# Perturbed Holomorphic Cylinders, Exponential Estimates, and Flow Lines

Dylan Cant  Stanford University<EMAIL_ADDRESS>Dedicated to Ollie

(Date: Friday, October 8, 2021)

###### Abstract.

A convergence result for perturbed holomorphic cylinders in $\mathbb{R}^{2n}$
is proved; a sequence of perturbed holomorphic cylinders converges to a
configuration of two holomorphic disks joined by a flow line. A key step in
the computation is played by an a priori exponential estimate.

###### 2020 Mathematics Subject Classification:

Primary 32Q65, 53D05 Secondary 53D40

The author would like to thank Daren Chen for many useful discussions
surrounding this result.

##  1\. Introduction

######  Definition 1 (geometric set-up).

Let $V_{n}$ be a $C^{\infty}$ convergent sequence of vector fields on
$\overline{B}(1)^{2n}\subset\mathbb{R}^{2n}$, $\epsilon_{n}\to 0$, and
$u_{n}:[-r_{n}-1,r_{n}+1]\times\mathbb{R}/\mathbb{Z}\to\overline{B}(1)$
satisfy the following partial differential equation:

(1) $\partial_{s}u_{n}+J_{0}\partial_{t}u_{n}=\epsilon_{n}V_{n},$

where $J_{0}$ is the standard complex structure on
$\mathbb{R}^{2n}\simeq\mathbb{C}\times\mathbb{C}\times\cdots\times\mathbb{C}$.

flow line for $V$ Figure 1. Degeneration of perturbed holomorphic cylinders.
Here $V=\lim_{n}V_{n}$.

We think of solutions to (1) as being perturbed holomorphic cylinders. If
$V_{n}$ is a gradient vector field, then this equation is _Floer’s equation_
(restricted to a finite length cylinder) (see [Flo89] and [Sal97]).

The main goal in this short paper is to explain how a sequence of maps $u_{n}$
satisfying Definition 1 has a subsequence that converges (in a certain sense)
to a flow line for $V=\lim_{n\to\infty}V_{n}$ joining two holomorphic disks
(see Figure 1). More precisely, we will show that if $\epsilon_{n}r_{n}$
converges to a finite number $\ell>0$, or $r_{n}\to\infty$ and
$\epsilon_{n}r_{n}\to\ell=0$, then (a subsequence of) $u_{n}$ will converge to
two holomorphic disks joined by a flow line for $V$ of length $2\ell$.

Here is the theorem we will prove.

######  Theorem 1.

Let $u_{n}$ be a sequence satisfying Definition 1. Suppose
$\epsilon_{n}r_{n}\to\ell$, and if $\ell=0$ then $r_{n}\to\infty$. Then there
exist numbers $\rho_{n}\to\infty$ so that $\epsilon_{n}\rho_{n}\to 0$,
$\rho_{n}<r_{n}$, and so that if we decompose the domain:

$[-r_{n},r_{n}]\times\mathbb{R}/\mathbb{Z}=([-r_{n},-r_{n}+\rho_{n}]\cup[-(r_{n}-\rho_{n}),r_{n}-\rho_{n}]\cup[r_{n}-\rho_{n},r_{n}])\times\mathbb{R}/\mathbb{Z},$

and pass to a subsequence, we have:

  1. (i)

the maps $u_{n}^{-}(s,t)=u_{n}(s-r_{n},t)$ and
$u_{n}^{+}(s,t)=u_{n}(s+r_{n},t)$, when restricted to
$[0,\rho_{n}]\times\mathbb{R}/\mathbb{Z}$ and
$[-\rho_{n},0]\times\mathbb{R}/\mathbb{Z}$, converge uniformly on to
holomorphic cylinders defined on $[0,\infty)\times\mathbb{R}/\mathbb{Z}$ and
$(-\infty,0]\times\mathbb{R}/\mathbb{Z}$ with removable singularities $x_{-}$
and $x_{+}$.

  2. (ii)

the rescaled maps

$v_{n}(s,t)=u_{n}(\epsilon_{n}^{-1}s,\epsilon_{n}^{-1}t)$

defined on the rescaled central cylinder
$[-\epsilon_{n}(r_{n}-\rho_{n}),\epsilon_{n}(r_{n}-\rho_{n})]\times\mathbb{R}/\epsilon_{n}\mathbb{Z}$
converges uniformly to a flow line $v_{\infty}$ of $V=\lim_{n}V_{n}$ of length
$2\ell$ joining $x_{-}$ and $x_{+}$.

See Figure 2.

$u^{-}_{\infty}$$u^{+}_{\infty}$$x_{-}$$x_{+}$$v_{\infty}$ Figure 2.
Converging to the limiting configuration.

######  Proof.

Theorem 1 is a combination of Lemmas 6, 7, and 8, proved below.

######  Remark 2.

Other researchers have considered this problem previously. See [Oh05] for a
similar compactness result and see [OZ10] and [OZ12] for an analysis of the
reverse _gluing_ process. The paper [OZ12] also shows exponential convergence
to a gradient flow segment. See also [MT09] for a related compactness result
which also exhibits the formation of gradient flow lines. These papers assume
the energy of the holomorphic cylinders is small, while in this paper we
assume that the image of $u_{n}$ lies inside of a ball $\overline{B}(1)$
(i.e., we assume a $C^{0}$ bound whereas they assume an $L^{2}$ bound on the
derivative).

A similar compactness phenomenon in the context of harmonic mappings is proved
in [CT99], where geodesic segments appear rather than gradient flow lines.

######  Remark 3.

In the course of our argument, we will need to pass to sub-cylinders a few
times to apply the a priori estimates from Section 2. As a consequence, our
description of the limit is only valid on
$[-r_{n},r_{n}]\times\mathbb{R}/\mathbb{Z}$. In applications, some other
information will be needed to control the behavior of $u_{n}$ on the ends
(e.g., $u_{n}$ is the restriction of a map defined on a larger Riemann surface
$\Sigma_{n}$ equipped with an embedding
$[-r_{n}-1,r_{n}+1]\times\mathbb{R}/\mathbb{Z}\to\Sigma_{n}$, or $u_{n}$ has
totally real boundary conditions on the two boundary circles).

######  Remark 4.

If $\epsilon_{n}r_{n}$ converges to $\infty$, then we expect any convergence
statement to require topological restrictions on $V$ (e.g., if $V$ is a
gradient vector field for a Morse function, then $u_{n}$ will converge to a
broken flow line; or, if $V_{n}$ is a Morsification of a Morse-bott vector
field $V$, then $u_{n}$ will converge to a Morse-Bott broken flow line, etc).

######  Remark 5.

The set-up in Definition 1 is a bit of a “toy problem” since we do not
consider more general almost complex structures $J$. The author expects more
general almost complex structures can be handled in a similar way. The
estimates/computations will involve many additional junk terms involving the
derivatives of $J$ (i.e., some things which cancel exactly in this paper will
not cancel exactly with a more general $J$). Hopefully the ideas conveyed in
this paper will still be of use to readers interested in more general almost
complex structures.

##  2\. A Priori Estimates

In this section we will prove various a priori estimates for maps $u_{n}$
satisfying Definition 1. Our first result is a _uniform a priori estimate_ for
$u_{n}$ and its derivatives. The elliptic estimates for the “delbar” operator
$\overline{\partial}=\partial_{s}+J_{0}\partial_{t}$ imply that for every
$k\in\mathbb{N}$ and $\delta>0$ there is a constant
$C^{\mathrm{ell}}_{\delta,k}$ so that for any smooth $\mathbb{R}^{2n}$-valued
function $u$ we have:

(2) $\left\lVert u\right\rVert_{W^{k+1,2}(s+\Omega(\delta))}\leq
C^{\mathrm{ell}}_{\delta,k}(\left\lVert(\partial_{s}+J_{0}\partial_{t})u\right\rVert_{W^{k,2}(s+\Omega(2\delta))}+\left\lVert
u\right\rVert_{W^{k,2}(s+\Omega(2\delta))}),$

where $\Omega(\delta)=[-\delta,\delta]\times\mathbb{R}/\mathbb{Z}$. See [RS01,
Lemma C.1] for a short proof. In particular, the fact that $V_{n}$ converges
in $C^{\infty}$ and $\left|u_{n}\right|$ is everywhere less than $1$, together
with the Sobolev embedding theorem, implies the following following result:

######  Lemma 2.

Assuming the set-up in Definition 1, we have the following a priori estimates
for the behavior of $u_{n}$ on the sub-cylinder
$[-r_{n},r_{n}]\times\mathbb{R}/\mathbb{Z}$:

(3)
$\limsup_{n\to\infty}\sup_{(s,t)\in[-r_{n},r_{n}]\times\mathbb{R}/\mathbb{Z}}\left|\nabla^{k}u_{n}(s,t)\right|\leq
C_{k},$

where $C_{k}$ depends only on $k$.

######  Proof.

It is straightforward to show that

$\limsup_{n\to\infty}\left\lVert
u_{n}\right\rVert_{W^{k,2}(s+\Omega(2^{-k}))}\leq C_{k}^{\prime}.$

Then the Sobolev embedding theorem (see [MS12, Theorem B.1.11]) gives

$\left|\nabla^{k}u(s,t)\right|\leq\left\lVert
u_{n}\right\rVert_{C^{k}(s+\Omega(2^{-k-2}))}\leq
C^{\mathrm{Sob}}_{k}\left\lVert
u_{n}\right\rVert_{W^{k+2,2}(s+\Omega(2^{-k-2}))}.$

These two observations complete the proof.

