(\pi z)$
for any (entire) function $g_{\beta}$ is as well. Moreover computing the limit
in (5) with $f_{\beta}$ gives $f_{\beta}^{\prime}(0)$ while computing it with
$F_{\beta}$ gives $f_{\beta}^{\prime}(0)+\pi g_{\beta}(0)$. So to compute
$\overline{\log Z(\beta)}$ via the replica trick requires picking a preferred
analytic continuation. At this stage it is unclear how this should be done, or
indeed if it can be.

Our goal here is to point out the relevance of the moment problem to the
general discussion of an ensemble interpretation of gravity, and the
particular question about averages of non-analytic functions of $Z(B)$ such as
the quenched free energy. Suppose we choose the boundary geometry $B$ such
that $Z(B)\geq 0$ for a unitary theory. This would for instance be the case if
$Z(B)$ were the thermal partition function, in which case we would denote it
by $Z(\beta)$ as we have above. If gravity computes an ensemble average over
unitary theories, then the sequence

$(\mathcal{P}_{n})_{n\geq
0}\equiv\left(1,\mathcal{P}(B),\mathcal{P}(B^{2}),\cdots\right)$ (6)

represents the integer moments of a measure $\mu_{B}$ on the positive real
axis:

$\mathcal{P}(B^{n})=\overline{Z(B)^{n}}\equiv\int_{0}^{\infty}\mathrm{d}\mu_{B}(Z)\,Z^{n}.$
(7)

This implies an infinite set of consistency conditions on this sequence,
including

$\mathcal{P}(B)=\overline{Z(B)}\geq
0,\qquad\mathcal{P}(B^{2})-\mathcal{P}(B)^{2}=\overline{\left(Z(B)-\overline{Z(B)}\right)^{2}}\geq
0,\qquad\cdots$ (8)

If such a measure exists, it may or may not be unique. For example,
$(n+1)_{n\geq 0}=(1,2,3,\cdots)$ is not the moment sequence of any measure on
$\mathbb{R}$ (let alone on $\mathbb{R}^{+}$), the moments $(n!)_{n\geq 0}$ are
unique, among all measures with support on $\mathbb{R}$, to the measure
$\mathrm{d}\mu(x)=\mathrm{d}x\,e^{-x}$ which has support on $\mathbb{R}^{+}$
only, and the measures [20]

$\mathrm{d}\mu(x)=\mathrm{d}x\frac{1}{\sqrt{2\pi}\,x}\exp\left[-\frac{1}{2}\left(\log
x\right)^{2}\right]\left[1+\lambda\,\sin\left(2\pi\log
x\right)\right]\,,~{}~{}~{}\lambda\in[-1,1]$ (9)

with support on $\mathbb{R}^{+}$ all have the same moment sequence
$\left(e^{n^{2}/2}\right)_{n\geq 0}$. Necessary and sufficient conditions on
$(\mathcal{P}_{n})_{n\geq 0}$ such that a measure exists and is unique are
reviewed in §2.

As an application of these conditions, we show in §3.1 that $\mu_{B}$ does not
exist for the multi-boundary partition functions of the
$\widehat{\text{CGHS}}$ model, a particular 2D dilaton-gravity theory. The
same conclusion holds more generally for any gravitational theory in which the
connected correlator $\mathcal{P}(B^{n})_{\textsf{conn}}$ vanishes for all but
finitely many $n$. Thus either $\widehat{\text{CGHS}}$ gravity is not dual to
an ensemble of unitary quantum theories, or its prediction for
$\mathcal{P}(B^{n})$ is incomplete. This latter possibility is indeed realized
in two examples of JT supergravity as we will further discuss in §3.1.

On the other hand if we know that $\mu_{B}$ exists, whether or not there is an
unambiguous answer for $F_{q}$ (given the gravitational data) depends on the
uniqueness of $\mu_{B}$. This can be inferred from the data
$\overline{Z(\beta)^{n}}$ on the integers alone, i.e. from the moments of
$\mu_{B}$ (see Theorem 2 in §2). The auxiliary machinery of analytic
continuation required by the replica trick (5) is immaterial: when $\mu_{B}$
is unique by moments then unambiguously

$\overline{\log Z}=\int_{0}^{\infty}\mathrm{d}\mu_{B}(Z)\,\log(Z)\,,$ (10)

and when $\mu_{B}$ is not unique by moments then uniquely determining
$\overline{\log Z}$ from the $\overline{Z^{n}}$ is not possible. In the latter
case there does not exist a preferred analytic continuation in (5); the limit
is ambiguous and depends on what exactly is meant by “average” on the left-
hand side of (5). In the former case we should note that we have merely proven
that the unique determination of $\overline{\log Z}$ (or more generally any
$\overline{f(Z)}$) from the $\overline{Z^{n}}$ is possible in principle. We
have not given an algorithm which computes $\overline{\log Z}$ from the
$\overline{Z^{n}}$; reconstructing a measure with infinite support from its
moments is a difficult task in general (e.g. [21]).

The above motivates the study of the uniqueness problem in the case of JT
gravity, where the ensemble dual is known to exist (§3.2). However, this
requires information about the asymptotic behavior of the moments, namely
$\overline{Z(\beta)^{n}}$ as $n\to\infty$. This is not available in the
current, perturbative formulation of JT. We will show in the appendix that
both the perturbative genus expansion in $e^{-S_{0}}$, and the low-temperature
expansion in $1/\beta$ break down as $n\to\infty$. As a result, without a
nonperturbative definition of the theory it is not possible to establish the
uniqueness of $\mu_{\beta}$. This nonperturbative definition (of JT or its
dual matrix integral) has in turn been argued in [10] to be non-unique.

##  2 The Stieltjes moment problem

Among the questions posed in the “Stieltjes moment problem” are the following:
given a sequence of real numbers $(\mathcal{P}_{n})_{n\geq 0}$, when does
there exist a (non-negative Borel) measure on the positive real axis for which
$\mathcal{P}_{k}$ is its $k$th moment, and when is this measure unique? These
questions were formulated and answered (in some form) by Stieltjes [20] in
1894, and are still an active research topic today. Among other things one is
interested in necessary or sufficient conditions for the existence or
uniqueness of a solution (famous sufficient (non-)uniqueness conditions
include Carleman’s condition [22], reviewed in Theorem 3 below, and Krein’s
condition [23]), but also in questions such as how to characterize the set of
measures with equal moments (see e.g. [24, 25, 26, 27]). There is a large body
of literature on this topic: two classic references are [21, 28], a recent
book is [29] and a review of checkable111I.e. those that are formulated
directly in terms of a given moment sequence or – for non-uniqueness results –
a given solution; these are most useful for our problem (1). (non-)uniqueness
criteria is [30]. Extensions of the original Stieltjes moment problem exist
where the measure is restricted to have support on a given closed subset of
$\mathbb{R}$, as do generalizations to multisequences and multivariate
measures with support on a closed subset of $\mathbb{R}^{n}$. The moment
problem on $\mathbb{R}$ is known as the Hamburger moment problem while the
moment problem on an interval is known as the Hausdorff moment problem.

Here we simply collect three theorems – necessary and sufficient conditions
for the existence and uniqueness of a solution to a univariate Stieltjes
moment problem – that mathematicians have proven (refs. [20], [31, 32] and
[22] respectively). Some definitions first: we will further abbreviate
$\mathcal{P}\equiv(\mathcal{P}_{0},\mathcal{P}_{1},\mathcal{P}_{2},\cdots)$
and define the shifted sequence
$\textsf{E}\mathcal{P}\equiv(\mathcal{P}_{1},\mathcal{P}_{2},\mathcal{P}_{3},\cdots)$.
Corresponding to a sequence $\mathcal{P}$ we define for each positive integer
$N$ the symmetric $N\times N$ Hankel matrix $H_{N}(\mathcal{P})$ where
$(H_{N})_{ij}=\mathcal{P}_{i-1+j-1},1\leq i,j\leq N$. Finally $\mathcal{P}$ is
called positive semidefinite if $H_{N}(\mathcal{P})$ is positive semidefinite
for all $N$, which we denote by $\mathcal{P}\succeq 0$.

######  Theorem 1 (necessary and sufficient for existence)

A sequence $\mathcal{P}$ corresponds to the
moments of a measure on the positive real axis if and only if $\mathcal{P}$
and $\textsf{E}\mathcal{P}$ are positive semidefinite, i.e.

$\mathcal{P}\succeq
0~{}~{}~{}and~{}~{}~{}\textnormal{$\textsf{E}\mathcal{P}$}\succeq 0\,.$ (11)

If $\mathcal{P}$ satisfies Theorem 1, it is called a “Stieltjes moment
sequence”. As two examples of these positivity conditions, consider the
positivity of $H_{2}(\mathcal{P})$ and $H_{2}(\textsf{E}\mathcal{P})$. Setting
$\mathcal{P}_{0}=1$ the former states
$\mathcal{P}_{2}-(\mathcal{P}_{1})^{2}\geq 0$. If
$\mathcal{P}_{1}=\overline{x}$ and $\mathcal{P}_{2}=\overline{x^{2}}$ are the
first and second moments of a (positive) measure then this inequality is
satisfied because
$\mathcal{P}_{2}-(\mathcal{P}_{1})^{2}=\overline{(x-\overline{x})^{2}}$, the
variance, is the expectation value of a positive random variable. More
generally $\overline{p(x)^{2}}$ must be positive for any polynomial $p$, which
is equivalent to the statement that $\mathcal{P}$ is positive semidefinite
(e.g. [29]). From $H_{2}(\textsf{E}\mathcal{P})\succeq 0$ we get
$\mathcal{P}_{3}\mathcal{P}_{1}-(\mathcal{P}_{2})^{2}\geq 0$, which must be
positive if the measure is supported on the positive real axis because
$\mathcal{P}_{3}=\overline{x^{3}}$ and therefore
$\mathcal{P}_{3}\mathcal{P}_{1}-(\mathcal{P}_{2})^{2}=\overline{x\left(\overline{x}^{1/2}x-\overline{x^{2}}/\overline{x}^{1/2}\right)^{2}}$.
Requiring that $\overline{x\,q(x)^{2}}$ is positive for all polynomials $q$
generates the positivity conditions on the $H_{N}(\textsf{E}\mathcal{P})$ and
vice versa. For the Hamburger moment problem only $\mathcal{P}$ is required to
be positive semidefinite: a sequence $\mathcal{P}$ corresponds to the moments
of a measure on the whole real axis if and only if $\mathcal{P}$ is positive
semidefinite (Hamburger’s theorem [33]). For the Hausdorff moment problem on
$[0,1]$ we require $\mathcal{P}\succeq 0$ and
$\textsf{E}\mathcal{P}\succeq\textnormal{{E}}(\textsf{E}\mathcal{P})$.

######  Theorem 2 (necessary and sufficient for uniqueness)

A Stieltjes moment sequence $\mathcal{P}$
corresponds to the moments of exactly one measure on the positive real axis if
and only if the smallest eigenvalue of either the $H_{N}(\mathcal{P})$ or the
$H_{N}(\textnormal{$\textsf{E}\mathcal{P}$})$ tends to 0 as
$N\rightarrow\infty$, i.e.

