HAUSDORFF OPERATORS ON SOME SPACES OF HOLOMORPHIC FUNCTIONS ON THE UNIT DISC

A. R. Mirotin

<EMAIL_ADDRESS>
Abstract. We introduce Hausdorff operators over the unit disc and give
conditions for boundedness of such operator in Bloch, Bergman, and Hardy
spaces on the disc. Identity approximation by Hausdorff operators is also
considered.

Key words and phrases. Hausdorff operator, Bloch space, Bergman space, Hardy
space, Möbius transformation, identity approximation.

Mathematics Subject Classification 47G10, 47B38, 46E30

##  1 Introduction and preliminaries

In [1] Hausdorff operators were defined over locally compact groups $G$ via
their automorphism groups $\mathrm{Aut}(G)$. But for the circle group
$\mathbb{T}$ this definition leads to almost trivial Hausdorff operators due
to the almost triviality of $\mathrm{Aut}(\mathbb{T})$. The main idea of this
note is as follows. If we want to fix the aforementioned problem we should
consider spaces of functions on the boundary $\mathbb{T}=\partial\mathbb{D}$
of the unit disc $\mathbb{D}$ with holomorphic extension into the disc and
employ the reach group $\mathrm{Aut}_{0}(\mathbb{D})$ of involutive Möbius
automorphisms of the disc instead of $\mathrm{Aut}(\partial\mathbb{D})$.

The one-dimensional Hausdorff operators were introduced by Hardy [11, Chapter
XI]. The modern theory of Hausdorff operators was inspired by [15]; see
surveys [16], [17]. Hardy introduced his operators as a continuous analog of
Hausdorff means; similarly, to get a mean of a holomorphic function $f$ on
$\mathbb{D}$ it is natural to integrate $f$ over all natural holomorphic
mixing of the disc, i. e., over the elements of the group
$\mathrm{Aut}(\mathbb{D})$ of Möbius transformations of $\mathbb{D}$ (see
theorems 4 and 5 below). So, we arrive at the following

Definition 1. For a holomorphic function $f$ on $\mathbb{D}$ the value of a
Hausdorff operator at $f$ is as follows:

$(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\varphi_{w}(z))d\mu(w),\
z\in\mathbb{D},$

where $\mu$ is some fixed positive Radon measure on $\mathbb{D}$, $K$ is some
fixed $\mu$-measurable function on $\mathbb{D}$, and $\varphi_{w}$ is an
element of $\mathrm{Aut}(\mathbb{D})$ of the form

$\varphi_{w}(z)=\frac{w-z}{1-\overline{w}z}\quad(w\in\mathbb{D}).$

We shall denote the set of such elements by $\mathrm{Aut}_{0}(\mathbb{D})$.

Example. If $\mu$ is concentrated on a subsequence
$\\{w_{n}\\}\subset\mathbb{D}$ we get a class of operators of the form

$(\mathcal{H}_{d}f)(z):=\sum_{n=0}^{\infty}d_{n}f\left(\frac{w_{n}-z}{1-\overline{w_{n}}z}\right)$

(discrete Hausdorff operators over $\mathbb{D}$). If the sequence
$\\{w_{n}\\}$ is finite this is a sort of so-called functional operators (see
[3]).

Remark. In [4] operators in $L^{p}(dH)$ of the form

$\mathbb{K}f(z)=\int_{\mathbb{D}}K(w)f(\varphi_{z}(w))dH(w)$

where $dH$ is the Möbius invariant area measure were introduced under the name
of Hausdorff-Berezin operators. This class of operators is different from our
as the previous example showes.

It should be noted that different operators of Hausdorff type on spaces of
holomorphic functions in the disc or the half-plane were considered in [10],
[2], [6], [7], [8], [9], [5] but all of them are not special cases of the
definition 1.

##  2 Boundedness of Hausdorff operators

The following lemma is crucial for the proofs of our main results.