Next, we will prove an _exponential type estimate_ that shows $u_{n}$ becomes
very thin as we approach the center of the cylinder.

######  Definition 6.

Introduce the _center of mass_ $q_{n}:[-r_{n}-1,r_{n}+1]\to B(1)$ given by

$q_{n}(s)=\int_{\mathbb{R}/\mathbb{Z}}u_{n}(s,t)\,\mathrm{d}t.$

Assuming that $u_{n}$ satisfies Definition 1, then we compute

(4)
$q_{n}^{\prime}(s)-\epsilon_{n}V_{n}(q_{n}(s))=\epsilon_{n}\int_{\mathbb{R}/\mathbb{Z}}V_{n}(u_{n}(s,t))-V_{n}(q_{n}(s))\,\mathrm{d}t.$

We think of of (4) as a perturbation of the flow line equation for $V_{n}$.

Introduce the function

(5)
$\gamma_{n}(s)=\frac{1}{2}\int_{\mathbb{R}/\mathbb{Z}}\left|u_{n}(s,t)-q_{n}(s)\right|^{2}\,\mathrm{d}t.$

Our goal is to show that
$\gamma_{n}^{\prime\prime}(s)-\delta^{2}\gamma_{n}(s)\geq 0$ for some
$\delta>0$. As explained in the pre-print [Can21, Section 4.4] the estimate
$\gamma^{\prime\prime}-\delta^{2}\gamma\geq 0$ is related to exponential decay
estimates.

######  Remark 7.

For simplicity of notation, we introduce the norm:

$\left\lVert
f\right\rVert^{2}=\int_{\mathbb{R}/\mathbb{Z}}\left|f(t)\right|^{2}\,\mathrm{d}t.$

######  Lemma 3.

There is a constant $\delta>0$, independent of $n$ and $u_{n}$, so that
$\gamma_{n}$ satisfies the following differential inequality

(6)
$\gamma_{n}^{\prime\prime}(s)-\delta^{2}\gamma_{n}(s)\geq\frac{3}{4}\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\frac{1}{4}\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2}$

for $s\in[-r_{n},r_{n}]$ and for $n$ sufficiently large. More precisely, there
is $N$ depending on $\left\\{\epsilon_{n}\right\\}$ so that (6) holds for
$n\geq N$ and $s\in[-r_{n},r_{n}]$.

######  Proof.

We compute

$\gamma_{n}^{\prime\prime}(s)=\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\int_{\mathbb{R}/\mathbb{Z}}\left\langle
u_{n}-q_{n},\partial_{s}\partial_{s}(u_{n}-q_{n})\right\rangle\mathrm{d}t.$

We have

(7)
$\partial_{s}(u_{n}-q_{n})=-J_{0}\partial_{t}(u_{n}-q_{n})+\epsilon_{n}(V_{n}(u_{n})-\int_{\mathbb{R}/\mathbb{Z}}V_{n}(u_{n})\,\mathrm{d}t),$

hence

$\partial_{s}\partial_{s}(u_{n}-q_{n})=-\partial_{t}\partial_{t}(u_{n}-q_{n})+\epsilon_{n}(\partial_{s}-J_{0}\partial_{t})[V_{n}(u_{n})-\int_{\mathbb{R}/\mathbb{Z}}V_{n}(u_{n})\,\mathrm{d}t].$

Integration by parts then yields

(8)
$\gamma_{n}^{\prime\prime}(s)=\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2}+\epsilon_{n}\int\left\langle
u_{n}-q_{n},\beta_{n}\right\rangle\mathrm{d}t,$

where

$\beta_{n}=(\partial_{s}-J_{0}\partial_{t})[V_{n}(u_{n})-\int_{\mathbb{R}/\mathbb{Z}}V_{n}(u_{n})\,\mathrm{d}t].$

We rewrite $\beta_{n}$ as follows:

$\displaystyle V_{n}(u_{n})-\int_{\mathbb{R}/\mathbb{Z}}V_{n}(u_{n})$
$\displaystyle=V_{n}(u_{n})-V_{n}(q_{n})+\int_{\mathbb{R}/\mathbb{Z}}V_{n}(u_{n})-V_{n}(q_{n})\,\mathrm{d}t$
$\displaystyle=A_{n}\cdot(u_{n}-q_{n}),$

where

(9) $A_{n}=\int_{0}^{1}\mathrm{d}V_{n}((1-\tau)q_{n}+\tau
u_{n})\mathrm{d}\tau-\int_{\mathbb{R}/\mathbb{Z}}\int_{0}^{1}\mathrm{d}V_{n}((1-\tau)q_{n}+\tau
u_{n})\mathrm{d}\tau\mathrm{d}t.$

As a consequence of the a priori estimates (3) we conclude that $A_{n}$ is
uniformly bounded in $C^{1}$ on the cylinder
$[-r_{n},r_{n}]\times\mathbb{R}/\mathbb{Z}$. For $n$ sufficiently large (say
$n\geq N$), we can bound its derivatives by a constant independent of $u_{n}$.

Since

$\beta_{n}=((\partial_{s}-J_{0}\partial_{t})A_{n})\cdot(u_{n}-q_{n})+A_{n}\cdot((\partial_{s}-J_{0}\partial_{t})(u_{n}-q_{n})),$

we conclude that for $s\in[-r_{n},r_{n}]$

$\left|\beta_{n}\right|\leq
C_{1}(\left|u_{n}-q_{n}\right|+\left|\partial_{s}(u_{n}-q_{n})\right|+\left|\partial_{t}(u_{n}-q_{n})\right|),$

where $C_{1}$ is independent of $n\geq N$ and $u_{n}$. By applying the Cauchy-
Schwarz inequality, and the inequality $2ab\leq a^{2}+b^{2}$, we conclude from
(8) that

$\displaystyle\gamma_{n}^{\prime\prime}(s)$
$\displaystyle\geq\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2}$
$\displaystyle\hskip 28.45274pt-\epsilon_{n}C_{2}(\left\lVert
u_{n}-q_{n}\right\rVert^{2}+\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2}),$

where $C_{2}$ is independent of $n\geq N$ and $u_{n}$. The next step in the
argument is to invoke the _Poincaré Inequality_ , which states that there is a
constant $c_{\mathrm{pc}}>0$ so that

$\text{$f:\mathbb{R}/\mathbb{Z}\to\mathbb{R}^{N}$ has mean
zero}\implies\left\lVert f\right\rVert^{2}\leq
c_{\mathrm{pc}}\left\lVert\partial_{t}f\right\rVert^{2}.$

A simple proof is available using Fourier analysis. Noting that $u_{n}-q_{n}$
has mean zero (by construction), we conclude that we can pick $n$ sufficiently
large so that $\epsilon_{n}C_{2}$ is less than
$\min\left\\{0.25c_{\mathrm{pc}}^{-1},0.25\right\\}$, which yields

$\gamma_{n}^{\prime\prime}(s)-\frac{1}{4c_{\mathrm{pc}}}\gamma_{n}(s)\geq\frac{3}{4}\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\frac{1}{4}\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2},$

provided $s\in[-r_{n},r_{n}]$. Setting $\delta^{2}=(4c_{\mathrm{pc}})^{-1}$
yields the desired result.

As in [Can21], we use the differential inequality (6) to conclude the
following _exponential a priori estimate_ on the $W^{1,2}$ size of
$u_{n}-q_{n}$:

######  Lemma 4.

Suppose that $u_{n}$ satisfies Definition 1 and $q_{n}$ is the center of mass
given by Definition 6. There is a constant $C$, independent of $u_{n}$ or
$r_{n}$, so that

$\displaystyle\int_{s-0.5}^{s+0.5}$ $\displaystyle\left\lVert
u_{n}-q_{n}\right\rVert^{2}+\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2}\mathrm{d}s$
$\displaystyle\hskip 142.26378pt\leq
C(e^{-\delta(r_{n}+s)}+e^{-\delta(r_{n}-s)})$

for $s\in[-r_{n}+1,r_{n}-1]$ and for $n$ sufficiently large. Notice that the
left hand side is the $W^{1,2}$ norm (squared) over the sub-cylinder
$[s-0.5,s+0.5]\times\mathbb{R}/\mathbb{Z}$. Here $\delta>0$ is the constant
appearing in the differential inequality (6).

######  Proof.

Let $\gamma_{n}$ be the quantity introduced in (5). Without loss of
generality, suppose that $n$ is sufficiently large so that the conclusion of
Lemma 3 holds, say $n\geq N$.

It is clear that $\gamma_{n}(-r_{n})$ and $\gamma_{n}(r_{n})$ are both bounded
by some constant $c>0$ (thanks to the estimates (3)). Then
$\beta_{n}=\gamma_{n}-c(e^{-\delta(r_{n}+s)}+e^{-\delta(r_{n}-s)})$ satisfies
$\beta_{n}^{\prime\prime}-\delta^{2}\beta_{n}\geq 0$, and $\beta_{n}(-r_{n})$
and $\beta_{n}(r_{n})$ are both non-positive.

