# Quantum localization corrections from the Bethe-Salpeter equation

Afifa Yedjour  Faculté de Physique, USTO MB, B.p.1505 El M’Naour, 31000.Oran,
Algérie.  Abdelaali Boudjemaa  Department of Physics, Faculty of Exact
Sciences and Informatics, Hassiba Benbouali University of Chlef, P.O. Box 78,
02000, Chlef, Algeria<EMAIL_ADDRESS>
###### Abstract

We investigate coherent matter wave transport in isotropic 3D speckle
potentials by using the Bethe-Salper equation and the self-consistent theory
of localization. This model constitutes an efficient tool to properly evaluate
corrections to Boltzmann diffusion by taking into consideration quantum
interference terms between the multiple-scattering paths. We calculate
analytically and numerically the static current density, the density of
states, the dipolar contribution and the reduced diffusion coefficient. Our
results reveal that quantum corrections to diffusive transport, known as weak
localization may not only lead to shift the above quantities but affect also
the position of the mobility edge.

Keywords: Cold atoms, Optical speckle potentials, Bethe-Salper equation,
Quantum localization correction, Quantum transport, Diffusion coefficient.

##  I Introduction

It has been established that the transport properties of a quantum particle in
a disordered system are intrinsically determined by the interference of
several scattering paths, which can lead to a spacial localization ref1 ; ref2
; ref3 ; ref4 . The quantum particle then remains localized around its initial
position, and it leads to a total suppression of transport, giving rise to
arrest the diffusion and hence, cancel the conductivity ref5 ; ref6 .

Experimentally, the localization has been observed in many systems including
light waves Wier ; Sche ; Sto ; Sch ; Lah , microwaves Dal ; Chab , sound
waves Weav , electron gases ref24 , and coherent matter waves ref7 ; ref8 .
Moreover, ultracold atoms open a new scenario for studying disorder-induced
localization, due to high degree of interactions. Experimental observation of
the Anderson transition of coherent matter waves in a disordered optical
potential has been reported in Refs.ref7 ; ref8 . In three dimensions (3D),
experimental observation of Anderson localization in matter waves of dilute
Bose and Fermi gases in a speckle potential has been reported in ref10 ; ref9
.

One could argue that in disordered media the macroscopic transport properties
namely diffusion, weak and strong localization, depend on the statistical
properties of the disorder potential ref11 . The most interesting feature in
the statistical properties of transport is the return probability to a given
point in which all scattering paths are closed loops ref11 ; kunn . A weak
localization comes from the fact that the probability of the wave to return to
its initial position is possible through loop paths. Each loop can generate
two multiple scattering paths along which exactly the same phase accumulates
in successive scattering events results in constructive interferences of the
matter wave. This coherent effect which is valid for any specific realization
of the disorder potential leading to a diffusive transport for which the
diffusion coefficient is reduced. On the other hand, if the disorder is strong
the propagation of a coherent wave is stopped after a certain time in any
dimension indicating that the diffusion coefficient vanishes. Therefore, the
return probability decreases exponentially from a certain point in space with
a characteristic length known as the localization length ref12 . Recently,
quantum transport and Anderson localization of atomic matter waves in 3D
anisotropic disordered potentials have been investigated in Piraud ; Piraud1 ;
ref19 .

In the present paper, we use the Bethe-Salper equation and the self-consistent
Born approximation to study the transport properties of ultracold atoms
exposed to isotropic 3D speckle potentials far below the quantum degenerate
regime (low densities). This model helps deal with the transport phenomenon of
disordered BEC. We calculate in particular quantum corrections stemming from
the interference effects known as weak localization to classical transport.
Starting from the quantum kinetic theory, we calculate the current density and
determine all relevant quantities such as the spectral function, the dipolar
contribution, and the reduced diffusion coefficient that are necessary to
describe the average diffusion process. Some useful analytic expressions are
obtained in some limits.

Furthermore, we present our numerical solutions of the Bethe-Salpeter equation
for the transport phenomenon of BEC subjected to a laser speckle potential. By
means of the self-consistent Born approximation we evaluate iteratively the
self-energy and the spectral function. This allows us to identify numerically
the current density, the density of state, the fraction of the localized atoms
and the diffusion coefficient. The main results emerging from this analysis is
that corrections due to the interference effects may alter the overall
behavior of these quantities and thus, affect the coherent transport regime.
We show that the diffusion constant which depends on the atomic energy and on
the disorder amplitude falls to zero due to the weak localization effects. The
weak localization to the diffusion constant can be measured by releasing a
confined atomic cloud and monitor its long-time spread inside the speckle
field by time-of-flight or in-situ imaging techniques kunn . The increasing of
the atomic energy leads to modify the position of the mobility edge. The most
advantage of our method is that it does not require cutoffs to eliminate short
wave paths that diverge even when scattering extends to infinite wave numbers
in contrast to the theories of Refs.kunn ; Piraud . Our results provide new
insights into the theory of Anderson localization and diffusion of matter wave
in isotropic optical disorders.

The rest of the paper is organized as follows. In section II we introduce the
basic concepts of quantum transport of matter waves in disordered media and
present the Bethe-Salpeter equation and its the eigenfunction. We obtain also
an analytic expression describing the quantum localization corrections for the
diffusion constant in the case of an isotropic scattering. Section III is
devoted to the numerical solutions of the Bethe-Salpeter equation for the
transport phenomenon of BEC subjected to a laser speckle potential and compare
them with classical regime. The current density and the reduced diffusion
constant are also compued and compared with the analytical results. In section
IV we present a brief summary.

##  II Coherent transport in a disordered medium

We consider a cloud of noninteracting cold atoms that is described by the
single particle Hamiltonian $H=p^{2}/2m+V({\bf r})$, where $V({\bf r})$ is a
static random potential introduced after the harmonic confinement in the
transport directions has been switched off as in the experiments realized in
ref7 ; ref8 . In such a regime of low densities, the evolution of the
condensate is described by a linear time-dependent Schrödinger equation with a
random potential. This latter is assumed to satisfy the following statistical
properties: $\langle V(\mathbf{r})\rangle=0$, and $\langle
V(\mathbf{r})V(\mathbf{r}^{\prime})\rangle=R(\mathbf{r}-\mathbf{r}^{\prime})$,
where $\langle\bullet\rangle$ denotes the disorder ensemble average and
$R(\mathbf{r}-\mathbf{r}^{\prime})$ is the disorder correlation function. In
the following we deal with coherent transport of such noninteracting cold
atoms in an isotropic disordered environment using the Bethe-Salpeter
equation.
We set $\hbar=1$ throughout the manuscript.

###  II.1 Solutions of Bethe-Salpeter equation

Let us start by considering two particles of Green function $\phi_{\bf
p,p^{\prime}}(\varepsilon,\Omega,\ {\bf q})$ which describes the probability
density of matter wave in a disorder potential. We seek for the hydrodynamic
expression which is defined as kunn ; Piraud ; ref15 :

$\phi_{\bf p,p^{\prime}}(\varepsilon,\Omega,\ {\bf
q})\equiv\overline{\langle{\bf p}_{+}|G(\varepsilon_{+})|{\bf
p}^{\prime}_{+}\rangle\langle{\bf
p}^{\prime}_{-}|G^{\dagger}(\varepsilon_{-})|{\bf p}_{-}\rangle},$ (1)

where $G$ is the retarded Green operator,
$\varepsilon_{\pm}=\varepsilon\pm\Omega/2$ , ${\bf p}_{\pm}={\bf p\pm q}/2$,
${\bf p^{\prime}_{\pm}=p^{\prime}\pm q}/2$ with ($q$ and $\Omega$ are the
Fourier conjugate variables of space and time, respectively ref15 ). Our
target is to analyze the behavior of the density diffusion for large distance
and long times. We anticipate that at long time ($\Omega\rightarrow 0)$ and
large distance $q\rightarrow 0$, the probability reads:

$\phi_{\bf p,p^{\prime}}(\varepsilon,\Omega,\ {\bf
q})=K\frac{L_{\varepsilon,{\bf p}}(\Omega,{\bf q})L_{\varepsilon,{\bf
p^{\prime}}}(\Omega,{\bf q})}{-i\Omega+D(\varepsilon)q^{2}},$ (2)

where $K$ is a normalized constant which will be determined later,
$L_{\varepsilon,\bf p}(\Omega,{\bf q})$ is the eigenfunction of the Bethe-
Selpeter equation associated with the hydrodynamic diffusion, and
$D(\varepsilon)$ is the diffusion constant. The density diffusion $\phi_{\bf
p,p^{\prime}}(\varepsilon,\Omega,\ {\bf q})$ is controlled by the Bethe-
Salpeter equation, which can be formally written as Piraud ; Piraud1 :

$\phi=\overline{G}\otimes\overline{G}^{\dagger}+\overline{G}\otimes\overline{G}^{\dagger}U\phi.$
(3)

The first term in Eq. (3) represents the intensity propagation with
uncorrelated disordered potential. The second term accounts for quantum
corrections, involves the vertex function $U_{\bf p,p^{\prime}}$ which
includes all correlations between different amplitudes in the density
propagation. The Green function is defined as : $G({\bf
p},\varepsilon)=[\varepsilon-\varepsilon_{\bf p}\ -\Sigma(\varepsilon,\ {\bf
p})]^{-1}$, where $\varepsilon_{p}=p^{2}/2m$ is the kinetic energy of the
particle, and $\Sigma(\varepsilon,{\bf p})$ is the self-energy which has
already been examined in ref22 .

The product of the average propagators
$\overline{G}^{\dagger}(\varepsilon_{-},{\bf
p}_{-})\otimes\overline{G}(\varepsilon_{+},{\bf p}_{+})$ can be reformulated
in momentum space by using the identity
$\overline{G}^{*}*\overline{G}=\big{(}\overline{G}^{*}-\overline{G}\big{)}/\big{(}\overline{G}^{-1}-\overline{G}^{*-1}\big{)}$
kunn  as:

$\overline{G}(\varepsilon_{+},{\bf
p}_{+})\overline{G}^{*}(\varepsilon_{-},{\bf
p}_{-})=\frac{\overline{G}(\varepsilon_{+},{\bf p}_{+})\
-\overline{G}^{*}(\varepsilon_{-},{\bf
p}_{-})}{\varepsilon_{-}-\varepsilon_{+}-({\bf p-\frac{1}{2}q})^{2}+({\bf
p+\frac{1}{2}q})^{2}-\Sigma^{*}(\varepsilon_{-},{\bf
p}^{-})+\Sigma(\varepsilon_{+},{\bf p}^{+})},$ (4)

which can be rewritten in a compact form as:

$\overline{G}(\varepsilon_{+},{\bf
p}_{+})\overline{G}^{*}(\varepsilon_{-},{\bf p}_{-})=\frac{\Delta G_{\bf
p}(\varepsilon,\Omega,\ {\bf
q})}{-\Omega+2\,({\bf{p}\cdot\bf{q}})+\Delta\Sigma_{p}(\varepsilon,\Omega,{\bf
q})},$ (5)

where $\Delta G_{\bf p}(\varepsilon,\Omega,\ {\bf
q})=\overline{G}(\varepsilon_{+},{\bf
p}_{+})-\overline{G}^{*}(\varepsilon_{-},{\bf p}_{-})$, and
$\Delta\Sigma_{p}(\varepsilon,\Omega,\ {\bf q})=\Sigma(\varepsilon_{+},{\bf
p}_{+})-\Sigma^{*}(\varepsilon_{-},{\bf p}_{-})$.
This leads to the standard quantum kinetic equation:

$\displaystyle\left[-\Omega+2({\bf p\cdot
q})+\Delta\Sigma_{p}(\varepsilon,\Omega,\ {\bf q})\right]\phi_{\bf
p,p^{\prime}}(\varepsilon,\Omega,\ {\bf q})$ $\displaystyle=$
$\displaystyle\Delta G_{\bf p}(\varepsilon,\Omega,\ {\bf q})\delta_{\bf
p,p^{\prime}}$ (6) $\displaystyle+$ $\displaystyle\Delta G_{\bf
p}(\varepsilon,\Omega,\ {\bf q})\sum_{p^{\prime\prime}}U_{\bf
p,p^{\prime\prime}}(\Omega,\ {\bf q})\phi_{\bf
p^{\prime\prime},p^{\prime}}(\varepsilon,\Omega,\ {\bf q}).$

As shown by Vollhardt and Wölfle ref17 , the irreducible vertex $U_{\bf
p,p^{\prime}}$ and the self-enegy $\Sigma_{\bf p}$ are related with each other
through the Ward identity

$\Delta\Sigma_{p}(\varepsilon,\Omega,\ {\bf q})=\sum_{p^{\prime}}\Delta G_{\bf
p^{\prime}}(\varepsilon,\Omega,\ {\bf q})U_{\bf
p^{\prime},p}(\varepsilon,\Omega,\ {\bf q}),$ (7)

which describes the flux conservation.
Upon integrating over $p$, we obtain a simpler form of the continuity equation
that describes the local conservation of the probability density

$\displaystyle-\Omega\ P_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf q})\ +{\bf
q}\cdot{\bf j}_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf q})\ \ =\Delta
G_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf q}),$ (8)

where $\Omega=i\partial_{t}$ and ${\bf q}=i{\bf\nabla}$. This conservation
equation which is similar to that obtained in Ref.kunn  is valid for any
initial distribution.

The probability density is written as: $P_{\bf
p^{\prime}}(\varepsilon,\Omega,\ {\bf q})=\sum_{p}\phi_{\bf
p,p^{\prime}}(\varepsilon,\Omega,\ {\bf q})$ and the current density reads:
${\bf j}_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf q})=\sum_{p}2{\bf
p}\cdot{\bf\phi}_{\bf p,p^{\prime}}(\varepsilon,\Omega,\ {\bf q})$.
One can easily check from Eq. (6) that

$\sum_{pp^{\prime}}\phi_{\bf p,p^{\prime}}(\varepsilon,\Omega,\
q=0)=\frac{\sum_{p}\Delta G_{\bf p}(\varepsilon,\Omega,\ q=0)}{-\Omega},$ (9)

where the factor $1/\Omega$ indicates that the total integrated density is
conserved. The Ward identity (7) implies that $L_{\varepsilon,\bf p}=\Delta
G_{\bf p}(\varepsilon)$ is a good choice for the eigenfunction at least for
large distance $q\rightarrow 0$ and long time, $\Omega\rightarrow 0$.
Therefore, we write

$L_{\varepsilon,\bf p}(\Omega=0,{\bf q})=\Delta G_{\bf p}(\varepsilon,{\bf
q})-\arrowvert G(\varepsilon,{\bf p})\rvert^{2}\Gamma({\bf p,q})+O(q^{2}),$
(10)

where $\Gamma({\bf p,q})$ is supposed to be linear in $\bf{q}$ and can in
isotropic media be written as: $\sim{\bf j}({\bf p})2({\bf p}\cdot{\bf q})$.
Replacing Eq.(2) into Eq.(6) for $\Omega=0$, one gets the following
expression:

$\displaystyle\left[2({\bf p\cdot q})+\Delta\Sigma_{p}(\varepsilon,{\bf
q})\right]\big{[}\Delta G_{\bf p}(\varepsilon,{\bf q})\ -\arrowvert G({\bf
p},\varepsilon)\rvert^{2}{\bf j}(\varepsilon,{\bf p})2({\bf p}\cdot{\bf
q})\big{]}\ L_{\varepsilon,\bf p^{\prime}}({\bf q})$ $\displaystyle=$
$\displaystyle D(\varepsilon)q^{2}\Delta G_{\bf p}(\varepsilon,{\bf
q})\delta_{\bf p,p^{\prime}}$ $\displaystyle+$ $\displaystyle\Delta G_{\bf
p}(\varepsilon,{\bf q})\sum_{p^{\prime\prime}}U_{\bf
p,p^{\prime\prime}}(\varepsilon,{\bf q})\big{[}\Delta G_{\bf
p^{\prime\prime}}(\varepsilon,{\bf q})$ $\displaystyle-$
$\displaystyle\arrowvert G(\varepsilon,{\bf p^{\prime\prime}})\rvert^{2}{\bf
j}(\varepsilon,{\bf p^{\prime\prime}})2({\bf p^{\prime\prime}}\cdot{\bf
q})\big{]}L_{\varepsilon,\bf p^{\prime}}({\bf q}).$

Expanding the identity (7) linearly in $q$ and inserting it into Eq.(II.1), we
obtain

$\displaystyle\left[2({\bf p\cdot q})+\Delta\Sigma_{p}(\varepsilon,{\bf
q})\right]\Delta G_{\bf p}(\varepsilon,{\bf q})\ -\Delta G_{\bf
p}(\varepsilon,{\bf q}){\bf j}({\bf p})2({\bf p}\cdot{\bf q})$
$\displaystyle=$ $\displaystyle\Delta G_{\bf p}(\varepsilon,{\bf
q})\Delta\Sigma_{\bf p}(\varepsilon,{\bf q})$ $\displaystyle-$
$\displaystyle\Delta G_{\bf p}(\varepsilon,{\bf
q})\sum_{p^{\prime\prime}}U_{\bf p,p^{\prime\prime}}(\varepsilon,{\bf
q})\arrowvert G(\varepsilon,{\bf p^{\prime\prime}})\rvert^{2}{\bf
j}(\varepsilon,{\bf p^{\prime\prime}})2({\bf p^{\prime\prime}}\cdot{\bf q}).$

After some algebra, the quantum kinetic equation takes the form

$2({\bf p\cdot q})+\sum_{p^{\prime\prime}}U_{\bf
p,p^{\prime\prime}}(\varepsilon,\ {\bf q})\arrowvert G(\varepsilon,{\bf
p^{\prime\prime}})\rvert^{2}{\bf j}(\varepsilon,{\bf p^{\prime\prime}})2({\bf
p^{\prime\prime}}\cdot{\bf q})={\bf j}(\varepsilon,{\bf p})2({\bf p}\cdot{\bf
q}).$ (13)

The current density is immediately found to be given by Eq.(13) with ${\bf q}$
replaced by ${\bf p}$

$\displaystyle{\bf j}(\varepsilon,{\bf p})=1+\sum_{p^{\prime\prime}}\frac{{\bf
p}^{\prime\prime}\ \cdot\ {\bf p}}{p^{2}}U_{\bf
p,p^{\prime\prime}}(\varepsilon,0)\arrowvert G(\varepsilon,{\bf
p^{\prime\prime}})\rvert^{2}{\bf j}(\varepsilon,{\bf p^{\prime\prime}}).$ (14)

This result is independent of the small momentum ${\bf q}$ since in deriving
Eqs. (II.1)-(14) we already kept only linear terms in ${\bf q}$, we therefore
put ${\bf q}=0$ in $U_{\bf p,p^{\prime\prime}}(\varepsilon,\ {\bf q})$ ref17 ;
VW ; ref18 . Equation (14) clearly shows that in lowest-order perturbation
theory, $j(\varepsilon,{\bf p})=1$. The term $({\bf p}^{\prime\prime}\cdot{\bf
p}/p^{2})U_{\bf p,p^{\prime\prime}}$ defines the cross section of all
interference contributions in multiple scattering. Analytical solutions of
Eq.(14) for 3D speckle potentials are shown in Appendix.

In order to find the eingenfunction of the Bethe-Salpeter equation, we employ
the following transformation ref18 :

$\Delta G_{\bf p}(\varepsilon,{\bf q})=G(\varepsilon,{\bf p+q}/2)\
-G^{*}(\varepsilon,{\bf p-q}/2)=2i\text{Im}G_{\bf p}(\varepsilon)\ +\
\frac{\partial\text{Re}\,G_{\bf p}(\varepsilon)}{\partial{\bf p}}\cdot{\bf
q}.$ (15)

Using the fact that

$\frac{\partial\text{Re}\,G_{\bf p}(\varepsilon)}{\partial{\bf p}}\cdot{\bf
q}=2\,{\bf p}\cdot{\bf q}\frac{\partial\text{Re}\,G_{\bf
p}(\varepsilon)}{\partial p^{2}},$

then introducing Eq.(15) into Eq. (10), one obtains for the eigenfunction of
the Bethe-Salpeter equation

$\displaystyle L_{\varepsilon,\bf p}({\bf q})=2i\,\text{Im}G_{\bf
p}(\varepsilon)\ +2\ {\bf p}\cdot{\bf q}\frac{\partial\text{Re}\,G_{\bf
p}(\varepsilon)}{\partial p^{2}}-\ \arrowvert G(\varepsilon,{\bf
p})\rvert^{2}2\ ({\bf p}\cdot{\bf q}){\bf j}(\varepsilon,{\bf p}),$ (16)

which admits an exact solution in linear order of $q$.

The spectrale function $\text{Im}G_{\bf p}(\varepsilon)$ contains all the
information about the diffusion of the particles in a disordered medium. It is
given by

$-\text{Im}G_{\bf p}(\varepsilon)=\frac{\text{Im}\Sigma_{\bf
p}(\varepsilon)}{\arrowvert{\varepsilon-\varepsilon_{\bf p}\ -\Sigma_{\bf
p}(\varepsilon)}\rvert^{2}},$ (17)

where the self-energy has the following form $\Sigma_{\bf
p}(\varepsilon)=\sum_{\bf p^{\prime}}\ U_{\bf
p,p^{\prime}}[\varepsilon-\varepsilon_{\bf p^{\prime}}-\Sigma_{\bf
p^{\prime}}(\varepsilon)]^{-1}$ which is a negative function (represents the
energy width of the spectral function). Thus,

$\arrowvert G(\varepsilon,{\bf p})\rvert^{2}=\frac{\text{Im}G_{\bf
p}(\varepsilon)}{\text{Im}\Sigma_{\bf p}(\varepsilon)}.$ (18)

The scattering mean-free time is defined as:

$\tau_{\bf p}(\varepsilon)=-\frac{\hbar}{2\ \text{Im}\Sigma_{\bf
p}(\varepsilon)}.$ (19)

Upon inserting Eqs.(17)-(19) into Eq.(16) and removing $-2i$, one finds:

$L_{\varepsilon,\bf p}({\bf q})=-\text{Im}G_{\bf p}(\varepsilon)\ -i({\bf
p}\cdot{\bf q})\ \gamma({\bf p},\varepsilon),$ (20)

where

$\gamma(\varepsilon,{\bf p})=\ \text{Im}G_{\bf p}(\varepsilon)\,2\tau_{\bf
p}(\varepsilon){\bf j}(\varepsilon,{\bf p})\ -\frac{\partial\text{Re}\,G_{\bf
p}(\varepsilon)}{\partial p^{2}}\ +O(q^{2}),$ (21)

is called the dipolar contribution which generates the flow of energy and
thus, provides corrections that support a current density. The solution of the
Bethe Salpeter equation $L_{\varepsilon,\bf p}({\bf q})$ at small $q$ is
governed by the spectral function ($A(\varepsilon,{\bf p})=-\text{Im}G_{\bf
p}(\varepsilon)/\pi$).