$\lim_{N\rightarrow\infty}\lambda_{\textnormal{{min}}}(H_{N}(\mathcal{P}))=0~{}~{}~{}\text{or}~{}~{}~{}\lim_{N\rightarrow\infty}\lambda_{\textnormal{{min}}}(H_{N}(\textnormal{$\textsf{E}\mathcal{P}$}))=0\,.$
(12)

Notice that for any sequence $\mathcal{P}$, the sequence
$\left(\lambda_{\textsf{min}}(H_{1}(\mathcal{P})),\lambda_{\textsf{min}}(H_{2}(\mathcal{P})),\cdots\right)$
is decreasing. This can be seen from the identity
$\lambda_{\textsf{min}}(H)=\underset{||c||=1}{\text{min}}\\{H_{ij}c_{i}c_{j}\\}$
and the fact that $H_{N}$ is a leading principal submatrix of $H_{N+1}$.

Employing Theorem 2 requires certain control over the smallest eigenvalue of
$H_{N}(\mathcal{P})$ and $H_{N}(\textsf{E}\mathcal{P})$ as
$N\rightarrow\infty$. A similar remark pertains to Theorem 1. However Theorem
1 could be usefully employed to prove that $\mathcal{P}$ is not a moment
sequence; we can, perhaps numerically, verify the (non-)positivity of
$H_{N}(\mathcal{P})$ and $H_{N}(\textnormal{$\textsf{E}\mathcal{P}$})$ for $N$
as large as is feasible.

It is more stringent for a solution to a Hamburger moment problem to be
unique: a Hamburger moment sequence $\mathcal{P}$ corresponds to the moments
of exactly one measure on the whole real axis if and only if the smallest
eigenvalue of the $H_{N}(\mathcal{P})$ tends to 0 as $N\rightarrow\infty$. A
solution to the Hausdorff moment problem is always unique.

######  Theorem 3 (sufficient for uniqueness)

A Stieltjes moment sequence $\mathcal{P}$ corresponds to the moments of
exactly one measure on the positive real axis if

$\sum_{n=1}^{\infty}\mathcal{P}_{n}^{-1/2n}=\infty\,.$

Qualitatively, the moments should not diverge too quickly. Notice that this is
merely a sufficient uniqueness condition, it is not necessary – there can be
no characterization of uniqueness based on the growth rate of the moments (see
[29]). A stronger but more user-friendly result that follows is:

######  Corollary 1 (stronger sufficient uniqueness condition)

A Stieltjes moment sequence $\mathcal{P}$ corresponds to the moments of
exactly one measure on the positive real axis if there exists a positive
constant $c$ such that

$\mathcal{P}_{n}\leq c^{n}(2n)!$ (13)

for all positive integers $n$.

For the Hamburger moment problem it is sufficient that the even order moments
are bounded, but more stringently so: $\mathcal{P}_{2n}\leq d^{n}(2n)!$ for a
$d>0$ and all $n$.

Connected components – In the context of the ensemble interpretation of
gravity, the $\mathcal{P}_{n}$ above are the full $n$-point correlators
$\mathcal{P}(B^{n})$. In terms of the connected components we have the
following sufficient condition for the uniqueness of the measure on $Z(B)$
(assuming it exists in the first place):

$\mathcal{P}_{\textsf{conn},\ell}<c^{\ell}\,\frac{(2\ell)!}{B_{\ell}}~{}~{}~{}\text{for
a }c>0\text{ and all }\ell\,.$ (14)

Here $B_{\ell}$ is the $\ell$th Bell number, which counts the number of
partitions of a set of size $\ell$ into subsets of any size. For the Hamburger
moment problem we have a similar sufficient bound:
$\mathcal{P}_{\textsf{conn},\ell}<d^{\ell}\,\ell!/B_{\ell}$ for a $d>0$ and
all $\ell$.

To see why (14) is sufficient, one can start from the expression of the full
$n$-point correlator in terms of the connected components:

$\mathcal{P}_{n}=\sum_{\sum_{\ell=1}^{n}\ell\,k_{\ell}=n}(\text{counting
factor})~{}~{}~{}\mathcal{P}_{\textsf{conn},1}^{k_{1}}\mathcal{P}_{\textsf{conn},2}^{k_{2}}\cdots\mathcal{P}_{\textsf{conn},n}^{k_{n}}\,,$
(15)

where the counting factor counts the amount of ways that a set of $n$ elements
can be partitioned into $k_{1}$ groups of one, $k_{2}$ groups of two, … and
$k_{n}$ groups of $n$.222A closed-form expression is
$n!/\prod_{i=1}^{n}(k_{i})!(i!)^{k_{i}}$. Since the sequence

$\left(\frac{(2\ell)!}{B_{\ell}}\right)^{1/\ell}\,,~{}~{}\ell=1,2,\cdots$ (16)

is increasing, we deduce from (14) that for all $\ell\leq n$

$\mathcal{P}_{\textsf{conn},\ell}^{k_{\ell}}<c^{\ell
k_{\ell}}\left(\frac{(2\ell)!}{B_{\ell}}\right)^{k_{\ell}}\leq\left(c^{n}\frac{(2n)!}{B_{n}}\right)^{\ell
k_{\ell}/n}\,.$

Using this in (15) leads to (13).

Eq. (14) involves the Bell numbers, which are bounded as follows [34]:

$B_{\ell}<\left(0.792\,\ell/\log(\ell+1)\right)^{\ell}~{}~{}~{}\text{for all
}\ell\in\mathbb{N}\setminus\\{0\\}\,.$ (17)

Using this and $\ell!>\sqrt{2\pi\ell}\,(\ell/e)^{\ell}$ we can write a
stronger but more familiar-looking sufficient condition,

$\exists
c>0:\forall\ell\in\mathbb{N}\setminus\\{0\\}:~{}~{}\mathcal{P}_{\textsf{conn},\ell}^{1/\ell}<c\,\ell\log(\ell+1)\,.$
(18)

Multivariate measures – Finally we mention additional consistency conditions
on the ensemble average interpretation of gravity (1). These can be obtained
by taking $B=\underset{i=1}{\overset{m}{\bigsqcup}}\,B_{i}^{n_{i}}$ in (1) and
considering the multisequence that arises, which is identified with the
moments of a multivariate measure:

$\displaystyle\mathcal{P}\left(\underset{i=1}{\overset{m}{\bigsqcup}}\,B_{i}^{n_{i}}\right)\equiv\mathcal{P}_{n_{1}n_{2}\cdots
n_{m}}=\int_{0}^{\infty}\mathrm{d}\mu_{\sqcup
B}(\boldsymbol{Z})~{}Z_{1}^{n_{1}}Z_{2}^{n_{2}}\cdots Z_{m}^{n_{m}}\,.$ (19)

This sets up a multidimensional moment problem on $(\mathbb{R}^{+})^{m}$
rather than a univariate one on $\mathbb{R}^{+}$. Less is known about the
solution to this considerably more difficult problem, which is tied to the
unknown classification of the non-negative polynomials on
$(\mathbb{R}^{+})^{m}$ for $m>1$ by the Riesz-Haviland theorem [35]. For
example, Stieltjes’ existence theorem (Theorem 1 above) – specifically the
“if” direction – does not generalize to the $m>1$ problem. See [36, 37, 29]
for a discussion of known results, including mention of a necessary existence
criterium (also an infinite set of positivity conditions, called “complete
positivity”) and Nussbaum’s [38] sufficient existence criterium, and [39] for
a collection of (non-)uniqueness criteria. In this last regard it is worth
noting Petersen’s theorem [40], which states that if
$\mathcal{P}_{n_{1}n_{2}\cdots n_{m}}$ is a (Stieltjes) moment multisequence
which is such that all the marginal (moment) sequences
$\mathcal{P}_{n_{1}00\cdots 0},\mathcal{P}_{0n_{2}00\cdots
0},\cdots,\mathcal{P}_{00\cdots 0n_{m}}$ are determinate (i.e. there is a
unique univariate measure with these moments), then
$\mathcal{P}_{n_{1}n_{2}\cdots n_{m}}$ is determinate. The marginals in this
case are all of the type $\left(\mathcal{P}(B^{n})\right)_{n\geq 0}$ for some
choice of $B$. So if we would know that $\mathcal{P}_{n_{1}n_{2}\cdots n_{m}}$
is a Stieltjes moment multisequence for all $m\geq 1$, the determinacy of
these sequences would follow from the determinacy of the $m=1$ sequences (for
all choices of $B$).

##  3 Examples

###  3.1 $\widehat{\text{CGHS}}$ (-like theories)

The $\widehat{\text{CGHS}}$ model [41] is a simplified version of CGHS
dilaton-gravity [42] involving a metric, two scalars and a $U(1)$ gauge field
in two dimensions. The dilaton acts as a Lagrange multiplier in the path
integral, selecting flat geometries. In two dimensions these are the disk and
the cylinder only. The other scalar is constant on-shell and sets the
temperature of black hole solutions. The path integral over all two-geometries
that have a boundary consisting of $n$ disconnected circles, all of equal
length $\beta$, is (from Eq. (15), cf. [19])

$\mathcal{P}_{n}(\beta)=\mathcal{P}_{\textsf{disk}}(\beta)^{n}\sum_{n^{\prime}=0}^{\lfloor
n/2\rfloor}\binom{n}{2n^{\prime}}(2n^{\prime}-1)!!\,r(\beta)^{n^{\prime}}\,,$
(20)

where [43, 44]

$r=\frac{\mathcal{P}_{\textsf{cylinder}}}{\mathcal{P}_{\textsf{disk}}^{2}}\,,~{}~{}\mathcal{P}_{\textsf{disk}}=\frac{2\pi}{\beta^{2}}\,,~{}~{}\mathcal{P}_{\textsf{cylinder}}=\frac{2\pi^{2}}{\beta}\,.$
(21)

We have made a choice of units here for the boundary value of the dilaton and
for the normalization of the symplectic form in [43, 44] which corresponds
with [19].

Since the only connected contributions to (20) are the disk and the cylinder,
we can immediately infer the following Gaussian measure which has the
$\mathcal{P}_{n}(\beta)$ as its moments,

$\displaystyle\mathrm{d}\mu_{\beta}(Z)$
$\displaystyle=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(\frac{-(Z-\mu)^{2}}{2\sigma^{2}}\right)\mathrm{d}Z\,,$
(22) $\displaystyle\mu$
$\displaystyle=\mathcal{P}_{\textsf{disk}}\,,~{}~{}~{}\sigma^{2}=\mathcal{P}_{\textsf{cylinder}}\,.$
(23)

That this measure is unique follows from Corollary 1 in §2 (more precisely,
its variant for the connected correlators and the Hamburger moment problem
stated below (14)). Since this measure has support on negative values of $Z$,
no identification as an ensemble average over unitary theories can be made for
(20).