As a consequence, $\beta_{n}$ must be non-positive everywhere, otherwise we
would conclude a positive maximum, which is precluded by
$\beta_{n}^{\prime\prime}\geq\delta^{2}\beta_{n}$. In particular, we conclude
that

(10) $\gamma_{n}(s)\leq c(e^{-\delta(r_{n}+s)}+e^{-\delta(r_{n}-s)}),$

for all $s\in[r_{n}-1,r_{n}+1]$ and for $n\geq N$. Integrating this estimate
over $[s-0.5,s+0.5]$, we find a constant $C$ so that

(11) $\int_{s-0.5}^{s+0.5}\left\lVert
u_{n}-q_{n}\right\rVert^{2}\mathrm{d}s\leq\frac{C}{3}(e^{-\delta(r_{n}+s)}+e^{-\delta(r_{n}-s)}),$

for $n\geq N$. It remains to estimate the integrals of
$\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}$ and
$\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert$. To do so, we will use a
convolution trick. Fix a bump function $\rho:\mathbb{R}\to[0,1]$ supported in
$[-1,1]$ and equal to $1$ on $[-0.5,0.5]$. As explained in the caption to
Figure 3, this can be done with
$\left\lVert\rho^{\prime\prime}\right\rVert_{L^{1}}\leq 40$ and
$\left\lVert\rho\right\rVert_{L^{1}}\leq 2$.

$0.5$$-0.5\hphantom{-}$$1$$-1\hphantom{-}$ Figure 3. The bump function $\rho$.
We have $\left\lVert\rho\right\rVert_{L^{1}}\leq 2$. Moreover, this can be
achieved with $\left\lVert\rho^{\prime\prime}\right\rVert_{L^{1}}\leq
4\pi^{2}+\epsilon\leq 40$, because one can take a smooth
$\epsilon$-approximation of the $C^{2}$ function $0.5-0.5\cos(2\pi x)$, which
interpolates from $0$ to $1$ over an interval of size $0.5$ (we need two
copies of this function, one of which is reflected through the $y$-axis).

We then convolve both sides of the differential inequality (6) to obtain

(12)
$(\rho^{\prime\prime}\ast\gamma_{n})(s)-\delta^{2}(\rho\ast\gamma)(s)\geq\frac{3}{4}\rho\ast\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\frac{1}{4}\rho\ast\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2}.$

For functions $f$ supported in $[-1,1]$, it is simple to apply (10) to
estimate

$\left|f\ast\gamma\right|(s)\leq\max_{s^{\prime}\in[s-1,s+1]}\left|\gamma(s^{\prime})\right|\left\lVert
f\right\rVert_{L^{1}}\leq c\left\lVert
f\right\rVert_{L^{1}}(e^{-\delta(r+s)}+e^{-\delta(r-s)}).$

This observation, with $f=\rho$ and $f=\rho^{\prime\prime}$ applied to (12),
yields

$\rho\ast(\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2})\leq
4(40+2\delta^{2})c(e^{-\delta(r+s)}+e^{-\delta(r-s)}).$

Since $\rho=1$ on $[-0.5,0.5]$, we obtain

$\int_{s-0.5}^{s+0.5}\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2}\,\mathrm{d}s\leq\rho\ast(\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2}).$

and so, if we pick $C\geq 6(40+2\delta^{2})c$, then we conclude that

$\int_{s-0.5}^{s+0.5}\left\lVert\partial_{s}(u_{n}-q_{n})\right\rVert^{2}+\left\lVert\partial_{t}(u_{n}-q_{n})\right\rVert^{2}\,\mathrm{d}s\leq\frac{2C}{3}(e^{-\delta(r+s)}+e^{-\delta(r-s)}).$

Combining this with (11) completes the proof of the lemma.

We think of Lemma 4 as being an a priori exponential estimate for the
$W^{1,2}$ norm. We will now use the elliptic estimates (2) to obtain similar
estimates in the $C^{k}$ norm:

######  Lemma 5.

Suppose that $u_{n}$ satisfies Definition 1 and $q_{n}$ is the center of mass
given by Definition 6. For $n$ sufficiently large and each $k\geq 0$, there is
a constant $M_{k}$ independent of $u_{n}$ or $r_{n}$ so that

$\left|\nabla^{k}(u_{n}-q_{n})(s,t)\right|\leq
M_{k}(e^{-c(r_{n}+s)}+e^{-c(r_{n}-s)}),$

for $s\in[-r_{n}+1,r_{n}-1]$, where $c=\delta/2>0$ is the constant appearing
in the differential inequality (6).

######  Proof.

Let $\Omega_{k}=[2^{-k},2^{k}]\times\mathbb{R}/\mathbb{Z}$. It suffices to
show that there are constants $S_{k}$ independent of $u_{n}$ and $r_{n}$ so
that

(13) $\left\lVert u_{n}-q_{n}\right\rVert_{W^{k,2}(s+\Omega_{k})}\leq
S_{k}(e^{-c(r_{n}+s)}+e^{-c(r_{n}-s)}),$

for $n$ sufficiently large and $s\in[-r_{n}+1,r_{n}-1]$. As in the proof of
Lemma 2, the Sobolev embedding theorem will then imply the desired $C^{k}$
estimates. Pick $n\geq N$ for $N$ large enough so that the $k=1$ case of (13)
holds (Lemma 4). We will not need to increase $N$ any further.

Recall from (7) that $u_{n}-q_{n}$ satisfies the following equation:

$\partial_{s}(u_{n}-q_{n})+J_{0}\partial_{t}(u_{n}-q_{n})=\epsilon_{n}A_{n}\cdot(u_{n}-q_{n}),$

where $A_{n}$ is given by (9). Note that the derivatives of $A_{n}$ on
$[-r_{n},r_{n}]\times\mathbb{R}/\mathbb{Z}$ are uniformly bounded because of
Lemma 2.

The elliptic estimates (2) can be applied to conclude that111Here we assume
the norms have been defined so that  $\left\lVert
A_{n}\cdot(u_{n}-q_{n})\right\rVert_{W^{k,2}}\leq\left\lVert
A_{n}\right\rVert_{C^{k}}\left\lVert u_{n}-q_{n}\right\rVert_{W^{k,2}}.$ For
some choices of norm, this may hold only up to a combinatorial constant
depending on $k$, but this will not affect the argument.

$\left\lVert u_{n}-q_{n}\right\rVert_{W^{k+1,2}(s+\Omega_{k+1})}\leq
C_{k}^{\mathrm{ell}}(\left\lVert
A_{n}\right\rVert_{C^{k}(s+\Omega_{k})}+1)\left\lVert
u_{n}-q_{n}\right\rVert_{W^{k,2}(s+\Omega_{k})}.$

Thus we may recursively define $S_{k}$ by the formula

$S_{k+1}=C^{\mathrm{ell}}_{k}(B_{k}+1)S_{k},$

where $B_{k}$ is any constant controlling the $C^{k}$ size of $A_{n}$. This
establishes (13) and therefore completes the proof.

##  3\. Convergence to the limiting configuration

Our goal in this section is to prove that the sequence $u_{n}$ converges near
the ends of the cylinder to holomorphic disks with removable singularities
$x_{-}$ and $x_{+}$, and, away from the ends, $u_{n}$ converges to a flow line
for $V=\lim_{n\to\infty}V_{n}$ connecting $x_{-}$ to $x_{+}$. See Figure 1 for
an illustration of the idea.

As we will see below, in order to capture the limiting flow line, we will need
to rescale the map $u_{n}$, i.e., we instead consider the limiting behavior of

$v_{n}(s,t)=u_{n}(\epsilon_{n}^{-1}s,\epsilon_{n}^{-1}t)$

defined on the rescaled cylinder
$[-\epsilon_{n}r_{n},\epsilon_{n}r_{n}]\times\mathbb{R}/\epsilon_{n}\mathbb{Z}$.

Since the domain of $v_{n}$ becomes highly degenerate from a metric
perspective, we cannot apply most a priori estimates to $v_{n}$ (e.g., one
complication is that disks with a fixed radii $r$ become multiply covered with
arbitrarily large covering degree in the domain of $v_{n}$). In order to show
that $v_{n}$ converges, we will apply the Arzelà-Ascoli theorem.222In our
proof, we will actually apply the Arzelà-Ascoli theorem to the center of mass
of $v_{n}$. In our case, where $J=J_{0}$, there is a cancellation when
deriving the equation for the center of mass (namely
$\int_{\mathbb{R}/\mathbb{Z}}J_{0}\partial_{t}u\,\mathrm{d}t=0$) which
simplifies the argument. Unfortunately, the first derivative of $v_{n}$ grows
as $\epsilon_{n}^{-1}$ times the first derivative of $u_{n}$. This makes it
difficult to bound the derivatives of $v_{n}$ (which is a pre-requisite for
applying Arzelà-Ascoli).

However, the exponential estimates derived in Section 2 _can_ be used to bound
the derivative of $v_{n}$ on any subcylinder sufficiently far from the
boundary circles; we need only travel far enough that
$e^{-c(r-s)}+e^{-c(r+s)}\leq\epsilon_{n}$.

We begin by analyzing the behavior of $u_{n}$ near the ends of the cylinder.
We have the following result:

######  Lemma 6.

Suppose that $u_{n}$ satisfies Definition 1, and $r_{n}\to+\infty$. Consider
the translations333The maximal domain of $u^{-}_{n}$ is $[-1,2r_{n}+1]$ and
the maximal domain of $u^{+}_{n}$ is $[-2r_{n}-1,+1]$.

$u^{-}_{n}(s,t)=u_{n}(s-r_{n},t)\text{ and }u^{+}_{n}(s,t)=u_{n}(s+r_{n},t).$

After passing to a subsequence, $u^{-}_{n}$ converges on compact subsets of
$[0,\infty)\times\mathbb{R}/\mathbb{Z}$ to a limiting holomorphic444By
“holomorphic,” we mean that $\partial_{s}u+J_{0}\partial_{t}u=0$. map
$u^{-}_{\infty}$, and (similarly) $u^{+}_{n}$ converges on compact subsets of
$(-\infty,0]\times\mathbb{R}/\mathbb{Z}$ to a limiting holomorphic map
$u^{+}_{\infty}$.