Finally, the normalization constant $K$ can be determined by matching Eq. (9)
and Eq.(2) for $q=0$. This yields

$K\sum_{pp^{\prime}}\frac{\text{Im}G_{\bf p}(\varepsilon)\text{Im}G_{\bf
p^{\prime}}(\varepsilon)}{-i\Omega}=\frac{\sum\limits_{p}2i\text{Im}G_{\bf
p}(\varepsilon)}{-\Omega},$ (22)

we thus, immediatly deduce that

$K=\frac{2}{\sum\limits_{p}-\text{Im}G_{\bf p}(\varepsilon)}.$ (23)

Replacing the resulting constant (23) into Eq.(2), one gets for the quantum
propability

$\phi_{\bf p,p^{\prime}}(\Omega,\ {\bf
q})=\frac{2}{\sum\limits_{p}-\text{Im}G_{{\bf
p}}(\varepsilon)}\frac{L_{\varepsilon,\bf p}({\bf q})L_{\varepsilon,\bf
p^{\prime}}({\bf q})}{-i\Omega\ +\ Dq^{2}},$ (24)

which has a diffusion pole according to kunn ; Piraud ; ref18 . The sum
$\sum\limits_{p}-\text{Im}G_{\bf p}(\varepsilon)$ is linked to the density of
states per unit volume $\rho(\varepsilon)$ as:
$\sum\limits_{p}-\text{Im}G_{\bf p}(\varepsilon)=\pi\rho(\varepsilon)$.
For $p\longrightarrow\infty$, $\text{Im}\Sigma_{\bf p}(\varepsilon)$ vanishes,
we then expect that $j(p)\longrightarrow 1$. The dipolar contribution
$\gamma(\varepsilon,{\bf p})$ has the following form

$\gamma_{0}(\varepsilon,{\bf p})=\ \ \arrowvert G(\varepsilon,{\bf
p})\rvert^{2}\ -\frac{\partial\text{Re}\,G_{\bf p}(\varepsilon)}{\partial
p^{2}}.$ (25)

Here $\gamma_{0}(\varepsilon,{\bf p})$ describes multiple scattering as a
sequence of scattering events where both retarded and advanced amplitudes
travel along the same path. When $p\longrightarrow\infty$, the kinetic energy
of the atoms exceeds the exitation potential, a large isotropic waves travel
without feeling the disorder effect ref15 ; kunn .

###  II.2 Diffusion constant from quantum localization corrections for
isotropic scattering

We now concentrate on the calculation of the diffusion constant using the
quantum kinetic equation derived in the previous section. Let us write the
expression of the current density

${\bf j}_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf q})=2\sum_{p}\
{\bf{p}\cdot\phi_{\bf p,p^{\prime}}}(\varepsilon,\Omega,\ {\bf q}).$ (26)

Incorporating $\phi_{\bf p,p^{\prime}}$ from Eq.(24) into ${\bf j}_{\bf
p^{\prime}}(\varepsilon,\Omega,\ {\bf q})$, we find

${\bf j}_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf
q})=\frac{2}{\pi\rho(\varepsilon)}\frac{L_{\varepsilon,\bf p^{\prime}}({\bf
q})}{-i\Omega\ +\ Dq^{2}}\sum_{p}2{\bf p}\ L_{\varepsilon,\bf p}({\bf q}).$
(27)

Inserting the corrections manifesting in $L_{\varepsilon,\bf p}({\bf q})$ into
Eq. (27), we obtain

${\bf j}_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf
q})=\frac{2}{\pi\rho(\varepsilon)}\frac{L_{\varepsilon,\bf p^{\prime}}({\bf
q})}{-i\Omega+\ Dq^{2}}i\ {\bf q}\ \ \frac{2}{3}\sum_{p}\bigg{[}-\
\text{Im}G_{\bf p}(\varepsilon)2\tau_{\bf p}(\varepsilon){\bf
j}(\varepsilon,{\bf p})p^{2}\ -\ p^{2}\frac{\partial\text{Re}\,G_{\bf
p}(\varepsilon)}{\partial p^{2}}\bigg{]}.$ (28)

The factor $(1/3)$ that appears in Eq. (28) comes from the angular integral.

The probability density is defined as:

$P_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf q})=\sum_{p}\phi_{\bf
p,p^{\prime}}(\varepsilon,\Omega,\ {\bf q})=2\ \frac{L_{\varepsilon,\bf
p^{\prime}}({\bf q})}{-i\Omega\ +\ Dq^{2}}.$ (29)

According to Fick’s law, which relates the diffusive flux to the concentration
gradient in the diffusive regime ($\Omega,{\bf q}\rightarrow 0$), ${\bf
j}_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf q})=i\ {\bf
q}D(\varepsilon)P_{\bf p^{\prime}}(\varepsilon,\Omega,\ {\bf q})$, the
diffusion constant is extracted as:

$D(\varepsilon)\ =\
\frac{1}{\pi\rho(\varepsilon)}\frac{2}{3}\sum_{p}p^{2}\left(-\text{Im}G_{\bf
p}(\varepsilon)2\tau_{\bf p}(\varepsilon){\bf j}(\varepsilon,{\bf p})\ -\
p^{2}\frac{\partial\text{Re}\,G_{\bf p}(\varepsilon)}{\partial p^{2}}\right).$
(30)

This equation is appealing since it includes weak localization corrections. In
the regime of a weak disorder, $\text{Im}G_{\bf p}(\varepsilon)$ is strongly
peaked near $p=\sqrt{\varepsilon}$, yielding
$\pi\rho(\varepsilon)\approx\sqrt{\varepsilon}/(4\pi)$. Therefore,
$D(\varepsilon)$ reduces to:

$D(\varepsilon)=\frac{4\pi}{\sqrt{\varepsilon}}\frac{2}{3}\frac{4\pi}{(2\pi)^{3}}\pi\int_{0}^{\infty}dpp^{4}\delta(\varepsilon-p^{2})2\tau_{\bf
p}(\varepsilon){\bf j}({\varepsilon,\bf p}).$ (31)

Keeping in mind that the second term is negligible, we find:
$D(\varepsilon)=4\varepsilon\,j(\sqrt{\varepsilon})\tau_{p=\sqrt{\varepsilon}}/3$.
Asuming that $j(\sqrt{\varepsilon})=1$ at large $\varepsilon$, we recover the
familiar expression for classical wave namely $D(\varepsilon)=v^{2}\tau^{*}/3$
Shap ; Boudj , where $v$ is the wave velocity, and $\tau^{*}$ is a transport
mean free time which is straightforwardly associated with the transport mean
path $\ell^{*}$ via $\tau^{*}=\ell^{*}/v$.

In the next section, we will discuss the critical regime of transition by
numerically solving the above Bethe-Salper equation.

##  III Numerical results

In this section, we solve numerically the Bethe-Salpeter equation for the
transport phenomenon of BEC subjected to a laser speckle potential. To do so,
we apply the self-consistent Born approximation to calculate the self-energy
which is the key factor of the diffusion. The current density $j(p)/p$ is
computed iteratively, and then insert the result into Eq.(30) to obtain the
diffusion coefficient. The number of required iterations for convergence is
$100$ iterations which are enough to obtain the desired solution Boudj .

From now on, all energies, including $\Sigma$, are expressed in units of the
quantum correlation energy $\varepsilon_{\sigma}=1/(2m\sigma^{2})$ with
$\sigma$ being the correlation length, momenta are expressed as $p=k$, where
the de-Broglie wave number $k$ is scaled in units of $1/\sigma$ (we recall
that $\hbar$=1). We choose a constant value of the disorder amplitude
$(V_{R}=0.5\varepsilon_{\sigma})$ for which the condition of a perturbative
disorder to be valid $\varepsilon_{k}\gg V_{R}^{2}/\varepsilon_{\sigma}$
delande .

Figure 1: Static current density $j(p)/p$ as a function of $p\sigma$ for
$\varepsilon_{c}=0$.

In Figure 1 we report the numerical and analytical results of the current
density $j(p)/p$ as a function of the scaled wave number $p\sigma$. We see
that the current density reaches its maximum at $p\sigma\lesssim 1$, then it
decreases and becomes constant for large momenta. The analytical results and
the numerical simulation diverge from each other notably for small momenta. A
closer look at the same figure shows that the numerical simulation predicts a
hump in the current density at $p\sigma\sim 1$, indicating that more atoms can
participate in the multiple scattering. These atoms are dephazing from their
initial positions most probably due to the effect of a weak localization. Both
methods show that the current diffusion converge to $1$ at very large momentum
$(p\rightarrow\infty)$. In such a regime weak localization corrections
disappear and a classical diffusion takes place (see also Eq. (14)).

Figure 2: Spectral function for $p=0$ and $V_{R}=0.5\varepsilon_{\sigma}$.

To calculate the spectral function, we solve numerically Eq. (17). The results
are shown in Figure 2. We observe the appearance of a peak at
$\varepsilon=-0.5\varepsilon_{\sigma}$. The width half height of this peak
$\Delta\varepsilon=0.4\varepsilon_{\sigma}$ is inversely proportional to the
mean free time of the atoms in speckle. At low energies, the relevant
parameter of localization is the typical length $\ell_{c}$ which is associated
with a typical momentum $k_{c}=\sqrt{2m\varepsilon_{c}}$. Note that the width
of the spectral function at $\varepsilon=\varepsilon_{c}$ is of the order of
$\ell_{c}\sim\sigma/(V_{R}/\varepsilon_{\sigma})^{2}$ref23 . In our case, we
find $\ell_{c}\approx 4\sigma$. From the same figure we see that the mobility
edge is located at $\varepsilon_{c}/\varepsilon_{\sigma}=-0.04$ which implies
that $k_{c}/k_{\sigma}=0.2$. In the limit of weak disorder, we find
$k_{c}\ell_{c}=0.8$ satisfying the celebrated Ioffe-Regel criterion namely,
$(k_{c}\ell_{c}\sim 1)$ BART . For $\ell_{c}\gg 4\sigma$, the density atomic
exceeds the optical potential which means that $\ell_{c}$ varies faster than
the modulations of the speckle $\sigma$. In such a situation we fall to a
classical diffusion.