It follows from Theorem 1 that the sequence
$\mathcal{P}=(\mathcal{P}_{n})_{n\geq 0}$ is positive semidefinite for all
$\beta$, and that $\textsf{E}\mathcal{P}=(\mathcal{P}_{n})_{n\geq 1}$ is not
positive semidefinite for any $\beta$. For example, one may verify that
$H_{2}(\textsf{E}\mathcal{P})$ has a negative eigenvalue for all
$\beta>\beta_{c}\equiv 2^{1/3}$ (see §2 for the definition of the Hankel
matrices), the inverse temperature above which
$\mathcal{P}_{\textsf{cylinder}}>\mathcal{P}_{\textsf{disk}}^{2}$ and
contributions from wormholes start dominating $\mathcal{P}_{n}$. For $N\geq
2$, $H_{N}(\textsf{E}\mathcal{P})$ is positive semidefinite only on
successively smaller intervals
$\cdots\subset[0,\beta_{N+1}]\subset[0,\beta_{N}]\subset\cdots\subset[0,\beta_{c}]$
with $\lim_{N\to\infty}\beta_{N}=0$.

Marcinkiewicz’ theorem [45] states that the Gaussian distributions are the
only probability measures on $\mathbb{R}$ with a polynomial cumulant
generating function, i.e. with only a finite number of nonzero connected
correlators. This implies that gravitational theories which have more than two
but only finitely many non-vanishing connected correlators
$\mathcal{P}_{\textsf{conn},n}$, cannot be dual to an ensemble. In fact in
this case even more general (“nonphysical”) ensembles which allow for negative
values of $Z$ are not allowed.

The above conclusions hold only if the gravity computation for
$\mathcal{P}_{n}$ is reliable. Otherwise, one could only conclude that the
ensemble interpretation implies that the gravitational result is incomplete.
For instance, there are two examples of JT supergravity where there is no
connected gravitational contribution with more than two boundaries [11].
Hence, on the gravity side these theories fall into the same category as
$\widehat{\text{CGHS}}$ . On the other hand, they are shown in [11] to be dual
to two Altland-Zirnbauer matrix ensembles with parameters $\upalpha=0,2$ and
$\upbeta=2$. In these ensembles $Z(\beta)>0$. Hence it is the nonperturbative
contributions to $\mathcal{P}_{n}$, inferred from the matrix model side, that
allow the full answer to be compatible with an ensemble interpretation.

###  3.2 JT gravity

As mentioned in the Introduction, for JT gravity Saad, Shenker and Stanford
[10] showed the existence of an ensemble interpretation (3) which implies a
measure $\mu_{\beta}$ on $Z(\beta)$ in (7). A remaining question in the
context of the moment problem is whether $\mu_{\beta}$ is unique by moments or
not. Theorem 2 in §2 gives an answer to this question in principle. Naturally
the answer depends on the behavior of $\mathcal{P}_{n}$ (or
$\mathcal{P}_{\textsf{conn},n}$) as $n\rightarrow\infty$.

This knowledge is not available in the definition of $\mathcal{P}_{n}$ as an
asymptotic perturbative series because it breaks down when $n$ becomes large
enough. We prove this statement in detail in the appendix, for two well-known
ways in which the JT path integral can be expressed as an asymptotic series,
i.e. the genus expansion in $e^{-S_{0}}$ [10] and the $1/\beta$ expansion [46,
47] that is obtained by a partial resummation of the genus expansion. The
structure of the genus expansion and its breakdown are reminiscent of the
breakdown of string perturbation theory with large number of external legs
[48]. On the other hand, the $1/\beta$ expansion has a qualitatively different
structure. The leading term at any $n$ coincides with the Airy limit result
$\mathcal{P}_{n}^{\rm Airy}$, which is obtained by taking $S_{0}\to\infty$ but
keeping $x\equiv\beta e^{-2S_{0}/3}$ fixed.

We conclude that a solely perturbative definition of the JT path integral does
not supply us with enough information to determine $\overline{\log Z(\beta)}$
and hence the quenched free energy.333The need for a nonperturbative
completion in order to calculate the quenched free energy has also been
emphasized in [49]. However, it turns out even in the Airy limit where the
$\mathcal{P}_{n}^{\rm Airy}$ are all known (at least in principle) [50], it is
not easy to determine the uniqueness of the distribution. In particular, the
moments grow too fast with $n$ to satisfy the sufficient condition for
uniqueness in Corollary 1:

$\mathcal{P}_{n}^{\rm
Airy}(x)\underset{n\to\infty}{\sim}\frac{\exp(n^{3}x^{3}/24)}{\sqrt{2\pi}(nx)^{3/2}}.$
(24)

In a recent work [51] we took a different approach and calculated
$\overline{\log Z(\beta)}$ directly in the matrix ensemble, in the limit
$\beta\rightarrow\infty$. The result is valid up to perturbative corrections
in $1/\beta$ and doubly nonperturbative corrections of order
$\exp(-\\#_{1}\exp(\\#_{2}S_{0}))$ with $\\#_{1,2}={\mathcal{O}}(1)$.

##  4 Conclusions

The ensemble interpretation of gravity can be tested and refined using the
tools that have been developed to study the moment problem. We reviewed some
of these tools and applied them to a few simple models of 2D gravity. While
these examples were two-dimensional, the connection between the ensemble
interpretation and the moment problem holds in any spacetime dimension. As we
have seen in the examples of JT gravity and JT supergravity, these tools are
often useful in showing when the gravity results are incomplete or ambiguous
to determine the properties of the putative ensemble dual. Therefore, they can
point to important nonperturbative corrections.

## Acknowledgments

We thank Christian Berg and Konrad Schmüdgen for their help in navigating the
literature on the moment problem, Sebastian Fischetti for a conversation that
lead to this work, and Hamid Afshar, Netta Engelhardt, Victor Godet, Matt
Kleban, Alex Maloney and Charles Marteau for discussions. OJ thanks the CCPP
at NYU for their hospitality while this work was in progress. The work of PZ
was supported by Ministry of Science and Higher Education of the Russian
Federation, agreement № 075-15-2019-1620.

## Appendix: perturbative JT gravity
jointly with Peter Zograf

In this appendix we first review two ways of expressing the JT gravity path
integral as an asymptotic series, closely following [10, 19, 46]. These are
the genus expansion (25) and the low-temperature expansion (38). Along the way
we review some properties of the Weil-Petersson volume polynomials that are
relevant to JT gravity. Using bounds respected by the intersection numbers of
$\psi$-classes on $\overline{\mathcal{M}}_{g,n}$, the Deligne-Mumford
compactification of the moduli space of complete hyperbolic surfaces of genus
$g$ with $n$ punctures, we show in two subsections that both the genus
expansion and the low-temperature expansion fail (in a way that we will
clarify) when the amount of boundaries $n$ is taken to infinity.

One way to write the path integral in JT gravity which connects $n\geq 1$
circular boundaries, all of equal length $\beta$, is via the following
asymptotic series in the small parameter $e^{-2S_{0}}$ (the genus expansion):

$\mathcal{P}_{\textsf{conn},n}(\beta)=\sum_{g=0}^{\infty}e^{-(2g+n-2)S_{0}}Z_{g,n}(\beta)$
(25)

where

$Z_{g,n}(\beta)=\left(\prod_{i=1}^{n}\int_{0}^{\infty}\mathrm{d}b_{i}\,b_{i}\,Z_{\textsf{trumpet}}(b_{i},\beta)\right)V_{g,n}(\boldsymbol{b})$
(26)

for
$(g,n)\notin\\{(0,1),(0,2)\\},Z_{0,1}(\beta)=(\sqrt{2\pi}\beta^{3/2})^{-1}e^{2\pi^{2}/\beta},Z_{0,2}(\beta)=1/(4\pi)$
and $Z_{\textsf{trumpet}}(b,\beta)=(2\pi\beta)^{-1/2}e^{-b^{2}/(2\beta)}$ (the
same normalization conventions as in [10, 19] are chosen, in which
$V_{0,3}(\boldsymbol{b})=1$). $V_{g,n}(\boldsymbol{b})$ for all $g,n$ is an
even, symmetric polynomial of degree $2(3g+n-3)$ with positive coefficients
[52],

$V_{g,n}(\boldsymbol{b})=\sum_{|\boldsymbol{\alpha}|\leq
3g+n-3}c_{\boldsymbol{\alpha}}^{(g)}\,b_{1}^{2\alpha_{1}}b_{2}^{2\alpha_{2}}\cdots
b_{n}^{2\alpha_{n}}\,,$ (27)

where $|\boldsymbol{\alpha}|\equiv\sum_{i=1}^{n}\alpha_{i}$. It is the Weil-
Petersson volume of the moduli space of genus $g$ Riemann surfaces with $n$
geodesic boundaries of lengths $\boldsymbol{b}=(b_{1},b_{2},\cdots,b_{n})$.
Mirzakhani [52] gave a recursion relation that computes the
$c_{\boldsymbol{\alpha}}^{(g)}$. They can be written as

$c_{\boldsymbol{\alpha}}^{(g)}=\frac{1}{2^{|\boldsymbol{\alpha}|}\boldsymbol{\alpha}!\,p!}\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\psi_{2}^{\alpha_{2}}\cdots\psi_{n}^{\alpha_{n}}\omega_{1}^{p}$
(28)

where $p=3g+n-3-|\boldsymbol{\alpha}|$,
$\boldsymbol{\alpha}!\equiv\alpha_{1}!\alpha_{2}!\cdots\alpha_{n}!$,
$\omega_{1}=2\pi^{2}\kappa_{1}$ where $\kappa_{1}$ is the first Mumford-
Morita-Miller class and the $\psi_{j}$ are the $\psi$-classes on
$\overline{\mathcal{M}}_{g,n}$. Plugging (27) into (26) and performing the
integrals gives

$\displaystyle Z_{g,n}(\beta)$
$\displaystyle=\left(\frac{\beta}{2\pi}\right)^{n/2}\sum_{|\boldsymbol{\alpha}|\leq
3g+n-3}(2\beta)^{|\boldsymbol{\alpha}|}\,c_{\boldsymbol{\alpha}}^{(g)}\boldsymbol{\alpha}!\,.$
(29)

We have the following useful inequalities [53, 54]:

$\displaystyle
c_{\boldsymbol{\alpha}}^{(g)}(2\boldsymbol{\alpha}+1)!\,4^{|\boldsymbol{\alpha}|}$
$\displaystyle\leq c_{\boldsymbol{0}}^{(g)}=V_{g,n}(\boldsymbol{0})\,,$ (30)
$\displaystyle V_{g,n}(\boldsymbol{0})\leq V_{g,n}(\boldsymbol{b})$
$\displaystyle\leq e^{|\boldsymbol{b}|/2}V_{g,n}(\boldsymbol{0})\,,$ (31)

for all $\boldsymbol{\alpha},g,n$ and $\boldsymbol{b}$. At large $n$ and fixed
$g$ we have the result [55]

$V_{g,n}(\boldsymbol{0})\sim a_{g}\,C^{n}n^{(5g-7)/2}n!~{}~{}~{}\text{as
}n\rightarrow\infty\,,$ (32)

for some $a_{g},C>0$, while at large $g$ and fixed $n$ we have [56]

$V_{g,n}(\boldsymbol{0})\sim\frac{\alpha}{\sqrt{g}}(4\pi^{2})^{2g+n-3}(2g+n-3)!~{}~{}~{}\text{as
}g\rightarrow\infty\,,$ (33)

for a $(g,n)$-independent constant $\alpha$ (approximately equal to
$1/\sqrt{\pi}$ [57]).