Moreover, $u^{-}_{\infty}(s,t)$ is asymptotic as $s\to\infty$ to a point
$x_{-}\in\overline{B}(1)$ and $u^{+}_{\infty}(s,t)$ is asymptotic as
$s\to\infty$ to a point $x_{+}\in\overline{B}(1)$ (both limits are uniform in
$t$).

$u^{-}_{\infty}$$u^{+}_{\infty}$$x_{-}$$x_{+}$ Figure 4. The limits
$u_{\infty}^{\pm}$ and the removable singularities $x_{\pm}$.

######  Proof.

It is a direct consequence of Lemma 2 that the $k$th derivative of $u^{-}_{n}$
is bounded on compact subsets of $[0,\infty)$ as $n\to\infty$. Since $k$ is
arbitrary, the Arzelà-Ascoli theorem implies a subsequence of $u^{-}_{n}$
converges on compact sets in the $C^{\infty}$ topology to a limiting map
$u^{-}_{\infty}:[0,\infty)\times\mathbb{R}/\mathbb{Z}\to\overline{B}(1)$.

Since $u_{n}$ satisfies
$\partial_{s}u_{n}+J_{0}\partial_{t}u_{n}=\epsilon_{n}V_{n}$, and
$V_{n}\epsilon_{n}\to 0$, we conclude that $u_{\infty}^{-}$ is holomorphic.
The infinite cylinder $[0,\infty)\times\mathbb{R}/\mathbb{Z}$ is
biholomorphically equivalent to a punctured disk. A standard result in complex
analysis says that any bounded holomorphic function defined on a punctured
disk has a removable singularity at the puncture.555In short, the boundedness
assumption implies that the Cauchy Integral Formula can be applied to the
holomorphic function, without any correction (“residue”) necessary. Thus
$\lim_{s\to\infty}u_{\infty}^{-}(s,t)=x_{-}$ exists (uniformly in $t$). The
same argument works for $u^{+}_{n}$. This completes the proof.

######  Lemma 7.

Assume the setup and conclusion of Lemma 6. There exists a sequence
$\rho_{n}\to+\infty$ so that the restrictions

$u_{n}^{-}:[0,\rho_{n}]\times\mathbb{R}/\mathbb{Z}\to\overline{B}(1)\text{ and
}u_{n}^{+}:[-\rho_{n},0]\times\mathbb{R}/\mathbb{Z}\to\overline{B}(1)$

converge uniformly to the limits $u_{\infty}^{-}$ and $u_{\infty}^{+}$, in the
sense that

$\lim_{n\to\infty}\sup_{(s,t)\in[0,\rho_{n}]\times\mathbb{R}/\mathbb{Z}}\left|u_{n}^{-}(s,t)-u_{\infty}^{-}(s,t)\right|+\sup_{(s,t)\in[-\rho_{n},0]\times\mathbb{R}/\mathbb{Z}}\left|u_{n}^{+}(s,t)-u_{\infty}^{+}(s,t)\right|=0.$

Moreover, this choice can be arranged so $\epsilon_{n}\rho_{n}\to 0$ and
$\rho_{n}<r_{n}$. See Figure 5.

$-r_{n}\vphantom{-\rho_{n}}$$r_{n}\vphantom{-\rho_{n}}$$-r_{n}+\rho_{n}$$r_{n}-\rho_{n}$
Figure 5. Cutting the domain into regions as in Lemma 7. The region in between
$-r_{n}$ and $-r_{n}+\rho_{n}$ converges uniformly to $u^{-}_{\infty}$ (after
translating to the right by $r_{n}$). The region in between $r_{n}-\rho_{n}$
and $r_{n}$ converges uniformly to $u^{+}_{\infty}$ (after translating to the
left by $r_{n}$). In particular, the circles at $\pm(r_{n}-\rho_{n})$ converge
uniformly to the removable singularities $x_{\pm}$.

######  Proof.

Our definition of $\rho_{n}$ will be such that $\rho_{n}\in\mathbb{N}$. To
simplify the notation introduce the quantity:

$d_{n}(k)=\sup_{(s,t)\in[0,k]\times\mathbb{R}/\mathbb{Z}}\left|u_{n}^{-}(s,t)-u_{\infty}^{-}(s,t)\right|+\sup_{(s,t)\in[-k,0]\times\mathbb{R}/\mathbb{Z}}\left|u_{n}^{+}(s,t)-u_{\infty}^{+}(s,t)\right|.$

Consider the following three ($n$-dependent) conditions we can place on
$k\in\mathbb{N}$:

  1. (1)

$\epsilon_{n}k\leq\epsilon_{n}^{1/2}$,

  2. (2)

$d_{n}(k)\leq k^{-1}$,

  3. (3)

$k<r_{n}$,

Define $\rho_{n}$ to be the maximal integer $k$ which satisfies all three
properties. We will now show that $\rho_{n}$ has the desired properties.
Observe that if $k$ is a fixed number, then both properties will hold for $k$
as $n\to\infty$, since both maps $u_{n}^{\pm}$ converge on compact sets
($[0,k]\times\mathbb{R}/\mathbb{Z}$ are $[-k,0]\times\mathbb{R}/\mathbb{Z}$
are compact sets, and $k^{-1}$ is a fixed number). This shows that $\rho_{n}$
is eventually greater than every fixed number, and hence $\rho_{n}\to\infty$.

Clearly $\epsilon_{n}\rho_{n}$ tends to zero, since property (1) holds.

Moreover, since $\rho_{n}\to\infty$, property (2) implies that
$d_{n}(\rho_{n})\to 0$. This completes the proof.

We will now analyze the region
$[-r_{n}+\rho_{n},r_{n}-\rho_{n}]\times\mathbb{R}/\mathbb{Z}$. This is where
we will see the flow line appear.

######  Lemma 8.

Let $u_{n}$ be a sequence satisfying Definition 1 and let $q_{n}$ be the
center of mass as in Definition 6. Suppose that
$\lim_{n\to\infty}\epsilon_{n}r_{n}=\ell$, and if $\ell=0$, require that
$r_{n}\to+\infty$. Fix a choice of subsequence and numbers $\rho_{n}$ so that
Lemma 6 and Lemma 7 hold. After passing to another subsequence, if necessary,
the rescaled maps

$v_{n}(s,t)=u_{n}(\epsilon_{n}^{-1}s,\epsilon_{n}^{-1}t)$

defined on the rescaled cylinder
$[\epsilon_{n}(-r_{n}+\rho_{n}),\epsilon_{n}(r_{n}-\rho_{n})]\times\mathbb{R}/\epsilon_{n}\mathbb{Z}$
converge uniformly to a flow line $v_{\infty}$ for $V=\lim_{n\to\infty}V_{n}$
of length $2\ell$. The flow line starts at $x_{-}$ and ends at $x_{+}$.

######  Proof.

We begin with the a priori estimate for the difference $u_{n}-q_{n}$; recall
that we have proved in Lemma 5 that for $n$ sufficiently large and all $k$ we
have

(14) $\left|\nabla^{k}(u_{n}-q_{n})(s,t)\right|\leq
M_{k}(e^{-c(r_{n}+s)}+e^{-c(r_{n}-s)}),$

for $s\in[-r_{n}+1,r_{n}-1]$ and all $t$. In particular, since
$\rho_{n}\to\infty$, we conclude

$\lim_{n\to\infty}\sup_{(s,t)\in[-r_{n}+\rho_{n},r_{n}-\rho_{n}]\times\mathbb{R}/\mathbb{Z}}\left|u_{n}(s,t)-q_{n}(s)\right|=0.$

As this is a $C^{0}$ estimate, it remains true after rescaling, i.e., with
$p_{n}(s)=q_{n}(\epsilon_{n}^{-1}s)$ then

$\lim_{n\to\infty}\sup_{(s,t)\in[-\epsilon_{n}^{-1}(r_{n}-\rho_{n}),\epsilon_{n}(r_{n}-\rho_{n})]\times\mathbb{R}/\epsilon_{n}\mathbb{Z}}\left|v_{n}(s,t)-p_{n}(s)\right|=0.$

Thus to prove the lemma it suffices to show that $p_{n}$ converges uniformly
on its domain
$[-\epsilon_{n}^{-1}(r_{n}-\rho_{n}),\epsilon_{n}(r_{n}-\rho_{n})]$ to a flow
line for $V$ of length $2\ell$ joining $x_{-}$ to $x_{+}$.