Figure 3:  Dipolar contributions $\gamma(p,\varepsilon)$ and
$\gamma_{0}(p,\varepsilon)$ for $\varepsilon/\varepsilon_{\sigma}=-0.1$. Here
$\gamma(p,\varepsilon)$ is associated with coherent matter wave transport (in
the presence of the quantum corrections) and $\gamma_{0}(p,\varepsilon)$
corresponds to the incoherent matter wave transport (in the absence of the
quantum corrections).

The solution of Bethe-Salpeter equation for the dipolar contribution in the
regime of coherent matter wave transport (i.e. in the presence of the quantum
corrections, $\gamma(\varepsilon,p)$) and for the incoherent matter wave
transport (i.e. classical regime, $\gamma_{0}(p,\varepsilon)$ is captured in
figure 3. We see that the two curves are almost indistinguishable, yielding an
excellent agreement in the region $p\sigma<0.5$. Whereas for $p\sigma>0.5$,
the curves diverge form each other (the width of quantum particles curve is
broadened) due to the enhancement of the correlations induced by weak
localization. The narrow spectrum in the classical regime indicates the
absence of quantum correlation effects. In the presence of the quantum
corrections, the energetic atoms are shifted by the interference giving rise
to increase the intensity of the dipolar contribution results in a large
anisotropic momentum because of their deviations by the disorder effect Piraud
. Given that $p\sigma$ equivalent to $\upsilon_{p}/\upsilon_{\sigma}$, where
$v_{\sigma}=1/\ m\sigma$, we find that the dipolar contribution
$\gamma(\varepsilon,p)$ is dominated by atoms with velocity
$v_{\sigma}=0.66/m\sigma$. We detect on the other hand a fraction of atoms
faster than $v_{\sigma}=1.11/m\sigma$ contributing to the multiple diffusion.
Our results show a large spectrum in $\gamma(\varepsilon,p)$ compared to that
obtained in Ref.ref18  since we take into account higher-order contribution to
the current density $j(p)$.

Figure 4: The density of states per unit of volume as a function of
$\varepsilon/\varepsilon_{\sigma}$ for $p=0$. In the classical diffusion the
density of states is given as:
$\rho(\varepsilon)=4\pi\delta(\varepsilon-\varepsilon_{p})$.

These quantum corrections have an important impact on the distribution of
waves packet. This can be seen in the density of states
$\rho(\varepsilon)=\int\text{ Im}G(\varepsilon,p)d{\bf p}/(2\pi)^{3}$ which
measures the average number of states in the random medium per unit of volume.
Figure 4 depicts that at energies $\varepsilon\ll\varepsilon_{\sigma}$, the
probability is small but finite. Our numerical calculation predicts that
around $38\%$ of atoms are localized due to the interference effects. Here the
fraction of localized atoms is given by
$f_{\text{loc}}\approx\int_{-\infty}^{\varepsilon_{c}}\ d\varepsilon\text{
Im}G(\varepsilon,p=0)$ ref23 . At higher energies, the probability grows with
the energy. In such a situation, one has
$\text{Im}G(\varepsilon,p)=-\pi\delta(\varepsilon-\varepsilon_{p})$, hence the
atoms do not feel the disorder potential and the system attains the classical
diffusion. Consequently, the density of states behaves like the free-space
expression $\rho(\varepsilon)\sim\varepsilon^{1/2}$, where the dispersion
relation reads $\varepsilon=p^{2}/2m$ (see black line in figure 4). It is
clear that the two curves (quantum and classical) cross with each other at
$\varepsilon\simeq 2.25\varepsilon_{\sigma}$ pointing out that the atoms
diffuse in a classical way at energies $\varepsilon\gtrsim
2.25\varepsilon_{\sigma}$.

Figure 5: Diffusion coefficient as a function of the energy for several values
of the disorder amplitude and for $p=0$.

In Figure 5, we plot the diffusion constant as a function of the energy for
different disorder amplitudes. Our results reveal that the diffusion
coefficient $mD(\varepsilon)$ reaches its maximum at negative energies for any
disorder amplitude, then it decreases when the energy becomes close to zero.
Such a decay in the diffusion coefficient which depends also on the scattering
amplitude as is seen in figure 5, is due to the weak localization effects
induced by the fluctuations caused by the number of scatterers.

Figure 6: Standard deviation of the diffusion coefficient in a critical regime
for $V_{R}/\varepsilon_{\sigma}=0.5$.

Let us discuss more profoundly the transport properties from diffusion
coefficient. To this end, we use the standard deviation to approach the
transition point. Figure 6 shows three distinct regions. In the first region,
$-0.12<\varepsilon/\varepsilon_{\sigma}<-0.04$, $mD(\varepsilon)$ decreases
rapidly with the energy until it attains the mobilty edge
$\varepsilon_{c}=-0.04\varepsilon_{\sigma}$. The situation is different in the
second region, $-0.04<\varepsilon/\varepsilon_{\sigma}<0$, where the diffusion
coefficient increases linearly. A similar behavior holds true for a Gaussian
disorder potential ref23 . In this region, the states cease to extend over the
disorder potential and thus delocalize. In the third interval $\varepsilon>0$
and for small energies, $mD(\varepsilon)$ is saturated to a constante value
($mD(\varepsilon)\approx 0.25$) owing to the weak localization effects. This
phenomenon is important because the wave transport is quite sensitive to large
momentum which leads to increase the interferences between scattered atoms
evoking a deviation of fast atoms from its initial position and hence reducing
the constant diffusion. This is in stark contrast with the Dirac-peaks
potential ref24 , where the atoms are insensitive to the disorder effects.
Note that in the anisotropic case, the diffusion coefficient increases in
power law in the limit of low energies while it behaves as
$D\propto\varepsilon^{5/2}$ at higher energies Piraud .

##  IV Conclusion

In this paper, we examined the transport and localization of matter waves in
isotropic 3D speckle potentials using the Bethe-Salper equation and the self-
consistent theory of Anderson localization. We calculated in particular the
fundamental transport quantities such as the current intensity, the dipolar
contribution, the density of states, the spectral function, and the reduced
diffusion constant. We found that these quantities deviate from their
classical counterpart due to corrections arising from multiple scattering of
matter waves. At low energy, our numerical calculations predict that the weak
localization corrections increase the probability density and may shift the
dipolar contribution. The reduction of the diffusion constant which depends on
the scattering amplitude signals the occurrence of weak localization effects.
Our results pointed out also that the diffusion process continues with the
initially fast atoms until the transport means path becomes of the order of
the wavelength of the condensate.

## CRediT author statement

Afifa Yedjour: Conceptualization, Methodology, Software, Data curation,
Writing- Original draft preparation.
Abdelaali Boudjemaa: Visualization, Investigation, Writing-Review and Editing

## Acknowledgments

AY gratefully acknowledges the helpful discussions with Bart Van Tiggelen.

##  V Appendix

In this appendix we derive an explicit expression for the density current in
the case of a 3D laser speckle potential produced by diffraction which is
widely used with quantum gases experiments, its correlation function is given
by $\langle U({\bf R})\rangle=V_{R}^{2}\mathrm{sinc}^{2}(R/\sigma)$, where
$V_{R}$ is the amplitude disorder. For $V_{R}\rightarrow\infty$ and
$\sigma\rightarrow 0$, the speckle potential reduces to the uncorrelated
white-noise random potential. In Fourier space one can write:

$U_{\bf p,p^{\prime}}=V_{R}^{2}\int d{\bf R}\sum_{\bf
p^{\prime}}\dfrac{\sin^{2}\left(R/\sigma\right)}{\left(R/\sigma\right)^{2}}e^{i({\bf
p}-\ {\bf p}^{\prime})\cdot{\bf R}}.$ (32)

Inserting $U_{\bf p,p^{\prime}}$ into Eq. (14) and using the fact that
$\sum_{\mathbf{p^{\prime}}}\equiv\int d{\bf p^{\prime}}/(2\pi)^{3}$, one finds

$j(\varepsilon,{\bf p})=1+\frac{V_{R}^{2}}{(2\pi)^{3}}I({\bf p,\
p^{\prime}})\arrowvert G(\varepsilon,{\bf
p^{\prime}})\rvert^{2}j(\varepsilon,{\bf p^{\prime}}),$ (33)

where

$I({\bf p,\ p^{\prime}})=\int d{\bf R}\int d{\bf p}^{\prime}\frac{{\bf
p}\cdot\ {\bf
p}^{\prime}}{p^{2}}\dfrac{\sin^{2}\left(R/\sigma\right)}{\left(R/\sigma\right)^{2}}e^{i({\bf
p}-\ {\bf p}^{\prime})\cdot{\bf R}}.$ (34)

Setting $\int d\hat{p^{\prime\prime}}=\ dp^{\prime\prime}p^{\prime\prime 2}$,
then the integral in momentum space becomes

$\displaystyle\int\frac{d\hat{p^{\prime}}}{4\pi}({\bf p}\ \cdot\ {\bf
p}^{\prime})e^{i({\bf p}-\ {\bf p}^{\prime})\cdot{\bf R}}$ $\displaystyle=$
$\displaystyle{\bf p}\cdot\frac{1}{p^{\prime}}e^{\ i\ {\bf p}\cdot{\bf
R}}\cdot i\frac{\partial}{\partial{\bf R}}\int d\hat{p^{\prime}}e^{-\ i\ {\bf
p}^{\prime\prime}\cdot{\bf R}}$ $\displaystyle=$ $\displaystyle{\bf
p}\cdot\frac{e^{i\ {\bf p}^{\prime}\cdot{\bf
R}}}{p^{\prime}}i\frac{\partial}{\partial{\bf R}}\mathrm{sinc}(p^{\prime}\ R)$
$\displaystyle=$ $\displaystyle\frac{\hat{p}.\hat{R}}{R}e^{i\ {\bf p}\cdot{\bf
R}}i\frac{\partial}{\partial p^{\prime}}\mathrm{sinc}(p^{\prime}\ R).$

Now the integral over $R$ can be evaluated as:

$\displaystyle\int\frac{d\hat{R}}{4\pi}\ \hat{R}\ e^{i{\bf p}\cdot{\bf R}}$
$\displaystyle=$ $\displaystyle\frac{1}{R}\frac{1}{i}\frac{\partial}{\partial\
p}\int\frac{d^{3}\hat{R}}{4\pi}e^{\ i\ {\bf p}\cdot{\bf
R}}=\frac{\hat{p}}{R}\frac{1}{i}\frac{\partial}{\partial\ p}\mathrm{sinc}(p\
R).$ (36)

With this, Eq (34) can be rewritten as:

$I(p,\ p^{\prime})=4\pi\frac{\partial}{\partial\ p}\frac{\partial}{\partial\
p^{\prime}}\int_{0}^{\infty}dR\,\mathrm{sinc}^{2}\left(\dfrac{R}{\sigma}\right)\mathrm{sinc}(p\
R)\ \mathrm{sinc}(p^{\prime}\ R)=4\pi\frac{\partial}{\partial\
p}\frac{\partial}{\partial\ p^{\prime}}\frac{1}{pp^{\prime}}\
K(p,p^{\prime}),$ (37)

where

$K(p,p^{\prime})=\int_{0}^{\infty}dR\
\mathrm{sinc}^{2}\left(\dfrac{R}{\sigma}\right)\left(\frac{\sin
p^{\prime}R}{R}\right)\left(\frac{\sin R}{R}\right).$ (38)