We can derive the following two bounds on the terms in the genus expansion
(25) (valid for all $n$ and $g$) from the inequalities (31):

$\left(\sqrt{\frac{\beta}{2\pi}}\right)^{n}\leq\frac{Z_{g,n}(\beta)}{V_{g,n}(\boldsymbol{0})}\leq
f(\beta)^{n}\,,$ (34)

where

$f(\beta)=\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\textsf{trumpet}}(b,\beta)\,e^{b/2}\,.$
(35)

With the lower bound in (34) and the asymptotic behavior (33) at large $g$ we
can demonstrate that the genus expansion (25) is asymptotic (in the variable
$e^{-2S_{0}}$) for any fixed $n$:

$\displaystyle e^{-(2g+n-2)S_{0}}Z_{g,n}$ $\displaystyle\geq
e^{-(2g+n-2)S_{0}}\left(\frac{\beta}{2\pi}\right)^{n/2}V_{g,n}(\boldsymbol{0})$
(36) $\displaystyle\sim
e^{-(2g+n-2)S_{0}}\left(\frac{\beta}{2\pi}\right)^{n/2}\frac{\alpha}{\sqrt{g}}(4\pi^{2})^{2g+n-3}(2g+n-3)!\,,$
(37)

as $g\rightarrow\infty$. It follows that when
$g=\mathcal{O}(e^{S_{0}})\times\text{(function of $n$)}$ the higher-genus
terms in the expansion certainly start dominating.

Using (29) in (25) gives

$\displaystyle\mathcal{P}_{\textsf{conn},n}$
$\displaystyle=\left(\frac{x}{2\pi}\right)^{n/2}(2x)^{n-3}\sum_{\ell=0}^{\infty}\beta^{-\ell}\left(\sum_{g=0}^{\infty}(2x)^{3g}P_{\ell,g,n}\right)\,,$
(38) $\displaystyle P_{\ell,g,n}$
$\displaystyle\equiv\Theta\left(3g+n-3-\ell\right)\times
2^{-\ell}\sum_{|\boldsymbol{\alpha}|=3g+n-3-\ell}c^{(g)}_{\boldsymbol{\alpha}}\boldsymbol{\alpha}!\,,$

where $x\equiv\beta\,e^{-2S_{0}/3}$ and $\Theta$ is the unit step function. In
this expression we have traded the asymptotic genus expansion (25) in
$e^{-2S_{0}}$ for an asymptotic expansion in $1/\beta$.444That this expansion
is asymptotic (for fixed $n\geq 3$) can be seen as follows: at any fixed level
$\ell\geq n-3$, the coefficient of $\beta^{-\ell}$ will contain a term
$(2x)^{3G}2^{-\ell}c_{\boldsymbol{\alpha}}^{(G)}\boldsymbol{\alpha}!$ where
$3G=\ell+3-n+q$ with $q=0,1$ or 2, and $|\boldsymbol{\alpha}|=q$. If we take
$\ell\rightarrow\infty$, then $G\rightarrow\infty$. We can then use [56,
Theorem 4.1] to conclude that $c_{\boldsymbol{\alpha}}^{(G)}\sim
c_{\boldsymbol{0}}^{(G)}$ which behaves as in (33) as $G\rightarrow\infty$.
This behavior contains a $(2G)!$ which beats all other factors. The transition
happens when $\ell=\mathcal{O}(\beta^{3/2})$. Note that in contrast to the
expression (25), the sums over genus in (38) converge, since the partial sums
form an increasing sequence which is bounded above:

$\displaystyle(2x)^{3g}\sum_{|\boldsymbol{\alpha}|=3g+n-3-\ell}c^{(g)}_{\boldsymbol{\alpha}}\boldsymbol{\alpha}!$
$\displaystyle\leq\frac{(2x)^{3g}}{4^{3g+n-3-\ell}}V_{g,n}(\boldsymbol{0})\sum_{|\boldsymbol{\alpha}|=3g+n-3-\ell}\frac{\boldsymbol{\alpha}!}{(2\boldsymbol{\alpha}+1)!}$
$\displaystyle\leq\frac{(2x)^{3g}}{4^{3g+n-3-\ell}}V_{g,n}(\boldsymbol{0})\sum_{|\boldsymbol{\alpha}|=3g+n-3-\ell}(\boldsymbol{\alpha}!)^{-1}$
$\displaystyle=\frac{(2x)^{3g}}{4^{3g+n-3-\ell}}V_{g,n}(\boldsymbol{0})\frac{n^{3g+n-3-\ell}}{(3g+n-3-\ell)!}$
$\displaystyle\lesssim
h(n)\left[\pi^{4/3}\frac{2mx}{3}\right]^{3g}\frac{1}{g!}~{}~{}~{}\text{as
}g\rightarrow\infty\,,$ (39)

where $h$ is an inconsequential function and we’ve used the bound (30) and the
asymptotic behavior (33).

In the following two subsections we will investigate the two asymptotic series
(25) and (38) for the same quantities $\mathcal{P}_{\textsf{conn},n}$ (valid
in different parameter regimes, however) in the limit where $n$ becomes large.
Not surprisingly, we will conclude that the perturbation theory implicit in
the asymptotic series fails in both cases. By this we mean the following: at
fixed $n$ we can always take $S_{0}$ or $\beta$ large enough so that the first
(few) terms in (25) and (38) respectively provide a good approximation to the
quantity $\mathcal{P}_{\textsf{conn},n}$. Now instead, in view of the bounds
discussed in §2 related to the moment problem, we would like to know how
$\mathcal{P}_{\textsf{conn},n}$ behaves as $n\rightarrow\infty$ while the
other parameters are held fixed. Without resumming/nonperturbatively
completing (25) and (38) this knowledge cannot be obtained: as
$n\rightarrow\infty$, with $\beta$ and $S_{0}$ held fixed, the $g=1$
($\ell=1$) term eventually becomes dominant compared to the $g=0$ ($\ell=0$)
term. We now show this in detail.

### Genus expansion fails at large $n$

In the genus expansion (25) of $\mathcal{P}_{\textsf{conn},n}$ we consider the
ratio of the genus $g+1$ contribution to the genus $g$ contribution, and bound
it below as follows:

$\displaystyle\frac{e^{-(2(g+1)+n-2)S_{0}}Z_{g+1,n}}{e^{-(2g+n-2)S_{0}}Z_{g,n}}$
$\displaystyle=e^{-2S_{0}}~{}\frac{\displaystyle\sum_{|\boldsymbol{\alpha}|\leq
3g+n}(2\beta)^{|\boldsymbol{\alpha}|}\,c_{\boldsymbol{\alpha}}^{(g+1)}\boldsymbol{\alpha}!}{\displaystyle\sum_{|\boldsymbol{\gamma}|\leq
3g+n-3}(2\beta)^{|\boldsymbol{\gamma}|}\,c_{\boldsymbol{\gamma}}^{(g)}\boldsymbol{\gamma}!}\geq
e^{-2S_{0}}~{}\frac{\displaystyle\sum_{|\boldsymbol{\alpha}|\leq
3g+n-3}(2\beta)^{|\boldsymbol{\alpha}|}\,c_{\boldsymbol{\alpha}}^{(g+1)}\boldsymbol{\alpha}!}{\displaystyle\sum_{|\boldsymbol{\gamma}|\leq
3g+n-3}(2\beta)^{|\boldsymbol{\gamma}|}\,c_{\boldsymbol{\gamma}}^{(g)}\boldsymbol{\gamma}!}\,.$

The following inequality holds:

$c_{\boldsymbol{\alpha}}^{(g+1)}>\frac{62\pi^{6}}{315}(2g+n-2)c_{\boldsymbol{\alpha}}^{(g)}\,,$
(40)

for all $\boldsymbol{\alpha}$ (with $|\boldsymbol{\alpha}|\leq 3g+n-3$), $g$
and $n$. It follows that certainly when $n>\mathcal{O}(e^{2S_{0}})$ it is no
longer justified to neglect any higher-genus contributions.

To derive (40) we start with [56, (2.12)], and note

$(2\boldsymbol{\alpha}+1)!\,2^{|\boldsymbol{\alpha}|}\,c_{\boldsymbol{\alpha}}^{(g+1)}\equiv[\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}]_{g+1,n}>B_{\boldsymbol{\alpha}}>16a_{3}[\tau_{1}\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}]_{g,n+1}$
(41)

where $a_{3}=31\pi^{6}/30240$. We will use [58, 59]

$\displaystyle(2g+n-2)[\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}]_{g,n}$
$\displaystyle=\frac{1}{2}\sum_{l=1}^{3g+n-2}\frac{(-1)^{l-1}l\,\pi^{2l-2}}{(2l+1)!}[\tau_{l}\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}]_{g,n+1}\,.$
(42)

Now, the right-hand side of this equation equals the $l=1$ term plus a
negative contribution. This can be seen from the inequality
$[\tau_{l}\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}]_{g,n+1}>[\tau_{l+1}\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}]_{g,n+1}$
(see e.g. [54, Lemma 3.6] or [56, (4.5)]) and the fact that
$\left(l\,\pi^{2l-2}/(2l+1)!\right)_{l\geq 1}$ is a decreasing sequence.
Therefore

$(2g+n-2)[\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}]_{g,n}<\frac{1}{12}[\tau_{1}\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}]_{g,n+1}\,.$
(43)

Putting (41) and (43) together gives (40).