Recall that $q_{n}(s)$ satisfies the differential equation (4), and hence
$p_{n}(s)$ satisfies

$p_{n}^{\prime}(s)-V_{n}(p_{n}(s))=\int_{\mathbb{R}/\mathbb{Z}}V_{n}(u_{n}(\epsilon_{n}^{-1}s,t))-V_{n}(q_{n}(\epsilon_{n}^{-1}s))\,\mathrm{d}t.$

A simple computation à la fundamental theorem of calculus yields

$V_{n}(u_{n}(\epsilon_{n}^{-1}s,t))-V_{n}(q_{n}(\epsilon_{n}^{-1}s))=B_{n}\cdot(u_{n}(\epsilon_{n}^{-1}s,t)-q_{n}(\epsilon_{n}^{-1}s)),$

where $B_{n}$ is uniformly bounded in $n$. Applying (14) with $k=0$ yields

$\left|p_{n}^{\prime}(s)-V_{n}(p_{n}(s))\right|\leq Ce^{-c\rho_{n}}\to 0$

for some constant $C$ for all
$s\in[-\epsilon_{n}(r_{n}-\rho_{n}),\epsilon_{n}(r_{n}-\rho_{n})]$. Notice
that $p_{n}^{\prime}(s)$ is bounded, and
$\lim_{n\to\infty}\epsilon_{n}(r_{n}-\rho_{n})=\ell$. Then a straightforward
application of Arzéla-Ascoli shows that, after passing to a subsequence,
$p_{n}$ converges uniformly to a flow line
$v_{\infty}:[-\ell,\ell]\to\overline{B}(1)$ for $V$. To be a bit more precise
about the convergence, we have

$\lim_{n\to\infty}\sup_{s\in[-\epsilon_{n}(r_{n}-\rho_{n}),\epsilon_{n}(r_{n}-\rho_{n})]\cap[-\ell,\ell]}\left|p_{n}(s)-v_{\infty}(s)\right|=0.$

In particular, since $p_{n}$ has a bounded derivative, and
$\epsilon_{n}(r_{n}-\rho_{n})\to\ell$, we conclude
$p_{n}(\pm\epsilon_{n}(r_{n}-\rho_{n}))$ converges to $v_{\infty}(\pm\ell)$.
We have

$p_{n}(\pm\epsilon(r_{n}-\rho_{n}))=q_{n}(\pm(r_{n}-\rho_{n}))=\int
u_{n}(\pm(r_{n}-\rho_{n}),t)\to x_{\pm},$

and hence $v_{\infty}(\pm\ell)=x_{\pm}$, as desired. This completes the proof.

## References

  * [Can21] Dylan Cant, _Exponential convergence to non-degenerate Reeb chords_ , 2021\.
  * [CT99] Jingyi Chen and Gang Tian, _Compactification of moduli space of harmonic mappings_ , Commentarii Mathematici Helvetici 74 (1999), 201–237.
  * [Flo89] Andreas Floer, _Symplectic fixed points and holomorphic spheres_ , Communications in Mathematical Physics 120 (1989), 575–611.
  * [MS12] D. McDuff and D. Salamon, _J-holomorphic curves and symplectic topology_ , 2nd ed., American Mathematical Society, Colloquium Publications, 2012\.
  * [MT09] I. Mundet i Riera and G. Tian, _A compactification of the moduli space of twisted holomorphic maps_ , Advances in Mathematics 222 (2009), no. 4, 1117–1196.
  * [Oh05] Yong-Geun Oh, _Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group_ , Duke Math. J. 130 (2005), no. 2, 199–295. MR 2181090
  * [OZ10] Yong-Geun Oh and Ke Zhu, _Floer trajectories with immersed nodes and scale-dependent gluing_ , 2010.
  * [OZ12] by same author, _Thick-thin decomposition of floer trajectories and adiabatic gluing_ , 2012.
  * [RS01] J. Robbin and D. Salamon, _Asymptotic behaviour of holomorphic strips_ , Ann. I. H. Poincaré 18, 5 (2001), 573–612.
  * [Sal97] Dietmar Salamon, _Lectures on Floer homology_ , 1997.


# Matching of Descriptive Labels to Glossary Descriptions

Toshihiro Takahashi
IBM Research - Tokyo
<EMAIL_ADDRESS>
Takaaki Tateishi
IBM Research - Tokyo
<EMAIL_ADDRESS>
Michiaki Tatsubori
IBM Research - Tokyo
<EMAIL_ADDRESS>

###### Abstract

Semantic text similarity plays an important role in software engineering tasks
in which engineers are requested to clarify the semantics of descriptive
labels (e.g., business terms, table column names) that are often consists of
too short or too generic words and appears in their IT systems. We formulate
this type of problem as a task of matching descriptive labels to glossary
descriptions. We then propose a framework to leverage an existing semantic
text similarity measurement (STS) and augment it using semantic label
enrichment and set-based collective contextualization where the former is a
method to retrieve sentences relevant to a given label and the latter is a
method to compute similarity between two contexts each of which is derived
from a set of texts (e.g., column names in the same table). We performed an
experiment on two datasets derived from publicly available data sources. The
result indicated that the proposed methods helped the underlying STS correctly
match more descriptive labels with the descriptions.

##  1 Introduction

In general IT projects, such as database and business process migration, IT
engineers invest significant effort in verifying consistency between various
models, such as table schemata and diagrams that depict object relations.
Maintaining a glossary of domain terms is a best practice that helps alleviate
their workload. The glossary serves as a reference that maps descriptive
labels (such as business terms and table column names) to corresponding
glossary descriptions, which are typically English sentences that provide
explanations and contexts.

### Target Problem and Challenges

In this paper, we address such a common mapping problem between descriptive
labels and glossary descriptions. We refer to this problem as Descriptive
Labels to Descriptions (DLD). In DLD, we are given multiple datasets and
glossaries. Each dataset contains a list of descriptive labels, while each
glossary contains a list of glossary descriptions. The goal is to establish
mappings from each label to its corresponding description.

However, there are several technical challenges in mapping descriptive labels
to the glossary descriptions due to the nature of the descriptive labels such
as too short and too generic words included by the descriptive names.

### Our Approach and Research Questions

To tackle the technical challenges, in this paper, we propose a novel
framework to solve DLD problems effectively by enriching the short and/or
generic words and capturing the context of the generic and/or ambiguous words.
Our framework is designed to be flexible enough to employ variations of
underlying semantic text similarity (STS) models, enrichment methods, and
contextualization methods whereas our implementation is limited to reasonable
combinations of a traditional TFIDF model, the PromCSE model Jiang et al.
(2022) (BERT-based STS model), the Flan-T5 Chung et al. (2022) large language
model (LLM), and Wikidata. With our implementation we performed an experiment
on two glossaries: a business glossary derived from the financial industry
business ontology and more than 1000 pairs of column names and corresponding
descriptions obtained from the Kaggle webpages. We organized our experiment to
verify our hypothesis that there are many descriptive labels that cannot be
mapped to corresponding descriptions by commonly used text similarity models
due to cryptic words and our label enrichment and contextualization methods
help the underlying STS models work better for such problematic labels. More
specially, the research questions we address in this paper are as follows.

  * •

How much improvement is observed in the metrics Mean Reciprocal Rank (MRR) and
Hits@k with the label enrichment methods?

  * •

How much improvement is observed in MRR and Hits@k with the label
contextualization methods?

  * •

What kind of labels meet the out-of-vocabulary issue and the ambiguity issue?
And what kind of labels can be solved by the label enrichment methods, and can
be disambiguated by label contextualization methods?

The reason why we use the Mean Reciprocal Rank (MRR) and Hits@k is that the
DLD problem can be seen as a task of ranking descriptions corresponding to
given descriptive labels and these metrics are commonly used for recommender
systems.

### Contributions

Our contributions in this paper are as follows.

  * •

We formulate the DLD problem, and identified the technical issues: out-of-
vocabulary issue and ambiguity issue of cryptic words. We provide two
practical benchmark datasets: Kaggle and FIBO including these issues.

  * •

To solve out-of-vocabulary issue, we propose Label Semantic Enrichment method
which leverages external knowledge.

  * •

To solve ambiguity issue, we propos Set-based Collective Contextualization
method by leveraging Large Language Model.

  * •

In our experiments, we clarified how effectively our approaches solve the
issues.

The rest of this paper is organized as follows. In the next two sections, we
discuss related works, and provide motivating examples using practical
datasets. We then present our approach to address them effectively, and
formulate the DLD task. Subsequently, we demonstrate how the DLD task is
tackled using several approaches, presenting experimental results. Afterward,
we provide a comprehensive analysis of our experimental results. Finally, we
conclude the paper.

##  2 Related Work

In general, DLD is a task of linking a chunk of text in a group to one in
another group. In this sense, it could be considered as a natural language
processing problem, specifically semantic text similarity or entity linking
Shen et al. (2014, 2021). Also, we could consider it as a problem of aligning
between domain-specific labels or descriptive names, as a set of domain
concepts, to another ontology of a common glossary.

### Semantic Text Similarity

There have been many studies on STS Chandrasekaran and Mago (2021). Some
exploited general knowledge bases Li et al. (2020), as in our study. Such
knowledge-based methods measure the similarity of two terms on the basis of
the structural properties of the knowledge bases, such as the number of edges.
Our method, however, uses the knowledge bases only to enrich descriptive
names.

Other methods are corpus based such as word2vec Mikolov et al. (2013) and BERT
Devlin et al. (2019). Such methods leverage large corpora to compute word-
embeddings useful for measuring the similarity between terms on the basis of
the idea that similar words occur together. The same idea is also applied for
capturing the characteristics of sentences, as with Sentence-BERT Reimers and
Gurevych (2019) and PromCSE Jiang et al. (2022). Many search engines use this
type of method to retrieve and rank relevant sentences and webpages. We
designed our method in such a way that it benefits from the advances of the
corpus-based methods and pre-trained models.

### Entity Linking and Ontology Learning

Entity linking Shen et al. (2014), such as ColNet Chen et al. (2019) and TabEL
Bhagavatula et al. (2015), is a task of linking terms described in a document
to entities defined in a knowledge graph such as Wikidata. Ontology learning
Asim et al. (2018), matching Shvaiko and Euzenat (2011), or alignment Ardjani
et al. (2015) are similar tasks that automatically or semi-automatically
gather terms and relations from documents to create an ontology and discover
correspondence between ontologies. Background knowledge is known to
significantly improve the performance of ontology matching systems Portisch et
al. (2021). Compared with these problems, the DLD problem mentions neither
knowledge graph nor ontologies. The descriptive names in a DLD problem are not
within contextual sentences or linked to other concepts, entities, or values.
Glossaries as ontologies in a DLD problem are also not linked but just
descriptions of known concepts.