Putting

$M(p,p^{\prime})=\frac{\partial}{\partial\ p}\ \frac{\partial}{\partial\
p^{\prime\prime}}\ K(p,p^{\prime})=\int_{0}^{\infty}\
dR\,\mathrm{sinc}^{2}\big{(}R/\sigma\big{)}\cos(p\ R)\ \cos(p^{\prime\prime}\
R).$ (39)

Then employing the relation
$\int_{-\infty}^{+\infty}dRe^{iqR}/R=\pi\Theta(q)$, we obtain:

$M(p,p^{\prime})\ =\ \frac{\pi\sigma}{16}\left[2\ -\ p\ -\ p^{\prime}\ -\
2\lvert p\ -\ p^{\prime}\rvert\ +\lvert p\ +\ p^{\prime}\ -\ 2\rvert\ \ +\
\lvert 2\ -\ p\ +\ p^{\prime}\rvert\ \ +\ \lvert 2\ +\ p\ -\ p^{\prime}\rvert\
\right].$ (40)

Upon substituting Eq.(40) into Eq.(37), we find

$I(p,p^{\prime})\ =\ \frac{4\pi^{2}\sigma}{16}\left[I_{1}\ +\ I_{2}\ +\ I_{3}\
+\ I_{4}\ \right],$ (41)

where

$I_{1}=2\ \frac{p\ +\ p^{\prime}\ -\ 2}{p^{2}p^{\prime 2}}\left(-p^{2}\ -\
p^{\prime 2}\ +\ pp^{\prime}\ +2\ +p\ +\ p^{\prime}\
\right)\left(\Theta\left(p\ +\ p^{\prime}\ -\ 2\right)\ -\ 1\right),$ $I_{2}=\
-\ 4\ \frac{p\ -\ p^{\prime}\ }{p^{2}p^{\prime 2}}\left(p^{2}\ +\ p^{\prime
2}\ +\ pp^{\prime}\ \right)\left(\left(\Theta(p\ -\ p^{\prime}\ \right)\ +\
1\right)\left(1\ -\,\Theta\left(p^{\prime}\ -\ p\ \right)\right),$ $I_{3}=2\
\frac{p^{\prime}\ -\ p\ +\ 2}{p^{2}p^{\prime 2}}\left(p^{2}\ +\ p^{\prime 2}\
+\ pp^{\prime}\ -2\ -p\ +\ p^{\prime}\ \right)\left(\Theta\left(p^{\prime}\ -\
p\ +\ 2\right)\ -\ 1\right),$ $I_{4}=2\ \frac{p^{\prime}\ -\ p\ -\
2}{p^{2}p^{\prime 2}}\left(p^{2}\ +\ p^{\prime 2}\ +\ pp^{\prime}\ -2\ +p\ +\
p^{\prime}\ \right)\left(\Theta\left(p^{\prime}\ -\ p\ -\ 2\right)\ -\
1\right).$

Equation (14) was solved self-consistently in section III. Its behavior has
been displayed in Fig.1.

##  VI References

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# Scalable Learning of Latent Language Structure
With Logical Offline Cycle Consistency

Maxwell Crouse Ramon Astudillo Tahira Naseem Subhajit Chaudhury
Pavan Kapanipathi Salim Roukos Alexander Gray
{maxwell.crouse, ramon.astudillo, subhajit<EMAIL_ADDRESS>
{tnaseem, kapanipa<EMAIL_ADDRESS>
IBM Research

###### Abstract

We introduce Logical Offline Cycle Consistency Optimization (LOCCO), a
scalable, semi-supervised method for training a neural semantic parser.
Conceptually, LOCCO can be viewed as a form of self-learning where the
semantic parser being trained is used to generate annotations for unlabeled
text that are then used as new supervision. To increase the quality of
annotations, our method utilizes a count-based prior over valid formal meaning
representations and a cycle-consistency score produced by a neural text
generation model as additional signals. Both the prior and semantic parser are
updated in an alternate fashion from full passes over the training data, which
can be seen as approximating the marginalization of latent structures through
stochastic variational inference. The use of a count-based prior, frozen text
generation model, and offline annotation process yields an approach with
negligible complexity and latency increases as compared to conventional self-
learning. As an added bonus, the annotations produced by LOCCO can be
trivially repurposed to train a neural text generation model. We demonstrate
the utility of LOCCO on the well-known WebNLG benchmark where we obtain an
improvement of $2$ points against a self-learning parser under equivalent
conditions, an improvement of $1.3$ points against the previous state-of-the-
art parser, and competitive text generation performance in terms of BLEU
score.

##  1 Introduction

Large language models (LLMs) have brought dramatic gains to semantic parsing-
related tasks, allowing for more performant systems that require significantly
less effort to adapt from one domain to the next. However, while their impact
has been undeniable, they still face numerous challenges. First, LLMs are
originally trained for text-only, sequence-to-sequence problems. In contrast,
semantic parsing is inherently a text-to-structure problem, wherein the
objective is to take in text as input and produce a logical form that is most
commonly a tree or graph (see Figure 1 for an example). Beyond the need to
account for explicit structure, LLMs must also overcome a paucity of training
examples, which generally require costly expert-level knowledge to collect in
this space.

To better generalize to formal, structured representations and alleviate data-
scarcity concerns, many high performing text-to-structure and structure-to-
text models employ a form of bootstrapping. That is, they fine-tune an initial
model using whatever supervised data is available and then subsequently use
that model to annotate a large amount of unlabeled text to serve as additional
training data [27, 45, 30, 6, 49, 39, 39, 29, 4]. This form of data
augmentation is commonly referred to as self-learning, with the parsed data
being referred to as pseudo-labels or silver data.

Unfortunately, using fine-tuned models to generate data is not always
straightforward, since, without specific modifications (e.g., [49, 12]) most
pretrained neural models do not offer any well-formedness guarantees. While
some approaches that are applied to simpler datasets can sidestep this issue
by deriving synthetic examples from grammars induced from the supervised data
[23, 3], such a strategy is untenable in more realistic open-ended domains. In
addition to well-formedness concerns, self-learning models also introduce
noise in the labels and are known to saturate in performance relatively
quickly (only one round of self-learning labeling and training is used in
state-of-the-art systems). More elaborate approaches leveraging latent
variable models [47] are more robust to such noise and can improve silver data
quality over multiple update rounds; however, they require marginalizing over
latent discrete structures, which adds significant complexity and
computational overhead.

$<$$>$ $<$$>$ $<$$>$ Figure 1: Text-to-RDF example from the WebNLG dataset for
the sentence, "Aarhus Airport is in Tristrup, Denmark which is part of the
Central Denmark Region."

In this work, we introduce Logical Offline Cycle Consistency Optimization
(LOCCO), a novel semi-supervised method for training a semantic parser that is
designed to address the aforementioned issues. Our method predicts parses for
a corpus of text; however, rather than treating the predictions as gold data,
each prediction is weighted as a function of two scores: 1) an LLM-produced
cycle-consistency score that provides a strong signal as to how faithful a
predicted sample is to its original text and 2) a count-based prior
probability that gives higher scores to parses that are syntactically valid
and share common substructure with other sampled parses across the corpus. The
result is a model that is incentivized to produce less-noisy parses that are
both coherent with respect to the input text and structurally regular. LOCCO
has a principled theoretical foundation as stochastic variational inference
[20] and can also be related to offline reinforcement learning. Importantly,
our method is straightforward to implement, trivial to parallelize, and comes
with very little added computational cost to standard silver data training. In
addition to producing a strong semantic parser, the output annotations
produced by LOCCO can also be used to train a structure-to-text model.

Contributions: (a) We introduce LOCCO, a semi-supervised method for training a
neural semantic parser. (b) We demonstrate how the weakly-supervised output of
LOCCO can be repurposed to train a strong text generation model. (c) We
demonstrate the effectiveness of LOCCO on the well-known WebNLG 2020 [8]
benchmark, where we improve semantic parsing by $1.3$ points over the previous
state-of-the-art parser while also achieving competitive text generation
performance. (d) We compare LOCCO to similar semi-supervised models on the
standard ATIS semantic parsing benchmark and demonstrate competitive
performance without the need for expensive online sampling. (e) We perform an
ablation analysis to determine how each component of LOCCO contributes to
overall performance.

##  2 Related Work

###  2.1 Cycle Consistency and Latent Variable Optimization

End-to-end differentiable Cycle-Consistency (CC) losses concern two
probabilistic models relating two domains $p(x\mid z)$ and $p(z\mid x)$, e.g.,
text / image or text / text. The parameters of both distributions are learned
end-to-end via gradient descent to maximize

$\mathbb{E}_{p(z\mid x)}[p(x\mid z)]=\int_{z\in D_{z}}p(x\mid z)p(z\mid
x)dz\quad\mbox{ or }\quad\mathbb{E}_{p(z\mid x)}[p(x\mid z)]=\sum_{z\in
D_{z}}p(x\mid z)p(z\mid x)$

for continuous or discrete bottleneck variables, respectively. Approaches
either optimize for one bottleneck, i.e., $z$, or both $x$ and $z$
simultaneously. CC losses are often used in a semi-supervised fashion by
combining datasets where only $z$ or $x$ are available with datasets where
they are both available. CC has been shown to be successful in many areas of
application, including image transformation [50], machine translation [18, 9],
speech-to-text, and text-to-speech [21, 42]. For all of these domains, the
expectations over the output sets are intractable. For continuous domains such
as image or speech, it is possible to backpropagate directly through either
reparametrization or by collapsing the distribution over the mean111Although
not explicitly stated, the output of the composed networks can be interpreted
as the mean of a constant variance Laplace distribution, reducing to
$||x-\mathbb{E}_{p(x\mid\mathbb{E}[z\mid x])}[x]||_{1}$ [50]. For discrete
domains, such as text or formal languages, this is not possible and
approximations are needed like strong independence assumptions, straight-
through approximations [5, 22] as in [42], the score-function estimator (i.e.,
REINFORCE [44]) used in [18, 21], or collapsing the distribution to $K$-best
[9].

CC losses are related as well to semi-supervised end-to-end learning with
latent variables when those variables correspond to interpretable domains,
e.g., latent summarization models [34], trees [11, 47] and sequence labeling
[48]. Most approaches leverage amortized variational inference in the form of
Variational Autoencoders [26] and some modified Expectation Maximization [48].
They are restricted to particular structures (e.g. trees) and some require
strong independence assumptions [48]. Here, we propose an offline version of
variational inference without structure restrictions, that can learn a prior
over the latent even when gradient learning is not possible (e.g., rule
learning). We also integrate and outperform LLM approaches, which have
generally displaced latent variable models.