###  $1/\beta$ expansion fails at large $n$

In (38) we would like to compare the $\ell=1$ contribution to the $\ell=0$
contribution as $n\rightarrow\infty$ for arbitrary but fixed $x\equiv\beta
e^{-2S_{0}/3}$, i.e. we are interested in the ratio

$\displaystyle R_{n}(x)$
$\displaystyle\equiv\frac{1}{2\beta}\frac{\sum_{g=0}^{\infty}(2x)^{3g}\sum_{|\boldsymbol{\alpha}|=3g+n-4}c_{\boldsymbol{\alpha}}^{(g)}\,\boldsymbol{\alpha}!}{\sum_{g^{\prime}=0}^{\infty}(2x)^{3g^{\prime}}\sum_{|\boldsymbol{\gamma}|=3g^{\prime}+n-3}c_{\boldsymbol{\gamma}}^{(g^{\prime})}\,\boldsymbol{\gamma}!}=\frac{2\pi^{2}}{\beta}\frac{\mathcal{F}_{1}(x,n)}{\mathcal{F}_{0}(x,n)}\,,$
(44) $\displaystyle\mathcal{F}_{0}(x,n)$
$\displaystyle=\sum_{g=0}^{\infty}x^{3g}\sum_{|\boldsymbol{d}|=3g+n-3}\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}\equiv\sum_{g=0}^{\infty}x^{3g}\mathcal{F}_{0}^{(g)}(n)\,,$
(45) $\displaystyle\mathcal{F}_{1}(x,n)$
$\displaystyle=\sum_{g=0}^{\infty}x^{3g}\sum_{|\boldsymbol{d}|=3g+n-4}\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}\kappa_{1}$
$\displaystyle=\sum_{g=0}^{\infty}x^{3g}\sum_{|\boldsymbol{d}|=3g+n-4}\int_{\overline{\mathcal{M}}_{g,n+1}}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}\psi_{n+1}^{2}\equiv\sum_{g=0}^{\infty}x^{3g}\mathcal{F}_{1}^{(g)}(n)\,,$
(46)

where we’ve assumed $n\geq 4$ and used (28). From the bound (39) it follows
that the series defining $\mathcal{F}_{0}$ and $\mathcal{F}_{1}$ have infinite
radii of convergence in $x$. (A closed-form expression for
$\mathcal{F}_{0}(x,n)$ is known [50] but we will not use it in the following.)
Our strategy is to bound below the individual ratios
$\mathcal{F}_{1}^{(g)}(n)/\mathcal{F}_{0}^{(g)}(n)$ for each $g$. For $g=0$ we
have the following result due to Kontsevich,

$\int_{\overline{\mathcal{M}}_{0,n}}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}=\binom{n-3}{\boldsymbol{d}}$
(47)

from which it follows that

$\displaystyle\mathcal{F}_{1}^{(0)}(n)$
$\displaystyle=\frac{(n-2)(n-3)}{2}n^{n-4}\,,$ (48)
$\displaystyle\mathcal{F}_{0}^{(0)}(n)$ $\displaystyle=n^{n-3}\,.$ (49)

So

$\frac{\mathcal{F}_{1}^{(0)}(n)}{\mathcal{F}_{0}^{(0)}(n)}=\frac{(n-2)(n-3)}{2n}\,.$
(50)

At large $n$ this is bounded below by a linear function in $n$. We aim to show
the same is true for all other $g$ too. Using Witten’s notation
$\langle\tau_{d_{1}}\tau_{d_{2}}\cdots\tau_{d_{n}}\rangle_{g}\equiv\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}$
(where $|\boldsymbol{d}|=3g+n-3$), the Virasoro constraints imply [60, 61]

$\displaystyle 15\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\tau_{2}\rangle_{g}$
$\displaystyle=\sum_{j=1}^{n}\left(2d_{j}+3\right)\left(2d_{j}+1\right)\langle\tau_{d_{1}}\cdots\tau_{d_{j}+1}\cdots\tau_{d_{n}}\rangle_{g}+\frac{1}{2}\langle(\tau_{0})^{2}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g-1}$
$\displaystyle+\frac{1}{2}\sum_{\underaccent{\bar}{n}=I\sqcup
J}\langle\tau_{0}\prod_{i\in
I}\tau_{d_{i}}\rangle_{g^{\prime}}\langle\tau_{0}\prod_{j\in
J}\tau_{d_{j}}\rangle_{g-g^{\prime}}\,.$ (51)

Here $\underaccent{\bar}{n}$ is the collection $\\{1,2,\cdots,n\\}$ which we
are to partition into two disjoint subsets $I$ and $J$.

It follows that

$\displaystyle
15\mathcal{F}_{1}^{(g)}(n)>\sum_{|\boldsymbol{d}|=3g+n-4}\sum_{j=1}^{n}\left(2d_{j}+3\right)\left(2d_{j}+1\right)\langle\tau_{d_{1}}\cdots\tau_{d_{j}+1}\cdots\tau_{d_{n}}\rangle\,,$
(52)

where we have suppressed the index $g$ in the Witten notation because in the
following we will only be considering intersection numbers on
$\overline{\mathcal{M}}_{g,n}$. We’d like to compare this with

$\displaystyle\mathcal{F}_{0}^{(g)}(n)$
$\displaystyle=\sum_{|\boldsymbol{\alpha}|=3g+n-3}\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle$
$\displaystyle=\sum_{\text{0 of }\alpha\text{'s are
}0}\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle+\sum_{\text{1 of
}\alpha\text{'s is
}0}\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle+\cdots+\sum_{\text{$n-1$
of }\alpha\text{'s are
}0}\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle\,.$ (53)

The rewriting in the second equality is possible when $g\geq 1$ (when $g=0$ at
least 3 $\alpha$’s must be zero). Let us consider the first contribution in
(53). Every element of this sum can be written in $n$ ways as
$\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle=\langle\tau_{(\alpha_{1}-1)+1}\cdots\tau_{\alpha_{n}}\rangle=\langle\tau_{\alpha_{1}}\tau_{(\alpha_{2}-1)+1}\cdots\tau_{\alpha_{n}}\rangle=\cdots=\langle\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{(\alpha_{n}-1)+1}\rangle$.
Since each of these $n$ appear in the sum in (52) with a coefficient greater
than one, we have
$\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle<(\text{those
contributions in \eqref{F1gnsum}})/n$ for every
$\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle$ appearing in the
first sum in (53). Consider then an element in the second sum in (53) with,
say, $\alpha_{1}=0$. This can be written in $n-1$ ways as
$\langle\tau_{0}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}\rangle=\langle\tau_{0}\tau_{(\alpha_{2}-1)+1}\tau_{\alpha_{3}}\cdots\tau_{\alpha_{n}}\rangle=\langle\tau_{0}\tau_{\alpha_{2}}\tau_{(\alpha_{3}-1)+1}\cdots\tau_{\alpha_{n}}\rangle=\cdots=\langle\tau_{0}\tau_{\alpha_{2}}\cdots\tau_{(\alpha_{n}-1)+1}\rangle$,
all of which appear in the sum in (52) (and we had not considered previously),
namely as

$\displaystyle(2\alpha_{2}+1)(2\alpha_{2}-1)\langle\tau_{0}\tau_{(\alpha_{2}-1)+1}\tau_{\alpha_{3}}\cdots\tau_{\alpha_{n}}\rangle$
$\displaystyle+(2\alpha_{3}+1)(2\alpha_{3}-1)\langle\tau_{0}\tau_{\alpha_{2}}\tau_{(\alpha_{3}-1)+1}\cdots\tau_{\alpha_{n}}\rangle+\cdots$
$\displaystyle+(2\alpha_{n}+1)(2\alpha_{n}-1)\langle\tau_{0}\tau_{\alpha_{2}}\tau_{\alpha_{3}}\cdots\tau_{(\alpha_{n}-1)+1}\rangle$
(54)
$\displaystyle=\langle\tau_{0}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}\rangle\times\sum_{i=2}^{n}(2\alpha_{i}+1)(2\alpha_{i}-1)$
(55)
$\displaystyle\geq\langle\tau_{0}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}\rangle\times\sum_{i=2}^{n}\left(4\alpha_{i}-1\right)$
(56)
$\displaystyle=\left[12(g-1)+3n+1\right]\langle\tau_{0}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}\rangle$
(57)
$\displaystyle>n\langle\tau_{0}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}\rangle\,,$
(58)

and therefore, as before,
$\langle\tau_{0}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}\rangle<(\text{the
corresponding contributions in \eqref{F1gnsum}})/n$ for every
$\langle\tau_{0}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}\rangle$ in the second
sum in (53). Of course the same holds for contributions with any other single
$\alpha$ set to zero (and the corresponding contributions in (52) are
different from the ones we’ve considered before). Finally consider the general
scenario where $0\leq m\leq n-1$ $\alpha$’s are set to zero, say, the first
$m$. In the same way

$\displaystyle(2\alpha_{m+1}+1)(2\alpha_{m+1}-1)\langle(\tau_{0})^{m}\tau_{(\alpha_{m+1}-1)+1}\tau_{\alpha_{m+2}}\cdots\tau_{\alpha_{n}}\rangle$
(59)
$\displaystyle+(2\alpha_{m+2}+1)(2\alpha_{m+2}-1)\langle(\tau_{0})^{m}\tau_{\alpha_{m+1}}\tau_{(\alpha_{m+2}-1)+1}\cdots\tau_{\alpha_{n}}\rangle$
$\displaystyle+\cdots+(2\alpha_{n}+1)(2\alpha_{n}-1)\langle(\tau_{0})^{m}\tau_{\alpha_{m+1}}\tau_{\alpha_{m+2}}\cdots\tau_{(\alpha_{n}-1)+1}\rangle$
$\displaystyle=\langle(\tau_{0})^{m}\tau_{\alpha_{m+1}}\tau_{\alpha_{m+2}}\cdots\tau_{\alpha_{n}}\rangle\times\sum_{i=m+1}^{n}(2\alpha_{i}+1)(2\alpha_{i}-1)$
(60)
$\displaystyle\geq\langle(\tau_{0})^{m}\tau_{\alpha_{m+1}}\tau_{\alpha_{m+2}}\cdots\tau_{\alpha_{n}}\rangle\times\sum_{i=m+1}^{n}(4\alpha_{i}-1)$
(61)
$\displaystyle=\left[12(g-1)+3n+m\right]\langle(\tau_{0})^{m}\tau_{\alpha_{m+1}}\tau_{\alpha_{m+2}}\cdots\tau_{\alpha_{n}}\rangle$
(62)
$\displaystyle>n\langle(\tau_{0})^{m}\tau_{\alpha_{m+1}}\tau_{\alpha_{m+2}}\cdots\tau_{\alpha_{n}}\rangle\,.$
(63)

This proves that for $g\geq 1$,
$15\mathcal{F}_{1}^{(g)}(n)>n\mathcal{F}_{0}^{(g)}(n)$ for all $n\geq 4$.
Including the $g=0$ result, we have

$\frac{\mathcal{F}_{1}^{(g)}(n)}{\mathcal{F}_{0}^{(g)}(n)}>\frac{n}{15}$ (64)

for all $g\geq 0$ and all $n\geq 5$. This proves that the ratio (44) diverges
as $n\rightarrow\infty$ for any $x>0$.