### Named Entity Disambiguation

Named entity disambiguation on knowledge graphs including ontologies is a key
component for the success of semantic text similarity, entity linking, and
ontology learning. Likewise, our label enrichment method and set-based
collective contextualization are both considered as methods to disambiguate
the descriptive labels for solving the DLD problems. As in the case of our
label enrichment method, recent literatures Mulang’ et al. (2020); Bastos et
al. (2021) also leverage the triples obtained from Wikidata to improve the
performance of pre-trained models for the named entity disambiguation on
Wikipedia whereas we leverage the sentences obtained from Wikidata. In
addition, unlike our contextualization, they do not compute the context of a
set of entities and rely only on the context of each single entity.

The idea of using a knowledge base for disambiguation is also presented in the
literature Ho et al. (2010). It proposes the use of a knowledge base to
measure the similarity between texts for semantic text similarity and
integrate it with a corpus-based measurement. However, it does not include any
process for enriching and/or contextualizing given texts.

##  3 Motivating Examples

In software engineering tasks such as database and business process
migrations, engineers are often requested to clarify the semantics of
descriptive labels (e.g., business terms, table column names). For example,
during the migration from a legacy system, IT engineers thoroughly analyze the
original descriptive labels present in the existing data models. They then
create a new logical data model and establish mappings between the original
labels and the new ones. Understanding the semantics of the original labels in
the context of the new logical data model is crucial, and the glossary serves
as a valuable resource. However, the original data model often suffers from
incompleteness and inconsistencies, as its element names may differ from those
in the new logical data model, and an up-to-date or accessible dictionary may
be lacking. Moreover, the mapping process typically requires human involvement
and may not have an initial mapping in the first iteration.

We could leverage a state-of-the-art semantic text similarity (STS)
measurement to map each descriptive labels to a corresponding description.
However, the following nature of the descriptive names makes it difficult to
make the mapping using the STS measurement in high accuracy.

  * •

The descriptive labels (e.g., LOAN_AMT, ACS_DT) often contain cryptic (too
short, too generic, and too ambiguous) words such as AMT (amount), ACS
(access), and DT (date). These cryptic words are out-of vocabulary unlike
aliases or nicknames, and often makes it difficult to understand their
meanings.

  * •

The descriptive labels often appear only in database table schemata or program
variables. This prevents us from associating the descriptive labels with
documents.

  * •

Many descriptive labels used in the IT systems are specific to those IT
systems. We could not rely on mappings between the descriptive labels and
descriptions created for other IT systems.

The same situation arises even in standardized or commonly used domain-
specific ontologies such as financial industry business ontology (FIBO). FIBO
is an ontology for financial business applications. It defines a named entity
“ALL” the description of which is “the currency identifier for Lek (the
currency of Albania)”. However, it is a very general word and often used as in
the case of “all types of bank loans”. Likewise, we found many descriptions
that include the word “all”. Therefore, no STS model is effective for matching
the descriptive name with the correct definition.

As a preliminary experiment, we collected the named entities and corresponding
descriptions from FIBO, and ran PromCSE to compute the similarity scores
between all the pairs of the named entities and descriptions. The top-ranked
descriptions corresponding to “ALL” were

  * •

“collection representing the total membership, or úniverse,́ of people,
resources, products, services, events, or entities of interest for some
question, experiment, survey or statistical program” (0.309),

  * •

“location in physical space” (0.291),

  * •

“a collection of managed investments that are all managed by a single
investment institution” (0.274),

where the values enclosed with the parentheses are the similarity scores
reported from PromCSE. The correct description of “ALL” was ranked 490th
(0.063).

##  4 Approach

Our approach is as follows:

  * •

We employ a STS model to measure a similarity of two sentences.

  * •

We apply Label Semantics Enrichment module to enrich descriptive labels.

  * •

We take into account the context of both of descriptive labels and glossary
descriptions.

### Label Semantics Enrichment (LSE)

A basic strategy to measure a similarity between a descriptive label and a
glossary description is using STS models such as TFIDF, PromCSE and LLM. TFIDF
suffer from aliases because it’s built on word-level exact matching. PromCSE
and LLM also may suffer from minor aliases and cryptic words in descriptive
labels.

Furthermore we employ Label Semantics Enrichment (LSE) to solve this problem.
LSE module retrieves sentences relevant to the given the descriptive label by
using a external knowledge database such as Wikidata and Bing. If the
retrieved sentences include sufficiently various aliases and relevant phrases
of cryptic words, STS model will work more effectively.

### Set-based Collective Contextualization (SCC)

As mentioned earlier, descriptive labels can often be ambiguous. In the case
that two tables have columns with the same label but semantically different
meanings, the same glossary description will be assigned to columns that are
semantically different.

To address this problem, we propose incorporating the context of a set of
descriptive labels and a set of glossary descriptions. For example, when
matching column names from multiple tables to various glossaries, we consider
the collective context provided by a set of column names within the same table
and a set of glossary descriptions within the same glossary. By leveraging
this broader context, we can more effectively identify and disambiguate the
intended meanings of the columns.

##  5 Problem Setting

Now we are given a dataset ${\mathcal{D}}$ which contains several semantic
groups $D_{i}\in{\mathcal{D}}$. The i-th group $D_{i}=(L_{i},G_{i})$ has a set
of descriptive labels $L_{i}=\\{l_{i,1},l_{i,2},\cdots,l_{i,n}\\}$ and a
glossary (a set of glossary descriptions)
$G_{i}=\\{g_{i,1},g_{i,2},\cdots,g_{i,n}\\}$. $l_{i,p}\in L_{i}$ is p-th
descriptive label of $L_{i}$. $g_{i,p}\in G_{i}$ is p-th glossary description
of $G_{i}$. $g_{i,p}$ describes a meaning of $l_{i,p}$.

The ultimate goal is to establish a complete mapping between $(l_{i,p},L_{i})$
and its corresponding $(g_{i,p},G_{i})$. However, the difficulty of this
problem is heavily influenced by the number of groups $D_{i}$ and the size of
the label set $L$ and the glossary $G$ in each group $D$.

To standardize the difficulty, we consider N-choice problem here. In this
scenario, given a target label set $L$ and a target label $l\in L$, the
objective is to identify the corresponding pair of the target description and
its glossary $(g,G)$ from N candidates $\\{(g_{k},G_{k})\\}_{k=1}^{N}$. This
is done by measuring the similarity between the label side $(l,L)$ and the
glossary side $(g_{k},G_{k})$.

##  6 DLD Framework

In our framework, DLD-Similarity Score $\Psi$ between label side $(l,L)$ and
glossary side $(g,G)$ is defined as

$\displaystyle\Psi(l,g,L,G|\theta)$ $\displaystyle=$ $\displaystyle\Psi_{\rm
T}(l,g,L|\theta_{\rm\scriptscriptstyle{L}SE},\theta_{\rm\scriptscriptstyle{S}TS})$
$\displaystyle\times$ $\displaystyle\Psi_{\rm
C}(L,G|\theta_{\rm\scriptscriptstyle{S}CC}).$

Here $\Psi_{\rm T}$ is Text-Similarity Score, and $\Psi_{\rm C}$ is Context-
Similarity Score. In $\Psi_{\rm T}$,
$\theta_{\rm\scriptscriptstyle{L}SE}\in\\{{\rm on},{\rm off}\\}$ is a switch
to enable or disable LSE module, and
$\theta_{\rm\scriptscriptstyle{S}TS}\in\\{\texttt{T},\texttt{P},\texttt{L}\\}$
is a switch indicating which STS model is used from three variations: TFIDF,
PromCSE, and LLM. In $\Psi_{\rm C}$,
$\theta_{\rm\scriptscriptstyle{S}CC}\in\\{{\rm on},{\rm off}\\}$ is a switch
to enable or disable SCC module.

### Text-Similarity Score

Text-Similarity Score $\Psi_{\rm T}$ measures a similarity between $l$ and
$g$. It’s defined as

$\displaystyle\Psi_{\rm
T}(l,g,L|\theta_{\rm\scriptscriptstyle{L}SE},\theta_{\rm\scriptscriptstyle{S}TS})=$
$\displaystyle\max_{s\in{\rm LSE}(l|\theta_{\rm\scriptscriptstyle{L}SE})}{\rm
STS}(s,g,L|\theta_{\rm\scriptscriptstyle{S}TS}).$

$\Psi_{\rm T}$ collects relevant sentences $\\{s_{i}\\}$ by invoking ${\rm
LSE}$, and computes similarity score between each sentence $s_{i}$ and $g$,
and outputs max of them.

We have two variations of ${\rm LSE}$ function: enabled version and disabled
version. The enabled version ${\rm LSE}(l|{\rm on})$ collects sentences
$\\{s_{i}\\}$ relevant to $l$ from external knowledge, and returns them. The
disabled version ${\rm LSE}(l|{\rm off})$ just returns $l$.

We also have three variations of ${\rm STS}$ function: TFIDF version ${\rm
STS}(\cdot|\texttt{T})$, PromCSE version ${\rm STS}(\cdot|\texttt{P})$, and
LLM version ${\rm STS}(\cdot|\texttt{L})$. See 7 for more details.