###  2.2 Semantic Parsing and Text Generation

This work focuses on translating between natural language and formal language
domains (see [17] for a recent survey), e.g., parsing between text and a
knowledge graph expressing the semantics of the sentence (as in Figure 1). In
the area of parsing, there is a large corpus of literature in parse induction
[17] which often involves marginalization over latent structures. Although
related to the presented work, these works have two fundamental differences.
They are focused on the unsupervised case [10] with few works considering
semi-supervised learning. They often require strong independence assumptions,
e.g., context-free grammars. Beyond parsing, there are a large number of works
focused on joint learning of semantic parsing and text generation [15, 1, 13,
16]. Similar to our work is CycleGT [16], which learns using a CC loss based
on iterative back-translation [19]. Also relevant to our work is that of [13],
which jointly learns both transformations without a CC loss, instead applying
REINFORCE to approximate non-differentiable losses such as BLEU and METEOR.

##  3 Our Technique

###  3.1 Desiderata

The objective of this work is to provide an algorithm for parsing between
text, $x$, and formal structured representations, $z$ (i.e., text-to-structure
and structure-to-text). The method should be able to harness recent
developments in neural network pretraining, as well as available inductive
biases and learning algorithms in the formal domain. In short, we aim to

  * •

be able to leverage strong pretrained transformer models (e.g., BART [31] or
T5 [38]) to learn functions mapping $x\rightarrow z$ and $z\rightarrow x$

  * •

be able to scale training to large data sizes, which implies overcoming the
lack of paired $(x,z)$ data samples

  * •

be able to incorporate arbitrary constraints into the formal domain $D_{z}$,
which may not be amenable to gradient-based learning, and further update these
during training

For this we propose a simple semi-supervised algorithm, inspired by Stochastic
Variational Inference [20], that fulfills the desiderata above. The algorithm
reduces to conventional cycle-consistency or self-learning under some
simplifications but outperforms both algorithms under the same experimental
conditions.

###  3.2 Logical Offline Cycle Consistency Optimization

To begin, we assume access to some supervised data consisting of pairs of
plain text $x$ and formal, structured representations $z$, i.e.,
$(x,z)\in\mathcal{D}^{S}$. In addition, we also assume access to much larger
quantities of only text, i.e., $x\in\mathcal{D}^{U}$. We start from a
probability distribution over sentences that arises from marginalizing over
the space of all latent structures $D_{z}$, e.g., all knowledge-graphs.

$p(x;\theta)=\sum_{z\in D_{z}}p(x,z;\theta)$ (1)

Following the usual variational formulation [43], one can express this
marginalization in terms of the Evidence Lower Bound (ELBO) and reformulate it
in a way that resembles a cycle consistency loss

$\displaystyle\log p(x;\theta)$ $\displaystyle\geq\overbrace{\log
p(x;\theta)-\mathrm{KL}(q(z\mid x;\phi)\mid\mid p(z\mid
x;\theta))}^{\mathrm{ELBO}}$ (2) $\displaystyle=\mathbb{E}_{z\sim q(z\mid
x;\phi)}[\log p(x\mid z;\theta)]-\mathrm{KL}(q(z\mid x;\phi)\mid\mid
p(z;\theta))$ (3)
$\displaystyle=\mathbb{E}_{\scriptsize\underbrace{z\\!\sim\\!q(z\\!\mid\\!x;\\!\phi)}_{\mbox{text-
to-structure}}}[\underbrace{\log p(x\mid z;\theta)}_{\mbox{\footnotesize
structure-to-text}}+\underbrace{\log p(z;\theta)}_{\mbox{\footnotesize
reasoner}}]\quad+\underbrace{\mathrm{H}(q_{\phi})}_{\mbox{\footnotesize
encoding entropy}}$ (4)

where $\mathrm{KL}()$ is the Kullback-Leibler divergence and $\mathrm{H}()$
the entropy. Variational methods alternate between maximizing the ELBO with
respect to $\phi$, bringing it closer to the marginal log-likelihood for
current $\theta^{i}$, and maximizing it with respect to $\theta^{i}$. From Eq.
2 one can see that setting $q_{\phi}$ equal to the posterior $p(z\mid
x;\theta)$ will make the bound tight yielding Expectation Maximization [35].
In this context $q_{\phi}$ is an auxiliary distribution that is recomputed for
each update of $\theta$.

With neural networks the alternate optimization of $\phi$ and $\theta$ with
gradient ascent becomes costly. Stochastic Variational Inference (SVI) [20]
alleviates this with updates based on a subset of the data, but requires a
large number of optimization steps and presents optimization problems [25].
Amortized variational inference, best exemplified by Variational Autoencoders
(VAEs) [26], solves this problem by reusing $q_{\phi}$ across all steps of
optimization of $\theta$ and simultaneously updating $\theta$ and $\phi$ via
gradient ascent of Eq. 3. VAEs set a parameter-less prior $p(z)$ and do not
update it during training.

The approach proposed here takes the formulation in Eq. 4 and the following
design choices

  * •

$q(z\mid x;\phi)$ is parametrized by a large language model with pretrained
parameters $\Omega$ that maps natural language to formal descriptions, i.e., a
semantic parser

  * •

$p(x\mid z;\rho)$ is parametrized with a separate copy of $\Omega$. It acts as
a conditional language model and is frozen after initialization to prevent
adaptation to faulty structures (note here that $\theta$ has been replaced
with $\rho$ to reflect separate parameters)

  * •

$p(z;\theta)$ is a count-based model factorizing the space of possible
substructures (e.g., into edges). It incorporates prior knowledge about the
formal language, such as valid statements

  * •

as an initial step, all models $q(z\mid x;\phi)$, $p(x\mid z;\rho)$ and
$p(z;\theta)$ are fine-tuned or trained with the labeled dataset
$\mathcal{D}^{S}$ of $(x,z)$ pairs

  * •

as in SVI we then alternate optimizing $\phi$ and $\theta$, but on full passes
over the unlabeled $\mathcal{D}^{U}$. We also use a counts estimator for
$\theta$, not gradient, and add $\mathcal{D}^{S}$ for regularization

As detailed222For ease of explanation, gradient updates shown are just
Stochastic Gradient Descent in Algorithm 1, the approach thus combines
alternate updates of parameters of SVI, but with full passes over the entire
$\mathcal{D}^{U}\cup\mathcal{D}^{S}$ with a count-based update. This has both
negligible overhead and low variance due to the large amount of samples. Text
to structure is a many to one mapping, which makes a count-based model also a
good choice i.e. there are fewer labels than for the text counterpart. With a
uniform $p(z;\theta)$, LOCCO reduces to cycle-consistency, albeit with offline
updates and frozen conditional language model. With a uniform $p(x\mid
z;\rho)$ it reduces to conventional self-learning.

The gradient update of $q(z\mid x;\phi^{i})$ includes an expectation over a
set $z\in D_{z}$ that is exponentially large as a function of the input (e.g.,
graphs) and requires back-propagating through $p(x\mid z;\rho)$ and
$p(z;\theta)$. We overcome this with the score function estimator [44] which
yields following Monte Carlo approximation for the gradient333The entropy term
$\mathrm{H}(q_{\phi})$ was empirically observed to have no effect and was
removed

$\displaystyle\nabla_{\phi^{i}}\mathbb{E}_{z\sim q(z\mid x;\phi^{i})}[V(z,x)]$
$\displaystyle=\mathbb{E}_{q(z\mid x;\phi^{i})}[\ V(z,x)\nabla_{\phi^{i}}\log
q(z\mid x;\phi^{i})\ ]$
$\displaystyle\approx\frac{1}{N}\sum_{n=1}^{N}V(z_{n},x)\nabla_{\phi^{i}}\log
q(z_{n}\mid x;\phi^{i}),\quad z_{n}\sim q(z\mid x;\phi^{i-1})$

where we make the additional offline assumption of $\phi^{i}\approx\phi^{i-1}$
for the purpose of sampling, and

$V(z,x)=\log p(x\mid z;\rho)+\log p(z;\theta^{i-1})$

Algorithm 1 Logical Offline Cycle Consistency Optimization

procedure LOCCO($\mathcal{D}^{S},\mathcal{D}^{U},q(z\mid x;\phi),p(x\mid
z;\rho),p(z;\theta),\Omega,K$)

$\hphantom{\quad}\mathllap{\rho}\leftarrow\Omega$ $\triangleright$
Initialization

$\hphantom{\quad}\mathllap{\phi^{0}}\leftarrow\Omega$

for batch $B$ sampled without replacement from $\mathcal{D}^{S}$ do
$\triangleright$ Supervised Warm-up

$\hphantom{\quad}\mathllap{\rho}\leftarrow\rho+\eta\cdot\frac{1}{|B|}\sum_{(x,z)\in
B}\nabla_{\rho}\log p(x\mid z;\rho_{k})$

$\hphantom{\quad}\mathllap{\phi^{0}}\leftarrow\phi^{0}+\eta\cdot\frac{1}{|B|}\sum_{(x,z)\in
B}\nabla_{\phi^{0}}\log q(z\mid x;\phi^{0})$

end for

for $(x,z)\in\mathcal{D}^{S}$ do $\triangleright$ Parts counts
$\Theta_{s}^{0}$ initially set to 0

$\Theta_{s}^{0}\leftarrow\Theta_{s}^{0}+1$ for $s\in\mathrm{parts}(z)$

end for

$\theta_{s}^{0}=\frac{\Theta_{s}^{0}}{\sum_{s^{\prime}}\Theta_{s^{\prime}}^{0}}$

for CC iteration $i\in[1,K]$ do $\triangleright$ Semi-supervised Training

$\mathcal{R}_{i}\leftarrow\emptyset$ $\triangleright$ Offline Inference
Network Update

for $x\in\mathcal{D}^{U}$ do

for $z_{j}\sim q(z\mid x;\phi^{i-1})$ do $\triangleright$ Draw $N$ samples

$v_{j}=\log p(x\mid z_{j};\rho)+\log p(z_{j};\theta^{i-1})$

$\mathcal{R}_{i}\leftarrow\mathcal{R}_{i}\cup\\{\ (x,z_{j},v_{j})\ \\}$

$\Theta_{s}^{i}\leftarrow\Theta_{s}^{i}+1$ for $s\in\mathrm{parts}(z_{j})$

end for

end for

for batch $B$ sampled without replacement from
$\mathcal{D}^{U}\cup\mathcal{D}^{S}$ do

$\phi^{i}\leftarrow\phi^{i}+\eta\cdot\frac{1}{|B||N|}\sum_{x\in
B}\sum_{(x,z_{n},R_{n})\in\mathcal{R}_{i}}R_{n}\nabla_{\phi^{i}}\log
q(z_{n}\mid x;\phi^{i})$

end for

$\theta_{s}^{i}=\frac{\Theta_{s}^{i}}{\sum_{s^{\prime}}\Theta_{s^{\prime}}^{i}}$
$\triangleright$ Logic Prior Update

end for

end procedure

This amounts to updating $\phi^{i}$ with the samples from the previous
iteration model $q(z\mid x;\phi^{i-1})$ as if they were gold but weighted by
$V(z,x)$ to reflect their possible imperfection. This offline update allows
for trivial parallelization of sampling and very delayed communication between
the sampler and optimizer, which permits the use of normal disk storage for
$V(z,x)$ values (diplayed in Figure 2).