## References

  * [1] S. R. Coleman, “Black Holes as Red Herrings: Topological Fluctuations and the Loss of Quantum Coherence,” Nucl. Phys. B307 (1988) 867–882.
  * [2] J. M. Maldacena and L. Maoz, “Wormholes in AdS,” JHEP 02 (2004) 053, arXiv:hep-th/0401024 [hep-th].
  * [3] R. Jackiw, “Lower Dimensional Gravity,” Nucl. Phys. B252 (1985) 343–356.
  * [4] C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,” Phys. Lett. 126B (1983) 41–45.
  * [5] S. Sachdev and J. Ye, “Gapless spin fluid ground state in a random, quantum Heisenberg magnet,” Phys. Rev. Lett. 70 (1993) 3339, arXiv:cond-mat/9212030 [cond-mat].
  * [6] S. Sachdev, “Holographic metals and the fractionalized Fermi liquid,” Phys. Rev. Lett. 105 (2010) 151602, arXiv:1006.3794 [hep-th].
  * [7] A. Kitaev.  http://online.kitp.ucsb.edu/online/entangled15/kitaev/. talk at KITP program “Entanglement in Strongly-Correlated Quantum Matter” (2015).
  * [8] J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,” Phys. Rev. D94 no. 10, (2016) 106002, arXiv:1604.07818 [hep-th].
  * [9] A. Kitaev and S. J. Suh, “The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual,” JHEP 05 (2018) 183, arXiv:1711.08467 [hep-th].
  * [10] P. Saad, S. H. Shenker, and D. Stanford, “JT gravity as a matrix integral,” arXiv:1903.11115 [hep-th].
  * [11] D. Stanford and E. Witten, “JT Gravity and the Ensembles of Random Matrix Theory,” arXiv:1907.03363 [hep-th].
  * [12] L. V. Iliesiu, “On 2D gauge theories in Jackiw-Teitelboim gravity,” arXiv:1909.05253 [hep-th].
  * [13] D. Kapec, R. Mahajan, and D. Stanford, “Matrix ensembles with global symmetries and ’t Hooft anomalies from 2d gauge theory,” JHEP 04 (2020) 186, arXiv:1912.12285 [hep-th].
  * [14] H. Maxfield and G. J. Turiaci, “The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral,” JHEP 01 (2021) 118, arXiv:2006.11317 [hep-th].
  * [15] E. Witten, “Matrix Models and Deformations of JT Gravity,” Proc. Roy. Soc. Lond. A 476 no. 2244, (2020) 20200582, arXiv:2006.13414 [hep-th].
  * [16] A. Maloney and E. Witten, “Averaging over Narain moduli space,” JHEP 10 (2020) 187, arXiv:2006.04855 [hep-th].
  * [17] N. Afkhami-Jeddi, H. Cohn, T. Hartman, and A. Tajdini, “Free partition functions and an averaged holographic duality,” JHEP 01 (2021) 130, arXiv:2006.04839 [hep-th].
  * [18] D. Stanford, “More quantum noise from wormholes,” arXiv:2008.08570 [hep-th].
  * [19] N. Engelhardt, S. Fischetti, and A. Maloney, “Free energy from replica wormholes,” Phys. Rev. D 103 no. 4, (2021) 046021, arXiv:2007.07444 [hep-th].
  * [20] T. J. Stieltjes, “Recherches sur les fractions continues,” Annales de la Faculté des sciences de Toulouse : Mathématiques 1e série, 8 no. 4, (1894) J1–J122.
  * [21] J. Shohat and J. Tamarkin, The Problem of Moments.  Mathematical surveys and monographs. AMS, 1950.
  * [22] T. Carleman, Les fonctions quasi analytiques.  Gauthier-Villars, 1926.
  * [23] M. Krein, “On an extrapolation problem due to Kolmogorov,” Dokl. Akad. Nauk SSSR 46 (1945) 306–309.
  * [24] C. Berg, “Indeterminate moment problems and the theory of entire functions,” J. Comput. Appl. Math. 65 no. 1, (1995) 27–55.
  * [25] M. Krein and A. Nudel’man, The Markov Moment Problem and Extremal Problems.  AMS, 1977.
  * [26] R. Nevanlinna, “Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjessche Momentenproblem,” Ann. Acad. Scient. Fennicae (A) 18 no. 5, (1922) .
  * [27] H. L. Pedersen, “La paramétrisation de Nevanlinna et le problème des moments de Stieltjes indéterminé,” Expo. Math. 15 no. 3, (1997) 273–278.
  * [28] N. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis.  University mathematical monographs. Oliver & Boyd, 1965.
  * [29] K. Schmüdgen, The Moment Problem.  Graduate Texts in Mathematics. Springer, 2017.
  * [30] G. D. Lin, “Recent Developments on the Moment Problem,” J. Stat. Distrib. App. 4 no. 5, (2017) , arXiv:1703.01027 [math.PR].
  * [31] C. Berg and M. Thill, “A density index for the Stieltjes moment problem,” IMACS Ann. Comput. Appl. Math. 9 (1991) 185–188.
  * [32] C. Berg, Y. Chen, and M. E. H. Ismail, “Small eigenvalues of large Hankel matrices: The indeterminate case,” Math. Scand. 91 no. 1, (2002) 67–81.
  * [33] H. Hamburger, “Über eine Erweiterung des Stieltjesschen Momentenproblems I-III,” Math. Ann. 81, 82, 82 (1920, 1921, 1921) 235–319, 120–164, 168–187.
  * [34] D. Berend and T. Tassa, “Improved bounds on Bell numbers and on moments of sums of random variables,” Probab. Math. Stat. 30 no. 2, (2010) 185–205.
  * [35] E. K. Haviland, “On the Momentum Problem for Distribution Functions in More Than One Dimension. I, II,” Am. J. Math. 57, 58 (1935, 1936) 562–568, 164–168.
  * [36] B. Fuglede, “The multidimensional moment problem,” Expo. Math. 1 (1983) 47–65.
  * [37] C. Berg, “The multidimensional moment problem and semigroups,” Proc. Symp. Appl. Math. 37 (1987) 110–124.
  * [38] A. E. Nussbaum, “Quasi-analytic vectors,” Ark. Mat. 6 no. 2, (1965) 179–191.
  * [39] C. Kleiber and J. Stoyanov, “Multivariate distributions and the moment problem,” J. Multivar. Anal. 113 (2013) 7–18.
  * [40] L. C. Petersen, “On the relation between the multidimensional moment problem and the one-dimensional moment problem,” Math. Scand. 51 no. 2, (1983) 361–366.
  * [41] H. Afshar, H. A. González, D. Grumiller, and D. Vassilevich, “Flat space holography and the complex Sachdev-Ye-Kitaev model,” Phys. Rev. D 101 no. 8, (2020) 086024, arXiv:1911.05739 [hep-th].
  * [42] J. Callan, Curtis G., S. B. Giddings, J. A. Harvey, and A. Strominger, “Evanescent black holes,” Phys. Rev. D 45 no. 4, (1992) 1005, arXiv:hep-th/9111056.
  * [43] H. R. Afshar, “Warped Schwarzian theory,” JHEP 02 (2020) 126, arXiv:1908.08089 [hep-th].
  * [44] V. Godet and C. Marteau, “New boundary conditions for AdS2,” JHEP 12 (2020) 020, arXiv:2005.08999 [hep-th].
  * [45] J. Marcinkiewicz, “Sur une propriété de la loi de Gauss,” Math. Z. 44 (1938) 612–618.
  * [46] K. Okuyama and K. Sakai, “JT gravity, KdV equations and macroscopic loop operators,” JHEP 01 (2020) 156, arXiv:1911.01659 [hep-th].
  * [47] K. Okuyama and K. Sakai, “Multi-boundary correlators in JT gravity,” JHEP 08 (2020) 126, arXiv:2004.07555 [hep-th].
  * [48] S. Ghosh and S. Raju, “Breakdown of String Perturbation Theory for Many External Particles,” Phys. Rev. Lett. 118 no. 13, (2017) 131602, arXiv:1611.08003 [hep-th].
  * [49] C. V. Johnson, “Low Energy Thermodynamics of JT Gravity and Supergravity,” arXiv:2008.13120 [hep-th].
  * [50] A. Okounkov, “Generating functions for intersection numbers on moduli spaces of curves,” Int. Math. Res. Not. IMRN 2002 no. 18, (2001) 933–957, arXiv:math/0101201 [math.AG].
  * [51] O. Janssen and M. Mirbabayi, “Low-temperature entropy in JT gravity,” arXiv:2103.03896 [hep-th].
  * [52] M. Mirzakhani, “Weil-Petersson volumes and intersection theory on the moduli space of curves,” J. Amer. Math. Soc. 20 (2007) 1–23.
  * [53] M. Mirzakhani, “Growth of Weil-Petersson Volumes and Random Hyperbolic Surfaces of Large Genus,” J. Differ. Geom. 94 no. 2, (2013) 267–300, arXiv:1012.2167 [math.GN].
  * [54] L. Kefeng and X. Hao, “A remark on Mirzakhani’s asymptotic formulae,” Asian J. Math 18 no. 1, (2014) 29–52, arXiv:1103.5136 [math.AG].
  * [55] Y. I. Manin and P. Zograf, “Invertible cohomological field theories and Weil-Petersson volumes,” Ann. Inst. Fourier (Grenoble) 50 no. 2, (2000) 519–535.
  * [56] M. Mirzakhani and P. Zograf, “Towards large genus asymtotics of intersection numbers on moduli spaces of curves,” Geom. Funct. Anal. 25 (2015) 1258–1289, arXiv:1112.1151 [math.AG].
  * [57] P. Zograf, “On the large genus asymptotics of Weil-Petersson volumes,” arXiv:0812.0544 [math.AG].
  * [58] N. Do and P. Norbury, “Weil-Petersson volumes and cone surfaces,” Geom. Dedicata 141 no. 93, (2006) , arXiv:math/0603406 [math.AG].
  * [59] K. Liu and H. Xu, “Recursion formulae of higher Weil-Petersson volumes,” Int. Math. Res. Not. IMRN 2009 no. 5, (2008) 835–859, arXiv:0708.0565 [math.AG].
  * [60] R. Dijkgraaf, “Intersection theory, integrable hierarchies and topological field theory,” NATO Sci. Ser. B 295 (1992) 95–158, arXiv:hep-th/9201003.
  * [61] R. Dijkgraaf, H. Verlinde, and E. Verlinde, “Topological strings in $d<1$,” Nucl. Phys. B 352 no. 1, (1991) 59–86.


# Wav2vec-based Detection and Severity Level Classification of Dysarthria from
Speech

###### Abstract

Automatic detection and severity level classification of dysarthria directly
from acoustic speech signals can be used as a tool in medical diagnosis. In
this work, the pre-trained wav2vec 2.0 model is studied as a feature extractor
to build detection and severity level classification systems for dysarthric
speech. The experiments were carried out with the popularly used UA-speech
database. In the detection experiments, the results revealed that the best
performance was obtained using the embeddings from the first layer of the
wav2vec model that yielded an absolute improvement of 1.23% in accuracy
compared to the best performing baseline feature (spectrogram). In the studied
severity level classification task, the results revealed that the embeddings
from the final layer gave an absolute improvement of 10.62% in accuracy
compared to the best baseline features (mel-frequency cepstral coefficients).

Index Terms—  Dysarthria, Severity level classification, Wav2vec 2.0, MFCCs.

##  1 Introduction

Dysarthria is a neuro-motor disorder caused by neurological damage of the
motor component of speech production. Dysarthria occurs either due to a
neurological injury (i.e., cerebral palsy, stroke) or due to a
neurodegenerative disease (i.e., Parkinsons’s disease, Huntington’s disease).
Dysarthric speech is often associated with atypical speech prosody, and
reduced tongue flexibility and imprecise articulation, all of which impact
speech intelligibility [1]. Assessment of speech intelligibility is essential
in distinguishing the progression of dysarthria.