### Context-Similarity Score

Context-Similarity Score $\Psi_{\rm C}$ measures a contextual similarity
between $L$ and $G$. We have two variations: SCC enabled version and disabled
version.

The SCC enabled version $\Psi_{\rm C}(L,G|{\rm on})$ directly ask to LLM about
a probability of $L$ and $G$ being the same or different. See 7 for more
details.

The disable version $\Psi_{\rm C}(L,G|{\rm off})$ is always outputs $1$.

##  7 Implementation Details

We implemented our method using Python. It leverages Wikidata as external
knowledge in LSE module. It runs TFIDF, PromCSE and Flan-T5 Chung et al.
(2022) in STS module to measure a similarity score between two sentences, and
also uses Flan-T5 in SCC module to measure Context-Similarity Score between a
set of labels and a set of descriptions. The use of TFIDF and PromCSE is
straightforward, therefore, in the following sections, we describe about how
to use Wikidata in LSE module, and how to use Flag-T5 in STS and SCC module.

### Using Wikidata for LSE

In LSE module, we used Wikidata as external resource. The webpage of Wikidata
provides a search interface that we usually access using a Web browser. We
leverage this Web interface for the implementation. It sends queries of
descriptive labels to the search interface using the HTTP protocol, and parses
resulting webpages to extract entity IDs.

Wikidata also provides a SPARQL (Harris and Seaborne, 2013) endpoint as a
query service. We leverage this query service to collect label names and
descriptions of the collected entity IDs. We also use the label
property111http://www.w3.org/2000/01/rdf-schema#label and description
property222https://schema.org/description to generate sentences relevant to
the query phrase.

### LLM-based Semantic Text Similarity Model

We employed three STS models (TFIDF, PromCSE, and LLM) to compute the Text-
Similarity Score $\Psi_{\rm T}$. TFIDF and PromCSE can directly generate
similarity scores for the given pair of sentences. On the other hand, LLM
requires a natural language prompt as input, and also produces a natural
language answer.

We performed prompt engineering to create suitable inputs for LLM and
developed a method to extract scores from the generated answer. Figure 1 shows
the typical prompt example.

To translate the LLN answer to numerical score, ${\rm STS}(\cdot|\texttt{L})$
function collects top N tokens ${\mathscr{T}}=\\{t_{i}\\}_{i=1}^{N}$ with
their probability $p(t_{i})$ from LLM answer. It classifies the N tokens into
a set of “yes” tokens ${\mathscr{T}}_{\rm y}$, “no” tokens ${\mathscr{T}}_{\rm
n}$, and others. And computes probability ratio of “yes” and “no” as $s=p_{\rm
y}/(p_{\rm y}+p_{\rm n})$. Here $p_{\rm y}$ and $p_{\rm n}$ is sum of
probability which token is “yes” and “no”. They can be computed as follows:

$p_{\rm y}=\sum_{t\in{\mathscr{T}}_{\rm y}}p(t).$

⬇

1I have a dataset and a glossary. The given dataset has these columns.

2[Column names]

3 - School District Code

4 - County Code

5 ...

6[Question]

7Is "School District Code" same to the following concept in glossary?

8glossary description: "The code by which a school district is identified, as
utilized by the Department’s ..."

9[Answer (Yes/No)]

Figure 1: Prompt example for LLM-based STS

### LLM-based Context-Similarity Algorithm

We also employed LLM to measure the Context-Similarity between a set of labels
and a set of descriptions. Figure 2 shows the typical prompt example. A
scoring logic is same to the logic mentioned in 7.

⬇

1I have a dataset and a glossary. The given dataset has these columns.

2[Column names]

3 - ordered

4 - device_computer

5 ...

6The given glossary has these glossary terms.

7[Glossary terms]

8 - Indicates whether an appeal to the published decision has been received.

9 - Indicates if Design Review is part of the application process for this
permit.

10 ...

11[Question]

12Does these glossary terms describe the given column names?

13[Answer (Yes/No)]

Figure 2: Prompt example for SCC

##  8 Experimental Setup

As we described in Section 1, our experiment is organized to verify the
hypothesis that (1) there are many descriptive labels that cannot be mapped to
corresponding descriptions by commonly used text similarity models due to
cryptic words and (2) our label enrichment and contextualization methods help
the underlying STS models work better for such problematic labels. In this
section, we first describe how we prepared the datasets based on the publicly
available data sources. We then describe what metrics and why we used and how
we compared the different combinations of the STS models, the label enrichment
method, and the contextualization method.

### Dataset and Benchmark

Our experiment is performed on the two datasets derived from the Kaggle
webpages and the financial industry business ontology (FIBO). Table 1 shows
the statistics of these datasets.

The Kaggle dataset consists of 85 semantic groups that includes 1347
descriptive labels and corresponding descriptions in total where we consider
tables and column names we extracted from the Kaggle webpages as the semantic
groups and the descriptive labels. The smallest semantic group contains only
10 descriptive labels while the largest semantic group contains 36 descriptive
labels. We further investigated how many semantic groups include each
descriptive label to see if the dataset is suitable to evaluate the
effectiveness of the contextualization method. Table 2 summarizes the
descriptive labels and the numbers of corresponding semantic groups, which are
referred to as frequencies of the descriptive labels, where we selected only
the descriptive labels whose frequency is more than 7. For example, the
descriptive label “type” appears in 7 semantic groups and has different
meanings such as “type of wine”, “media type of animation film” and “flag if
company is private or public”.

FIBO defines concepts and relations used in financial domain, using the Web
Ontology Language (OWL) (Smith et al., 2004). It consists of 2086 named
entities each of which has its label and description specified by particular
XML tags such as the description and definition. We collected these labels and
descriptions as descriptive labels and descriptions of the dataset. We then
partitioned the pairs of the descriptive labels and descriptions into 44
semantic groups based on IRIs of the corresponding named entities. Unlike the
Kaggle dataset, there is no descriptive label that is included by multiple
semantic groups.

From these two datasets, we created 4 problems: Kaggle-10-choice,
Kaggle-50-choice, FIBO-10-choice, FIBO-50-choice as mentioned in Section 5 and
performed the experiment on these 4 problems.

Table 1: Statistics of the datasets | Kaggle | FIBO
---|---|---
entries | 1347 | 2086
semantic groups | 85 | 44
avg of # of words (label) | 1.64 | 4.05
max of # of words (label) | 6 | 16
min of # of words (label) | 1 | 1
avg of # of words (desc) | 14.4 | 16.1
max of # of words (desc) | 389 | 140
min of # of words (desc) | 2 | 3
Table 2: Frequency of descriptive labels of Kaggle label | frequency | label | frequency
---|---|---|---
name | 16 | country | 8
date | 14 | title | 8
age | 12 | id | 8
year | 9 | type | 7
county | 9 | status | 7
gender | 8 | description | 7

### Evaluation Method

In our evaluation, we compare the 12 combinations of models based on the 3
underlying STS models (TFIDF, PromCSE, LLM-based) and the presence or absence
of the label semantics enrichment (SLE) and the set-based collective
contextualization (SCC), which are represented by the following naming rule:
$\\{\texttt{T},\texttt{P},\texttt{L}\\}{\mathchar
45\relax}\\{\phi,\texttt{LSE}\\}{\mathchar 45\relax}\\{\phi,\texttt{SCC}\\},$
where T,P, and L represent TFIDF, PromCSE, and the LLM-based STS. For example,
‘T-LSE-SCC’ represents the combination of TFIDF, LSE, and SCC. ‘L’ represents
the LLM-based STS model not augmented with LSE or SCC.

To measure the success of matching the descriptive labels with the
descriptions, we employ the Mean Reciprocal Rank (MRR) and Hits@k as
performance metrics which are commonly used for evaluating the performance of
search engines and recommendation systems. This is because the DLD problem can
be seen as a task of ranking descriptions corresponding to given descriptive
labels.

##  9 Experimental Results

We present our comprehensive results against the 10-choice problems of the
Kaggle and FIBO datasets in Table 3 and against the 50-choice problems in
Table 4.

As a whole, regarding the first and second research questions, we proved that
the label semantics enrichment (LSE) and the set-based collective
contextualization (SCC) helps the underlying STS models produce better scores
in both MRR and Hits@k except that LSE often gave negative impact to PromCSE
and the LLM-based STS model in many cases.

Regarding the third research question, we observed that there were reasonably
many labels not correctly mapped only by the underlying STS models were
correctly mapped by the STS model augmented with LSE and SCC. In particular,
SCC successfully disambiguated generic words, and helped the underlying STS
model correctly mapped out-of-vocabulary labels which include highly cryptic
words. In addition, the L-SCC combination achieved over 99% in MRR for the
10-choice problem. The failed descriptive labels were difficult to be
correctly mapped even by humans.

We describe more details in the following sections one by one.