Parallelizable Scoring$\log p(x|z)$Text Generator$\log p(z)$Logic Model$z$ \-
Structure (Sampled)Semantic Parser$x$ \- Observed TextUpdate Variational
Parameters$\log p(x|z)+\log p(z)$$z$ \- Structure (Sampled)Semantic Parser$x$
\- Observed TextUpdate Reasoner$z$ \- Structure (Sampled)Logic Model$x$ \-
Observed Text$z\sim p(z|x)$backpropagationupdate counts Figure 2:
Parallelization details for LOCCO semi-supervised training

The large variance of $V(z,x)$ as an estimate is problematic [41], and thus in
our implementation we make the following two adjustments from the
reinforcement learning literature. First, we normalize the reward as

$A(z,x)=\frac{V(z,x)-\mu}{\sigma}$

where $\mu$ and $\sigma$ are the mean and standard deviation of the reward
across all $N$ samples drawn from $q(z|x;\phi^{i-1})$. Second, following [40]
we substitute $V(z,x)$ by a clipped surrogate objective

$\begin{gathered}r_{z_{n}}=\frac{q(z_{n}|x;\phi^{i})}{q(z_{n}|x;\phi^{i-1})}\\\
R(z,x)=\min(r_{z_{n}}A(z,x),\
\operatorname*{clip}(r_{z_{n}},1-\epsilon,1+\epsilon)\ A(z,x))\end{gathered}$

where $\epsilon$ is a small constant ($\epsilon=0.2$ in our experiments). This
clipped objective limits the change to $q(z|x;\phi)$ at each training
iteration, thus helping to avoid catastrophic forgetting.

The optimization of $\theta$ is carried out with a simple count-based maximum
likelihood estimator with smoothing factor $\tau$ and a strong factorization
into parts, e.g., subexpressions

$p(z;\theta)=\prod_{s\in\mathrm{parts}(z)}p(s;\theta)\quad\mbox{with}\quad
p(s;\theta)=\theta_{s}=\frac{\Theta_{s}}{\sum_{s^{\prime}\in\mathcal{D}_{S}}\Theta_{s^{\prime}}}$

$s\in\mathrm{parts}(z)$ are all subtrees of the input logical form, e.g., when
the target forms are sets of triples (as in WebNLG) a subtree corresponds to
an individual triple. $\Theta_{s}$ contains a count of the number of times
part $s$ was observed in the entire corpus and is initialized with $\tau$.
$\mathcal{D}_{S}$ is the set of all data types.

##  4 Experiments

We performed an extensive series of evaluations utilizing two datasets, the
english version of the WebNLG2020+ dataset [8] and the ATIS dataset as
processed as in [14]. Our primary goals were to determine if LOCCO produces an
effective semantic parser and to assess the contribution of each component of
LOCCO to semantic parsing performance. In addition, we were also interested to
learn if the outputs of LOCCO could be used to train a reasonable text
generation system. For WebNLG we include a comparison with recent systems in
both parsing and generation, including the state-of-the art. We also include a
self-learning baseline, component ablation, and investigation into the effect
of iterative training. For ATIS we assess the effect of training data size on
performance.

###  4.1 Datasets

WebNLG is a dataset where each example is a pairing of text with a set of RDF
triples. The dataset contains 13,211 training pairs, 1,667 validation pairs,
2,155 pairs for testing semantic parsing, and 1,779 pairs for testing text
generation. Its use in this work was motivated by it being a well-known, open-
domain benchmark with several systems to compare against that tests both
semantic parsing and text generation. For our WebNLG experiments, silver data
consisted of 50,000 sentences randomly selected from the TekGen corpus [1].
TekGen is a large-scale dataset intended to provide a more realistic testbed
for knowledge extraction. It is comprised of text instances from Wikipedia
that have been annotated with a state-of-the-art RDF triple semantic parser.
As our system is intended to operate with unlabeled data, we used only the
text from examples extracted from the corpus.

ATIS is a semantic parsing dataset where each example is a pairing of text
with a $\lambda$-calculus logical form. The dataset consists of 4,434 training
pairs, 490 validation pairs, and 447 test pairs. We reproduce the StructVAE
experimental setup in [47] where the training set is split into two disjoint
subsets of varying sizes. One of the subsets is treated as the gold dataset
(i.e., keeping both the text and logical form) and the other is considered the
silver dataset (i.e., keeping only the text). This both tests LOCCO’s
performance for different data sizes and shows how the approach generalizes to
more complex meaning representations than straightforward RDF-triples. We also
provide StructVAE results for completeness444It is important to note that
StructVAE preceded the use of LLMs and is thus at a clear disadvantage.

We performed minimal processing of both datasets. The parentheses of each
logical form were replaced with <SE> and </SE> tags to demarcate expression
boundaries, and each text-to-structure and structure-to-text example was
prompted with either "Text to Graph:" or "Graph to Text:", respectively. For
WebNLG, we applied the following transformations to each example: 1) The
subject, relation, and object were marked with <S>, <R>, and <O> tags,
respectively and 2) the camel-cased text of each triple element was split into
individual words based on capitalization.

For WebNLG, we used the provided evaluation scripts to assess performance. For
semantic parsing, there were four types of scored matches; however, for space,
we display only the Exact Match metric in our results section (we provide the
full table of results in the Appendix). For text generation, we provide
results for BLEU, METEOR, and chrF++, with BLEU being our primary metric. With
ATIS, we report exact-match accuracy, i.e., whether or not the generated form
exactly matched the target.

###  4.2 Training Details

For all experiments, we used pretrained BART-large [31] as our model. The
semantic parser was taken to be the model produced at the last iteration of
semi-supervised training. For each iteration, we evaluated the model on
validation data after every 2500 update steps and kept only the top performing
model. We list all hyperparameters in the Appendix.

For our text generation experiments, we aimed to keep the training setup as
simple as possible. We first used the final model from text-to-structure
training to generate a new set of data (following the same setup as each of
the prior iterations). Then, we flipped the generated annotations, converting
each pair $(x,z)$ into $(z,x)$. Following the conversion, we trained a BART-
large model from scratch on the sampled annotations in the same way as was
done for the semantic parsing experiments.

Method | F1 | Precision | Recall
---|---|---|---
LOCCO (Ours) | 0.736 | 0.729 | 0.749
ReGen [13] | 0.723 | 0.714 | 0.738
Grapher [33] | 0.709 | 0.702 | 0.720
Amazon AI [15] | 0.689 | 0.689 | 0.690
bt5 [2] | 0.682 | 0.670 | 0.701
CycleGT [16] | 0.342 | 0.338 | 0.349
Baseline [8] | 0.158 | 0.154 | 0.164
(a) Semantic parsing ranked by F1

Method | BLEU | METEOR | chrF++
---|---|---|---
ReGen [13] | 0.563 | 0.425 | 0.706
LOCCO (Ours) | 0.552 | 0.406 | 0.691
Amazon AI [15] | 0.540 | 0.417 | 0.690
OSU NLG [32] | 0.535 | 0.414 | 0.688
FBConvAI [46] | 0.527 | 0.413 | 0.686
bt5 [2] | 0.517 | 0.411 | 0.679
NUIG-DSI [36] | 0.517 | 0.403 | 0.669
cuni-ufal [24] | 0.503 | 0.398 | 0.666
CycleGT [16] | 0.446 | 0.387 | 0.637
Baseline [8] | 0.406 | 0.373 | 0.621
RALI [28] | 0.403 | 0.386 | 0.634
(b) Text generation ranked by BLEU

Table 1: WebNLG test set results for semantic parsing (F1 Strict) and text
generation (BLEU, METEOR, chrF++). Dashed line includes existing results
matching or outperfomring LOCCO

##  5 Results

###  5.1 WebNLG

Our main results can be found in Tables 1(a) and 1(b), which show the
performance of our model for both semantic parsing and text generation as
compared to other approaches. As can be seen in Table 1(a), LOCCO achieves
state-of-the-art performance on the semantic parsing task, with a notable
improvement (0.13 F1) over the next best model ReGen [13]. Importantly, our
model achieves these results without any special modifications to the
underlying large language model (e.g., constrained output, triple reordering,
etc.) that are common to the other approaches on this dataset [15, 2].

In Table 1(b), we see that our approach yields a reasonably performant text
generation system. It outperforms all other approaches (many of which were
specifically designed for RDF-to-text) but ReGen. This is significant, as the
text generation system we use is functionally a byproduct of our process for
producing a semantic parser. It has no tailored architectural features and is
simply trained using the data produced by our semantic parser.

Method | Reward Function | Semantic Parsing | Text Generation
---|---|---|---
|  | F1 | Precision | Recall | BLEU | METEOR | chrF++
LOCCO | $\log p(x|z)+\log p(z)$ | 0.736 | 0.729 | 0.749 | 0.552 | 0.406 | 0.691
LOCCO | $\log p(x|z)$ | 0.733 | 0.725 | 0.745 | 0.551 | 0.416 | 0.692
LOCCO | $\log p(z)$ | 0.716 | 0.710 | 0.728 | 0.519 | 0.405 | 0.676
Greedy SL | – | 0.715 | 0.708 | 0.726 | 0.507 | 0.401 | 0.663
Sampling SL | – | 0.718 | 0.712 | 0.728 | 0.524 | 0.407 | 0.677
Gold-Only | – | 0.691 | 0.684 | 0.703 | 0.526 | 0.406 | 0.678
Table 2: WebNLG ablation results for semantic parsing (in terms of Exact
Match) and text generation

####  5.1.1 Ablation Experiments

In addition to our main results, we also perform extensive ablation
experiments to determine the contributions of each element of our training
objective. In Table 2 we show various ablations to the reward function of our
model, self-learning (SL) where the annotated silver parses are drawn from
either greedy or sampling-based decoding, and gold-only training where no
silver data is used.

From the table, it can be seen that using silver data in any capacity leads to
improved performance over gold-only training. This is a promising result, as
it suggests that our approach could be used to improve the other state-of-the-
art models that did not train with external data, e.g., [13]. The results
indicate that greedy and count-based rewards produce roughly the same
performance. This is somewhat unsurprising, as the count-based model should
reward higher-probability triples that are sampled frequently (i.e., those
that would be produced by greedy decoding). The most important result is that
the combination of cycle-consistency and the count-based logic model produces
the best performance, better than either score individually.

Figure 3: Semantic parsing performance across training iterations as measured
by Exact Match F1

####  5.1.2 Performance Across Epochs

Though the main results were based on the model trained at the final
iteration, we were also interested in the performance of each model in the
intermediate iterations of training. The across-iteration performance is shown
in Figure 3, where we see that the unablated version of LOCCO demonstrates
consistently higher performance than the other versions. In addition,
consistent with our remarks in Section 1, we see that sampling-based self-
learning produces strong results at first but then degenerates over time.
Another interesting result is that greedy self-learning is largely equivalent
in performance to LOCCO when only the prior $p(z)$ is used for the reward.
Again, we suspect that this is due to the nature of our count-based model
which upweights logical forms with frequent triples, i.e., those considered
more likely by our neural model, and thus more likely to also be a part of the
greedy decoding.