Speech assessment is generally performed in voice clinics by speech-language
pathologists who conduct intelligibility tests to identify the potential
presence of dysarthria as well as its severity level [2]. Subjective
intelligibility tests are costly and laborious, and they are prone to biases
of pathologists due to their familiarity and experience with patients. Hence,
the design of an objective method for the severity assessment of dysarthric
speech is important. The assessment of dysarthric speech is carried out in two
phases consisting of (1) the identification of the presence of dysarthria and
(2) the estimation of the severity level of the disease. Both of these phases
are important diagnostic steps that are needed to make clinical decisions on
medication and therapy of the patient. This work focuses on both of the above-
mentioned phases by studying speech-based detection and severity level
classification of dysarthria. In both of these topics, the current study
focuses on the use of the pre-trained wav2vec model to extract the features
[3].

Automatic detection and severity level classification of dysarthria from
speech is enabled using data-driven approaches based on supervised learning.
This involves building machine learning models that are trained using speech
data which has been collected from patients and which has been labelled by
speech-language pathologists. Many automatic dysarthria detection and severity
level classification methods are based on acoustic features that characterize
the salient aspects in the production of dysarthric speech [4, 5]. Abnormal
variations of pronunciation, prosody and voice quality were investigated by
using the sentence-level features in [5]. In [6, 7], authors investigated the
single frequency filtering -based features for dysarthric speech detection and
4-class intelligibility classification (very low, low, medium, and high). In
[8], auditory distinctive features (based on models of mid-external ear and
basilar membrane) were proposed for the assessment of dysarthria. Their study
also showed that the combination of auditory features with mel-frequency
cepstral coefficients (MFCCs) improves the intelligibility estimation of
dysarthric speech. The short and long-term temporal measures based on the log-
energy dynamics and auditory inspired modulation spectral features were
investigated for dysarthric speech intelligibility assessment in [9]. Glottal
source features along with the OpenSmile features [10] were investigated both
in detection of dysarthric speech as well as in classification of the
intelligibility of dysarthric speech in [11, 12]. Linear weighted combination
of articulation, phonation, and prosody features of speech were used in
intelligibilty estimation in [13, 14]. Recently, in [15] authors explored the
use of different spectro-temporal representations, such as the spectrogram and
mel-spectrogram, and convolutional neural networks in intelligibility
assessment of dysarthric speech.

In a few recent years, pre-trained neural network models have become popular
for various speech technology tasks, such as automatic speech recognition
(ASR), speaker recognition and emotion recognition [3, 16, 17, 18]. In the
current study, we take advantage of an effective pre-trained model, wav2vec,
in speech-based detection and severity level classification of dysarthria. The
topic is motivated by recent findings reported in recognition of pathological
speech that have shown good performance for the wav2vec models [16, 19]. These
models are generally first pre-trained in an unsupervised manner on large
speech datasets and then fine-tuned on a small set to perform the required
task. In this work, we experiment with the wav2vec model shared on HuggingFace
[20].

The main contributions of the study are:

  * •

Layer-by-layer analysis of the utility of the wav2vec features in the
detection of dysarthria (healthy $vs$. dysarthric) and in the classification
of the severity level (very low, low, medium, and high) of dysarthria from
speech.

  * •

Systematic comparison between popularly used spectral features and the wav2vec
features.

##  2 The detection and severity level classification systems

This work studied the following two classification problems: (1) a binary
classification problem to distinguish dysarthric speech from healthy speech
(i.e., detection problem), and (2) a multi-class classification problem to
classify the severity level of dysarthria from speech to 4 classes (very low,
low, medium, and high). In both problems, a pre-trained wave2vec 2.0 [3, 16]
model is used as a feature extractor, and a support vector machine (SVM) is
used as a classifier. A schematic block diagrams of the systems built for the
two aforementioned problems is shown in Figure 1. The following two sub-
sections describe the feature extraction and classification methods in more
detail.

Fig. 1:  The proposed systems for (a) detection of dysarthric speech and for
(b) severity level classification of dysarthric speech.

###  2.1 Pre-trained feature extractor

In building the classification systems for both problems, we use the pre-
trained wav2vec 2.0 [3] model as a feature extractor. We utilize Facebook’s
wav2vec 2.0 model that has been trained with 960 hours of audio from the
Librispeech corpus [3]. The model was originally trained to be used in ASR,
which implies that the final layers of the network have learnt to extract
embeddings that contain information about phoneme identity of speech. However,
the embeddings from the first layers of the network also contain information
about phones, which makes them useful features also for many other speech-
related tasks than ASR [21].

We take use of the inputs of the first transformer block of the context
network of the wav2vec model, as well as of the output embeddings of each of
the 12 transformer blocks. As the embeddings are extracted for each non-
overlapping 20 ms frame of the signal, we compute the average over the frames
to obtain the final feature vectors. The dimensionality of the embeddings is
768, which is also the dimensionality of the features that we finally use in
our classifiers. These features will be referred to as the wav2vec features in
this paper. To refer to the individual layers, we use the layer numbers, i.e.,
”wav2vec-N” refers to the features associated to the $N$th layer of the model.

###  2.2 Classifiers

In this study, a binary SVM classifier is used to distinguish between healthy
and dysarthric samples. For the multi-class classification between the four
severity levels (very low, low, medium, and high), SVM with the one-vs-one
architecture [22] was used. Both in the binary classification and in the
4-class classification, we used radial basis function as kernel, and a
regularization parameter value of 1. For gamma, we used scaling according to
the dimensionality and variance of data, written as $\gamma=1/(D\cdot
Var(X))$, where $Var(X)$ is the variance of the training data, and D is the
dimension of the feature vectors.

##  3 Experimental setup

###  3.1 Database of dysarthric speech

This study uses the UA-speech database of dysarthric speech [4]. The database
consists of 765 isolated words recorded in three blocks (B1, B2, and B3) by 15
dysarthric speakers (four female and eleven male) diagnosed with cerebral
palsy and 13 healthy controls (four female and nine male). The overall
intelligibility of each of the dysarthric speakers has been assessed using the
subjective evaluations with 5 native listeners. Based on the intelligibility
ratings, the dysarthric speakers have been grouped into four severity
categories: very low (4 speakers), low (3 speakers), medium (3 speakers) and
high (5 speakers). Each block contains 255 words in which 155 words are common
to all three blocks, and the remaining 100 words differ across the blocks. An
eight-microphone array was used for speech recording. Speech was sampled at 16
kHz and each microphone was spaced at intervals of 1.5 in. The current study
was carried out using the speech utterances from all three blocks of each
speaker, recorded by microphone number 6 of the array. More details of the UA-
speech database can be found in [4].

###  3.2 Baseline features used for comparison

In order to compare the performance of the wav2vec features, three popularly
used spectral features (spectrogram, mel-spectrogram and MFCCs) were
considered as they were shown in [7] to provide good discrimination between
dysarthric and healthy speech. All these feature representations are derived
using the Librosa toolkit [23].

####  3.2.1 Spectrogram

The spectrogram features were computed by taking the logarithm of the
amplitude spectrum that was estimated using a 1024-point fast Fourier
transform (FFT) with the Hamming windowing in frames of 25 ms with a shift of
5 ms. Finally, the features of the spectrogram were averaged over the time
axis yielding a 513-dimensional feature vector per utterance.

####  3.2.2 Mel-Spectrogram

The mel-spectrogram features were computed by applying a mel-filterbank with
80 filters on amplitude spectrum. The resulting mel-spectrogram was mapped to
a decibel scale through logarithm. By averaging the mel-spectrogram features
the over time axis, a 80-dimensional feature vector was obtained per
utterance.

####  3.2.3 MFCCs

To compute the MFCCs, the discrete cosine transform (DCT) was used to
transform the mel-scale spectrum into mel-cepstrum. The first 13 cepstral
coefficients (including the $0^{th}$ coefficient) were considered and their
delta & double-delta coefficients were computed, which yielded a
39-dimensional MFCC feature vector per utterance.

Fig. 2:  Detection accuracy of dysarthric speech for the three baseline
features (spectrogram, mel-spectrogram and MFCCs) and for all the 13 wav2vec
features. The blue bars represent the features derived from the wav2vec model,
with the tick labels indicating the index of the corresponding layer. Heights
of the bars represent the mean accuracy. Fig. 3:  Severity level
classification accuracy of dysarthric speech for the three baseline features
(spectrogram, mel-spectrogram and MFCCs) and for all the 13 wav2vec features.
The blue bars represent the features derived from the wav2vec model, with the
tick labels indicating the index of the corresponding layer. Heights of the
bars represent the mean accuracy.

###  3.3 Training and testing

The binary detection experiments were conducted with leave-one-speaker-out
(LOSO) cross-validation, where speech signals of one speaker was considered as
test data and speech signals of the remaining 27 speakers were used for
training the SVM classifier. Both training and testing data were z-score
normalized using the mean and standard deviation of the training data. In each
iteration, the evaluation metrics were saved. This process was repeated for 28
iterations (equaling to the number of speakers), and finally the evaluation
metrics were averaged over the 28 iterations.

For the severity level classification experiments, three dysarthric speakers
were left out (one male speaker from ”very low” level of intelligibility and
two male speakers from ”high” level of intelligibility) to have the same
number of dysarthric speakers in each class. Experiments were carried out
using the 12 remaining dysarthric speakers. In each iteration, speech signals
of four speakers (one from each class) were considered as test data and speech
signals of the remaining speakers were used to train the classifier. By
considering one speaker from each class for testing the SVM classifier, a
total of 81 (3*3*3*3) iterations (training and testing process) were
performed.

###  3.4 Evaluation metrics

In order to evaluate the performance of the dysarthria detection systems, the
following four commonly used evaluation metrics were considered in this study:
accuracy (ACC), sensitivity (SE), specificity (SP), F1-score (F1) apart from
the confusion matrices. For the severity level classification systems, mean
accuracy and class-wise accuracies were used for assessing the performance of
the systems.

##  4 Results

This section reports the results obtained using the wav2vec features and the
baseline features. The results are reported in Section 4.1 for the detection
problem (i.e., the binary classification problem) , and in Section 4.2 for the
4-class severity level classification problem.

###  4.1 Detection of dysarthric speech

The obtained accuracies of the binary detection experiments are shown in
Figure 2 for the three baseline features and for all the 13 wav2vec features.
Table 1 presents the results for the baseline features and for the three best
wav2vec features by showing all the four selected metrics. It can be observed
that the wav2vec-1 outperformed all the other features in all metrics. In
particular, when compared to the best performing baseline feature
(spectrogram), wav2vec-1 showed an absolute improvement of 1.23% in accuracy.
This finding indicates that the first embedding layer of the wav2vec model
learns generic speech representations that are useful for distinguishing
dysarthric speech from healthy speech.