Table 3: Result of 10-choice problem

Kaggle10C | MRR | Hits@1 | Hits@3 | Hits@5
---|---|---|---|---
T | 0.735 | 0.670 | 0.687 | 0.687
T-LSE | 0.795 | 0.733 | 0.788 | 0.818
T-SCC | 0.741 | 0.682 | 0.687 | 0.687
T-LSE-SCC | 0.864 | 0.827 | 0.854 | 0.860
P | 0.872 | 0.800 | 0.931 | 0.972
P-LSE | 0.865 | 0.794 | 0.918 | 0.963
P-SCC | 0.987 | 0.978 | 0.996 | 0.999
P-LSE-SCC | 0.986 | 0.977 | 0.996 | 0.997
L | 0.982 | 0.970 | 0.995 | 0.997
L-LSE | 0.975 | 0.958 | 0.993 | 0.997
L-SCC | 0.994 | 0.990 | 0.998 | 0.999
L-LSE-SCC | 0.994 | 0.990 | 0.998 | 0.999
FIBO10C | MRR | Hits@1 | Hits@3 | Hits@5
T | 0.751 | 0.689 | 0.716 | 0.727
T-LSE | 0.874 | 0.833 | 0.881 | 0.907
T-SCC | 0.768 | 0.712 | 0.729 | 0.733
T-LSE-SCC | 0.925 | 0.898 | 0.933 | 0.937
P | 0.941 | 0.911 | 0.963 | 0.986
P-LSE | 0.957 | 0.934 | 0.973 | 0.990
P-SCC | 0.988 | 0.978 | 0.999 | 1.000
P-LSE-SCC | 0.990 | 0.981 | 1.000 | 1.000
L | 0.988 | 0.978 | 0.998 | 1.000
L-SCC | 0.993 | 0.986 | 1.000 | 1.000

  * •

This table shows the experiment results under the 12 settings (STS (TFIDF,
PromCSE, LLM), LSE on/off, SCC on/off) with Kaggle and FIBO datasets. Hit@k is
hit ratio represents how many times the true descriptions appeared in top-$k$
in the list. Larger is better. MRR is mean reciprocal rank. Larger is better.
Highest scores of each metric are denoted in bold.

Table 4: Result of 50-choice problem Kaggle50C | MRR | Hits@1 | Hits@5 | Hits@10
---|---|---|---|---
T | 0.654 | 0.604 | 0.686 | 0.687
T-SLE | 0.698 | 0.635 | 0.753 | 0.776
T-SCC | 0.691 | 0.671 | 0.687 | 0.687
T-SLE-SCC | 0.816 | 0.782 | 0.842 | 0.849
P | 0.726 | 0.624 | 0.846 | 0.916
P-SLE | 0.728 | 0.638 | 0.836 | 0.906
P-SCC | 0.960 | 0.933 | 0.990 | 0.995
P-SLE-SCC | 0.957 | 0.928 | 0.989 | 0.993
L | 0.939 | 0.903 | 0.984 | 0.993
L-SCC | 0.982 | 0.970 | 0.993 | 0.996
FIBO50C | MRR | Hits@1 | Hits@5 | Hits@10
T | 0.683 | 0.648 | 0.701 | 0.712
T-SLE | 0.810 | 0.772 | 0.848 | 0.868
T-SCC | 0.707 | 0.677 | 0.720 | 0.728
T-SLE-SCC | 0.877 | 0.841 | 0.916 | 0.928
P | 0.870 | 0.824 | 0.923 | 0.952
P-SLE | 0.907 | 0.871 | 0.950 | 0.969
P-SCC | 0.953 | 0.918 | 0.995 | 0.998
P-SLE-SCC | 0.965 | 0.938 | 0.998 | 1.000
L | 0.958 | 0.932 | 0.992 | 0.996
L-SCC | 0.976 | 0.957 | 0.999 | 1.000

### Effectiveness of LSE

We observed that LSE was effective only for TFIDF models as shown in Table 5,
but the negative impact to the other models are very limited. We think that
the pre-trained models, PromCSE and LLM, already have better or equivalent
capability compared with LSE.

Table 6 shows some examples of descriptive labels whose rankings were improved
or degraded by LSE from the results of Kaggle-10-choice problem. It shows that
LSE properly address the out-of-vocabulary problem encountered in the TFIDF
model, thereby successfully enriched cryptic or domain specific words.
Additionally, we provide some examples of descriptive labels from the Kaggle
result that were accurately ranked as top 1 by the LSE-enhanced model (T-LSE),
but failed to achieve the same ranking by the baseline model (T) in Table 6.

On the other hand, there were certain descriptive labels that were
successfully ranked by the baseline model (T), but failed by the LSE-enhanced
model (T-LSE). Table 6 shows some examples. For more general words, the
incorporation of LSE may introduce noise and hinder the accurate
identification of the corresponding glossary description.

Table 5: Improved MRR by LSE in Kaggle10 w/o LSE | MRR | w/ LSE | MRR | improved
---|---|---|---|---
T | 0.735 | T-LSE | 0.795 | 0.061
T-SCC | 0.741 | T-LSE-SCC | 0.864 | 0.123
P | 0.872 | P-LSE | 0.865 | -0.007
P-SCC | 0.987 | P-LSE-SCC | 0.986 | -0.001
L | 0.982 | L-LSE | 0.975 | -0.007
L-SCC | 0.994 | L-LSE-SCC | 0.994 | 0.000
Table 6: Example descriptive labels effected by LSE

Improved labels by LSE ‘AADT’, ‘Burglary’, ‘chlorides’, ‘DEROG’, ‘FGA’, ‘FTA’,
‘HAZMAT’, ‘isbn’, ‘isFlaggedFraud’, ‘iso3’, ‘lvdd’, ‘MentHlth’, ‘MVP’, ‘Riot’,
‘rpm’, ‘Sugarcanes’, ‘synopsis’, ‘total sulfur dioxide’, ‘Ward Interactions’

Degraded labels by LSE ‘Age’, ‘category’, ‘Category’, ‘country’, ‘date’,
‘description’, ‘Genre’, ‘id’, ‘Make’, ‘name’, ‘Rating’, ‘Score’, ‘sex’,
‘status’, ‘Status’, ‘Title’, ‘type’, ‘url’

### Effectiveness of SCC

Table 7 shows the improvement of the MRR scores by SCC where SCC improved the
MRR score in all the cases. In particular, it contributed to the improvement
of over 10% for the PromCSE models.

We analyzed the descriptive labels that were successfully ranked as top 1 as
the result of the improvement by SCC (P-LSE-SCC), but failed by the baseline
model (P-LSE). We found these descriptive labels can be classified into two
groups: general words and highly cryptic words. For general words, PromCSE
collected several descriptions from different glossaries with high
confidences. SCC worked as screening indicators in this situation. Table 8
shows some examples of the general labels whose rankings were improved by SCC.

In the case of highly cryptic words, PromCSE often struggled to find similar
descriptions from all candidates, because the highly cryptic words are out-of-
vocabulary even for PromCSE. Consequently, the Text-Similarity Scores of the
top-ranked descriptions were relatively low. However, the Context-Similarity
Score of the corresponding glossary was significantly high. SCC elevated the
confidence of these correct but low-ranked descriptions, pushing them to the
top of the list. Table 8 also shows some examples of the cryptic labels
improved by SCC.

Table 7: Improved MRR by SCC in Kaggle10 w/o SCC | MRR | w/ SCC | MRR | improved
---|---|---|---|---
T | 0.735 | T-SCC | 0.741 | 0.006
T-LSE | 0.795 | T-LSE-SCC | 0.864 | 0.069
P | 0.872 | P-SCC | 0.987 | 0.115
P-LSE | 0.865 | P-LSE-SCC | 0.986 | 0.122
L | 0.982 | L-SCC | 0.994 | 0.011
L-LSE | 0.975 | L-LSE-SCC | 0.994 | 0.019
Table 8: Example descriptive labels improved by SCC

General labels ‘Age’, ‘close’, ‘Close’, ‘date’, ‘Gender’, ‘high’, ‘id’, ‘ID’,
‘name’, ‘Name’, ‘Pos’, ‘Services’, ‘Status’, ‘Tags’, ‘Title’, ‘TITLE’, ‘type’,
‘Type’, ‘url’, ‘Value’, ‘y’, ‘Year’, ‘Years’

Cryptic labels ‘3PA’, ‘ACS/Map’, ‘AST’, ‘CholCheck’, ‘CLAGE’, ‘CLNO’, ‘DEROG’,
‘DREB’, ‘Fedu’, ‘FGS’, ‘FTA’, ‘FTM’, ‘G1’, ‘G2’, ‘G3’, ‘GD’, ‘GF’, ‘GO / SC
Num’, ‘hsc_p’, ‘MentHlth’, ‘Mjob’, ‘OREB’, ‘PTS’, ‘RC_ID’, ‘RH’, ‘SFY’

##  10 Concluding Remarks

We formulated the DLD problem as important and practical task, and identified
the technical issues: out-of-vocabulary issue and ambiguity issue of cryptic
words. We proposed a framework to solve the issues. To solve out-of-vocabulary
issue, we proposed Label Semantic Enrichment method by leveraging external
knowledge. To solve ambiguity issue, we proposed Set-based Collective
Contextualization method by leveraging Large Language Model. We provided two
practical benchmark datasets: Kaggle and FIBO including the issues, and
designed N-choice problem on the datasets. In our experiments, we clarified
how our approach are effective to solve the issues. We plan to release the
benchmark datasets under a reasonable license for future advances in this
technical area.

## Limitations

The empirical evaluation of our methods is mainly done on the datasets derived
from the publicly available data sources whereas we used pre-trained models of
Flan-T5 and PromCSE in the evaluation. Therefore, there might be overlapping
data sources, and hence the risk of data leakage. Even so, our evaluation
showed that both the label enrichment and the contextualization contributed to
the improvement of the TFIDF-based STS, which never rely on any external data
sources.

In our experiment, the LLM-based STS model outperformed the other models in
all the cases. However, we need to care about its inference time when we use
it in a practical situation, since the estimated total inference time for
completing the experiment was roughly 150 hours. On the other hand, the TFIDF
and PromCSE models were obviously more efficient than the LLM-based STS model.
Those total inference times based on our observation were 1.6 hours and 25
hours, respectively.

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