###  5.2 ATIS

$|\mathcal{D}^{S}|$ | LOCCO |  StructVAE [47] |  SOTA [7]
---|---|---|---
| Gold-Only | Self-Learning | $R(z,x)$ | Gold-Only | Self-Learning | $R(z,x)$ |
500 | 71.9 | 76.8 | 75.9 | 63.2 | 65.3 | 66.0 | –
1000 | 77.0 | 77.9 | 81.0 | 74.6 | 74.2 | 75.7 | –
2000 | 86.1 | 86.4 | 87.1 | 80.4 | 83.3 | 82.4 | –
3000 | 85.9 | 87.3 | 87.7 | 82.8 | 83.6 | 83.6 | –
4434 | 86.3 | – | – | 85.3 | – | – | 89.1
Table 3: Semantic parsing on ATIS for various training set sizes. The last row
reflects when all supervised data is used (i.e., there is no additional data
for semi-supervised training)

Table 3 shows our results on the ATIS semantic parsing dataset as compared to
StructVAE [47]. The first column shows the size of the gold dataset,
$\mathcal{D}^{S}$, while the remaining columns provide results for each
system. For both LOCCO and StructVAE we distinguish between training with
gold-only (i.e., only examples in $\mathcal{D}^{S}$ used), self-learning, and
$R(z,x)$ (i.e., when all silver examples are scored) settings. In addition, we
include the current state-of-the-art method [7] for reference.

Similar to StructVAE, LOCCO demonstrates performance gains over both the
supervised and self-learning settings (with the exception of
$|\mathcal{D}^{S}|=500$). This suggests that, like StructVAE, our approach is
producing more meaningful annotations of the unlabeled data than pure self-
learning. This is important, as the target meaning representation,
$\lambda$-calculus, is significantly more complex than the representations
required for WebNLG. Key to note is that our results are achieved with offline
sampling and scoring, while theirs requires sampling during training. Lastly,
we emphasize that our objective with this experiment was not to compare raw
performance, but was instead to determine if our approach yielded similar
gains as compared to the supervised and self-learning settings. While our
model demonstrated an overall improvement as compared to theirs, this is
likely attributable to our much stronger pretrained model (they use an LSTM
with GLOVE embeddings [37]) that provided a better baseline performance.

##  6 Conclusions

In this paper, we introduced Logical Offline Cycle Consistency Optimization
(LOCCO), a novel semi-supervised method for training a neural semantic parser.
Our method was inspired by Stochastic Variational Inference, and designed from
the ground up to be scalable, take advantage of powerful pretrained LLMs, and
be able to incorporate inductive biases relevant to the formal domain. We
demonstrated the effectiveness of our model on two standard benchmark
datasets, where it achieved strong performance for both semantic parsing and
text generation.

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##  Appendix A Appendix

###  A.1 Hyperparameters and Hardware Details

Hyperparameter | WebNLG | ATIS
---|---|---
Dropout | 0.0 | 0.0
Batch size | 8 | 8
Learning rate | 5e-6 | 5e-6
Training iterations | 3 | 3
Training epochs per iteration | 1 | 100
Patience | 5 | 5
Temperature | 1.0 | 1.0
Top-p | 0.95 | 0.95
Number of samples N | 5 | 5
Table 4: Hyperparameters for WebNLG and ATIS

For hyperparameter choices, the batch size and learning rate were chosen based
on common defaults for BART-large. Sampling-based parameters (i.e.,
temperature, top-p, and number of samples N) were similarly chosen to be those
commonly found in RL-based works. The decision to disable dropout was to
reduce a possible source of randomness in the results after observing it had
no effect on validation performance. We chose the number of epochs to run per
training iteration after observing that BART overfit the dataset (in the case
of WebNLG) or saturated in performance (in the case of ATIS) on validation
data around that point.

In terms of hardware, our experimental setup utilized a HPC cluster with CPU
and GPU machines running Red Hat Enterprise Linux release 8.7 (Ootpa). CPU
machines were used for all non-neural preprocessing and GPU machines were used
for model training. Both CPU and GPU machines had 2 CPU cores and 100GB of
RAM. GPU machines ran an NVIDIA V100 Tensor Core GPU with 40GB of GPU memory.

###  A.2 Full Results Tables

WebNLG measures semantic parsing performance with 4 different metrics, each
reflecting a degree of match quality. The 4 metrics are Exact, Entity Type,
Partial, and Strict. In Tables 5 and 6, we show results for all of these
metrics (and include the text generation metrics for ease of reference).

Method | Match | F1 | Precision | Recall
---|---|---|---|---
ReGen [13] | Exact | 0.723 | 0.714 | 0.738
| Entity Type | 0.807 | 0.791 | 0.835
| Partial | 0.767 | 0.755 | 0.788
| Strict | 0.720 | 0.713 | 0.735
Grapher [33] | Exact | 0.709 | 0.702 | 0.720
| Entity Type | – | – | –
| Partial | 0.735 | 0.725 | 0.750
| Strict | 0.706 | 0.700 | 0.717
Amazon AI [15] | Exact | 0.689 | 0.689 | 0.690
| Entity Type | 0.700 | 0.699 | 0.701
| Partial | 0.696 | 0.696 | 0.698
| Strict | 0.686 | 0.686 | 0.687
bt5 [2] | Exact | 0.682 | 0.670 | 0.701
| Entity Type | 0.737 | 0.721 | 0.762
| Partial | 0.713 | 0.700 | 0.736
| Strict | 0.675 | 0.663 | 0.695
CycleGT [16] | Exact | 0.342 | 0.338 | 0.349
| Entity Type | 0.343 | 0.335 | 0.356
| Partial | 0.360 | 0.355 | 0.372
| Strict | 0.309 | 0.306 | 0.315
Baseline [8] | Exact | 0.158 | 0.154 | 0.164
| Entity Type | 0.193 | 0.187 | 0.202
| Partial | 0.200 | 0.194 | 0.211
| Strict | 0.127 | 0.125 | 0.130
LOCCO (Ours) | Exact | 0.736 | 0.729 | 0.749
| Entity Type | 0.808 | 0.796 | 0.829
| Partial | 0.775 | 0.766 | 0.793
| Strict | 0.733 | 0.726 | 0.745
Table 5: Full WebNLG results for semantic parsing Method | Reward Function | Semantic Parsing | Text Generation
---|---|---|---
|  | Match | F1 | Precision | Recall | BLEU | METEOR | chrF++
LOCCO | $\log p(x|z)+\log p(z)$ | Exact | 0.736 | 0.729 | 0.749 | 0.552 | 0.406 | 0.691
|  | Entity Type | 0.808 | 0.796 | 0.829 |  |  |
|  | Partial | 0.775 | 0.766 | 0.793 |  |  |
|  | Strict | 0.733 | 0.726 | 0.745 |  |  |
LOCCO | $\log p(x|z)$ | Exact | 0.733 | 0.725 | 0.745 | 0.551 | 0.416 | 0.692
|  | Entity Type | 0.804 | 0.791 | 0.825 |  |  |
|  | Partial | 0.771 | 0.761 | 0.788 |  |  |
|  | Strict | 0.729 | 0.722 | 0.742 |  |  |
LOCCO | $\log p(z)$ | Exact | 0.716 | 0.710 | 0.728 | 0.519 | 0.405 | 0.676
|  | Entity Type | 0.798 | 0.786 | 0.818 |  |  |
|  | Partial | 0.760 | 0.751 | 0.777 |  |  |
|  | Strict | 0.712 | 0.705 | 0.723 |  |  |
Greedy SL |  | Exact | 0.715 | 0.708 | 0.726 | 0.507 | 0.401 | 0.663
|  | Entity Type | 0.786 | 0.775 | 0.805 |  |  |
|  | Partial | 0.755 | 0.745 | 0.770 |  |  |
|  | Strict | 0.708 | 0.702 | 0.719 |  |  |
Sampling SL |  | Exact | 0.718 | 0.712 | 0.728 | 0.524 | 0.407 | 0.677
|  | Entity Type | 0.785 | 0.775 | 0.802 |  |  |
|  | Partial | 0.755 | 0.747 | 0.769 |  |  |
|  | Strict | 0.713 | 0.707 | 0.723 |  |  |
Gold-Only |  | Exact | 0.691 | 0.684 | 0.703 | 0.526 | 0.406 | 0.678
|  | Entity Type | 0.762 | 0.750 | 0.783 |  |  |
|  | Partial | 0.732 | 0.722 | 0.749 |  |  |
|  | Strict | 0.684 | 0.677 | 0.696 |  |  |
Table 6: Full WebNLG ablation results

###  A.3 Dataset Examples

Here we provide additional examples from both of the datasets used in this
paper. In addition, we show how the $\mathrm{parts}(z)$ function breaks down a
logical form. Figure 4 provides examples from the WebNLG corpus while Figure 5
provides examples from the ATIS corpus.

####  A.3.1 WebNLG

Text x:
"The Aarhus is the airport of Aarhus, Denmark."
Logical Form z:
(<S> Aarhus Airport <R> city served <O> "Aarhus, Denmark")
parts(z):
{ (<S> Aarhus Airport <R> city served <O> "Aarhus, Denmark") }

Text x:
"The Acharya Institute of Technology’s campus is located in Soldevanahalli,
Acharya Dr. Sarvapalli Radhakrishnan Road, Hessarghatta Main Road, Bangalore,
India, 560090. It was established in 2000 and its director is Dr G.P.
Prabhukumar. It is affiliated to the Visvesvaraya Technological UNiversity in
Belgaum and has 700 postgraduate students."
Logical Form z:
(<S> Acharya Institute of Technology <R> affiliation <O> Visvesvaraya
Technological University)
(<S> Acharya Institute of Technology <R> campus <O> "In Soldevanahalli,
Acharya Dr. Sarvapalli Radhakrishnan Road, Hessarghatta Main Road, Bangalore –
560090.")
(<S> Acharya Institute of Technology <R> country <O> "India")
(<S> Acharya Institute of Technology <R> established <O> 2000)
(<S> Acharya Institute of Technology <R> motto <O> "Nurturing Excellence")
(<S> Acharya Institute of Technology <R> state <O> Karnataka)
(<S> Visvesvaraya Technological University <R> city <O> Belgaum)
parts(z):
{ (<S> Acharya Institute of Technology <R> state <O> Karnataka)
(<S> Visvesvaraya Technological University <R> city <O> Belgaum), … }

Figure 4: RDF examples from the WebNLG corpus

####  A.3.2 ATIS

Text x:
"is there a flight from ci1 to ci0 which connect in ci2"
Logical Form z:
(lambda $0 e (and (flight $0) (from $0 ci1) (to $0 ci0) (stop $0 ci2)))
parts(z):
{ (flight $0), (from $0 ci1), … }

Text x:
"where is ap0 locat"
Logical Form z:
(lambda $0 e (loc:t ap0 $0))
parts(z):
{ (loc:t ap0 $0), (lambda $0 e (loc:t ap0 $0)) }

Text x:
"show me the fare from ci1 to ci0"
Logical Form z:
(lambda $0 e (exists $1 (and (from $1 ci1) (to $1 ci0) (= (fare $1) $0))))
parts(z):
{ (= (fare $1) $0), (fare $1), … }

Figure 5: $\lambda$-calculus examples from the ATIS corpus