Confusion matrices for the dysarthria detection experiments are displayed in
Table 2 for the spectrogram feature and the best performing wav2vec feature
(wav2vec-1). It can be seen that there are less confusions between healthy and
dysarthric speech for the wav2vec-1 feature compared to the spectrogram
feature.

Table 1:  Dysarthria detection results for the three baseline features along with the best three wav2vec features. Here ACC refers to accuracy, SE refers to sensitivity, SP refers to specificity, and F1 refers to F1-score. Feature | ACC [%] | SE | SP | F1
---|---|---|---|---
Baseline features
Spectrogram | 92.72 | 0.92 | 0.94 | 0.93
Mel-spectrogram | 91.48 | 0.91 | 0.92 | 0.92
MFCCs | 89.23 | 0.86 | 0.93 | 0.90
Wav2vec features
wav2vec-1 | 93.95 | 0.93 | 0.95 | 0.94
wav2vec-2 | 91.96 | 0.90 | 0.94 | 0.92
wav2vec-3 | 91.97 | 0.90 | 0.95 | 0.92
Table 2:  Confusion matrices of dysarthria detection for the spectrogram
feature (the best performing baseline) along with wav2vec-1 (the best
performing wav2vec feature).

Spectrogram | wav2vec-1
---|---
| Healthy | Dysarthric |  | Healthy | Dysarthric
Healthy | 93.94 | 6.06 | Healthy | 94.85 | 5.15
Dysarthric | 8.34 | 91.66 | Dysarthric | 6.82 | 93.18

###  4.2 Severity level classification of dysarthric speech

The performance in term of accuracy are shown in Figure 3 for the three
baseline features and for all the 13 wav2vec features for the severity level
classification experiments. Table 3 shows the overall classification accuracy
together with the class-wise accuracies for the baseline features and for the
two best wav2vec features.

From Figure 3, it can be clearly seen that almost all the wav2vec features
(except for wav2vec-1) outperformed the three baseline features in terms of
the mean accuracy. Interestingly, unlike in the detection problem, the best-
performing wav2vec features were the ones obtained from the final layers. In
fact, there is a rising trend in the accuracy when moving from the first layer
towards the final layer. This result was expected because the severity of
dysarthria is associated with the intelligibility of speech (e.g., phoneme
identity) and the wav2vec pre-trained model is designed for the ASR tasks,
therefore the final layers can effectively learn information related to the
linguistic contents of speech.

Among the baseline features, it can be observed that the MFCCs and spectrogram
performed better than the mel-spectrogram (chance level is 25%). Compared to
the best baseline (MFCCs), wav2vec-12 and wav2vec-13 gave an absolute
improvement of 8.88% and 10.62%, respectively. The results also show that for
the wav2vec features, the class-wise accuracies are relatively less biased
towards the two extreme ends of the severity scale (”very low” and ”high”
levels of dysarthria severity) compared to the baseline features.

Table 3:  Dysarthria severity level classification accuracies and class-wise
accuracies for the three baseline features along with the two best wav2vec
features. Here ACC refers to accuracy and C refers to class.

Feature | ACC [%] | $C_{very-low}$ | $C_{low}$ | $C_{medium}$ | $C_{high}$
---|---|---|---|---|---
Baseline features
Spectrogram | 33.26 | 44.54 | 16.65 | 14.95 | 56.87
Mel-spectrogram | 26.21 | 32.10 | 13.71 | 16.09 | 42.94
MFCCs | 33.94 | 50.63 | 7.32 | 21.02 | 56.76
Wav2vec features
wav2vec-12 | 42.82 | 63.10 | 22.77 | 16.51 | 68.91
wav2vec-13 | 44.56 | 56.09 | 23.21 | 18.77 | 80.16

##  5 Summary and Conclusions

In this study, we explored the state-of-the-art pre-trained wav2vec model to
extract features in the context of dysarthric speech detection and in the
context of severity level classification of dysarthria. A comparison of the
wav2vec features with the popularly used spectral and cepstral-based baseline
features (spectrogram, mel-spectrogram, and MFCCs) was carried out both in the
detection problem and in the severity level classification problem.

The results of the dysarthric speech detection experiments indicated that the
features extracted from the first layer (wav2vec-1) outperformed the baseline
and other wav2vec features by showing an absolute improvement of 1.23% in
accuracy compared to the best performing baseline feature (spectrogram). This
indicates that the starting layers of the wav2vec model has learned generic
speech features that can be effectively used for the detection of dysarthric
speech. The results of the severity level classification experiments showed
that the features extracted from the final layers (wav2vec-12 and wav2vec-13)
performed better than the baseline and the other wav2vec features. Compared to
MFCCs (the best baseline feature), an absolute improvement of 8.88% and 10.62%
was given by wav2vec-12 and wave2vec-13, respectively. We argue that the
improved performance that was obtained using the features from the final
layers is due to the wav2vec model’s capability to extract features that
contain information related to the linguistic contents associated to speech
intelligibility (which is directly related to the severity level of
dysarthria).

Taken together, the experimental findings of the study indicate that the
classification systems seem to be generalizable to unseen speakers using the
features from the starting layers in the detection task, and the features from
the final layers in the multi-class classification task. However, further
research is required in order to study the generalizability of the wav2vec
features for other disorders and to study also the performance of these
features in cross-database scenarios.

## References

  * [1] Joseph R Duffy,  Motor speech disorders e-book: Substrates, differential diagnosis, and management,  Elsevier Health Sciences, 2019.
  * [2] Ray D Kent, Gary Weismer, Jane F Kent, and John C Rosenbek,  “Toward phonetic intelligibility testing in dysarthria,”  Journal of Speech and Hearing Disorders, vol. 54, no. 4, pp. 482–499, 1989.
  * [3] Alexei Baevski, Yuhao Zhou, Abdelrahman Mohamed, and Michael Auli,  “wav2vec 2.0: A framework for self-supervised learning of speech representations,”  in Advances in Neural Information Processing Systems, H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, Eds. 2020, vol. 33, pp. 12449–12460, Curran Associates, Inc.
  * [4] Heejin Kim, Mark Hasegawa-Johnson, Adrienne Perlman, Jon Gunderson, Thomas S Huang, Kenneth Watkin, and Simone Frame,  “Dysarthric speech database for universal access research,”  in Ninth Annual Conference of the International Speech Communication Association, 2008.
  * [5] Jangwon Kim, Naveen Kumar, Andreas Tsiartas, Ming Li, and Shrikanth S Narayanan,  “Automatic intelligibility classification of sentence-level pathological speech,”  Computer speech & language, vol. 29, no. 1, pp. 132–144, 2015\.
  * [6] Krishna Gurugubelli and Anil Kumar Vuppala,  “Perceptually enhanced single frequency filtering for dysarthric speech detection and intelligibility assessment,”  in ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2019, pp. 6410–6414.
  * [7] Krishna Gurugubelli and Anil Kumar Vuppala,  “Analytic phase features for dysarthric speech detection and intelligibility assessment,”  Speech Communication, vol. 121, pp. 1–15, 2020.
  * [8] Kamil Lahcene Kadi, Sid Ahmed Selouani, Bachir Boudraa, and Malika Boudraa,  “Fully automated speaker identification and intelligibility assessment in dysarthria disease using auditory knowledge,”  Biocybernetics and Biomedical Engineering, vol. 36, no. 1, pp. 233–247, 2016.
  * [9] Tiago H Falk, Wai-Yip Chan, and Fraser Shein,  “Characterization of atypical vocal source excitation, temporal dynamics and prosody for objective measurement of dysarthric word intelligibility,”  Speech Communication, vol. 54, no. 5, pp. 622–631, 2012.
  * [10] Florian Eyben, Martin Wöllmer, and Björn Schuller,  “Opensmile: The munich versatile and fast open-source audio feature extractor,”  in Proceedings of the 18th ACM International Conference on Multimedia, New York, NY, USA, 2010, MM ’10, p. 1459–1462, Association for Computing Machinery.
  * [11] NP Narendra and Paavo Alku,  “Dysarthric speech classification using glottal features computed from non-words, words and sentences.,”  in Interspeech, 2018, pp. 3403–3407.
  * [12] NP Narendra and Paavo Alku,  “Dysarthric speech classification from coded telephone speech using glottal features,”  Speech Communication, vol. 110, pp. 47–55, 2019.
  * [13] Panying Rong, Yana Yunusova, Jun Wang, Lorne Zinman, Gary L Pattee, James D Berry, Bridget Perry, and Jordan R Green,  “Predicting speech intelligibility decline in amyotrophic lateral sclerosis based on the deterioration of individual speech subsystems,”  PloS one, vol. 11, no. 5, pp. e0154971, 2016.
  * [14] Marc S De Bodt, Maria E Hernandez-Diaz Huici, and Paul H Van De Heyning,  “Intelligibility as a linear combination of dimensions in dysarthric speech,”  Journal of communication disorders, vol. 35, no. 3, pp. 283–292, 2002.
  * [15] HM Chandrashekar, Veena Karjigi, and N Sreedevi,  “Spectro-temporal representation of speech for intelligibility assessment of dysarthria,”  IEEE Journal of Selected Topics in Signal Processing, vol. 14, no. 2, pp. 390–399, 2019.
  * [16] Abner Hernandez, Paula Andrea Pérez-Toro, Elmar Nöth, Juan Rafael Orozco-Arroyave, Andreas Maier, and Seung Hee Yang,  “Cross-lingual self-supervised speech representations for improved dysarthric speech recognition,”  arXiv preprint arXiv:2204.01670, 2022.
  * [17] Omar Mohamed and Salah A Aly,  “Arabic speech emotion recognition employing wav2vec2. 0 and hubert based on baved dataset,”  arXiv preprint arXiv:2110.04425, 2021.
  * [18] Nik Vaessen and David A Van Leeuwen,  “Fine-tuning wav2vec2 for speaker recognition,”  in ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2022, pp. 7967–7971.
  * [19] Tamás Grósz, Dejan Porjazovski, Yaroslav Getman, Sudarsana Kadiri, and Mikko Kurimo,  “Wav2vec2-based paralinguistic systems to recognise vocalised emotions and stuttering,”  in Proceedings of the 30th ACM International Conference on Multimedia, 2022, pp. 7026–7029.
  * [20] Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, et al.,  “Huggingface’s transformers: State-of-the-art natural language processing,”  arXiv preprint arXiv:1910.03771, 2019.
  * [21] Lara Gauder, Leonardo Pepino, Luciana Ferrer, and Pablo Riera,  “Alzheimer disease recognition using speech-based embeddings from pre-trained models.,”  in Interspeech, 2021, pp. 3795–3799.
  * [22] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay,  “Scikit-learn: Machine learning in Python,”  Journal of Machine Learning Research, vol. 12, pp. 2825–2830, 2011\.
  * [23] Brian McFee, Colin Raffel, Dawen Liang, Daniel P Ellis, Matt McVicar, Eric Battenberg, and Oriol Nieto,  “librosa: Audio and music signal analysis in Python,”  in Proceedings of the 14th python in science conference. Citeseer, 2015, vol. 8, pp. 18–25.