Lemma 1. [1, Lemma 2] Let $(X;m)$ be a measure space, $\mathcal{F}(X)$ some
Banach space of $m$-measurable functions on $X$, $(\Omega,\mu)$ a
$\sigma$-compact quasi-metric space with positive Radon measure $\mu$, and
$F(w,z)$ a function on $\Omega\times X$. Assume that

(a) the convergence of a sequence in norm in $\mathcal{F}(X)$ yields the
convergence of some subsequence to the same function for $m$-a. e. $z\in X$;

(b) $F(w,\cdot)\in\mathcal{F}(X)$ for $\mu$-a. e. $w\in\Omega$;

(c) the map $w\mapsto F(w,\cdot):\Omega\to\mathcal{F}(X)$ is Bochner
integrable with respect to  $\mu$.

Then for $m$-a. e. $z\in X$ one has

$\left((B)\int_{\Omega}F(w,\cdot)d\mu(w)\right)(z)=\int_{\Omega}F(w,z)d\mu(w).$

###  2.1 The Bloch space

The Bloch space $\mathcal{B}$ of $\mathbb{D}$ is defined to be the space of
analytic functions $f$ on $\mathbb{D}$ such that

$\|f\|_{\mathcal{B}}:=\sup\\{(1-|z|^{2})|f^{\prime}(z)|:z\in\mathbb{D}\\}<+\infty.$

Then $\|\cdot\|_{\mathcal{B}}$ is a complete semi-norm on $\mathcal{B}$, and
$\mathcal{B}$ can be made into a Banach space by introducing the norm

$\|f\|^{\prime}_{\mathcal{B}}:=\|f\|_{\mathcal{B}}+|f(0)|.$

It is important that $\|\cdot\|_{\mathcal{B}}$ is Möbius invariant, that is,

$\|f\circ\varphi_{w}\|_{\mathcal{B}}=\|f\|_{\mathcal{B}}$

for all $\varphi_{w}\in\mathrm{Aut}_{0}(\mathbb{D})$ [13, p. 101].

Theorem 1. If $K(w)$ and $K(w)\log(1-|w|)$ belong to $L^{1}(\mu)$, then the
operator $\mathcal{H}_{K,\mu}$ is bounded on $\mathcal{B}$ and

$\|\mathcal{H}_{K,\mu}\|_{\mathcal{B}\to\mathcal{B}}\leq\int_{\mathbb{D}}|K(w)|\left(2+\frac{1}{2}\log\frac{1+|w|}{1-|w|}\right)d\mu(w).$

Proof. Note that the conditions of Lemma 1 hold for $X=\Omega=\mathbb{D}$,
$m=A$, where $dA(z)=dxdy/\pi$ stands for the normalized area (Lebesgue)
measure on $\mathbb{D}$, $\mathcal{F}(X)=\mathcal{B}$, and
$F(w,z)=K(w)f(\varphi_{w}(z))$. Indeed,

(a) if $f_{n}\in\mathcal{B}$ and $f_{n}\to 0$ strongly, then $f_{n}(0)\to 0$
and $f_{n}^{\prime}\to 0$ uniformly on $\\{z\in\mathbb{D}:|z|<r\\}$ for every
$r\in(0,1)$ and therefore for all $z\in\mathbb{D}$

$f_{n}(z)=\int_{0}^{z}f^{\prime}_{n}(t)dt+f_{n}(0)\to 0\quad(n\to\infty);$

(b) since $\|f\circ\varphi_{w}\|_{\mathcal{B}}=\|f\|_{\mathcal{B}}$, this is
obvious;

(c) this follows from the estimate

$\|F(w,\cdot)\|^{\prime}_{\mathcal{B}}\leq|K(w)|\left(2+\frac{1}{2}\log\frac{1+|w|}{1-|w|}\right)$

(see below).

So, by Lemma 1 for $dA$-a.e. $z\in\mathbb{D}$

$\mathcal{H}_{K,\mu}f(z)=\left((B)\int_{\mathbb{D}}K(w)f\circ\varphi_{w}d\mu(w)\right)(z),$
$None$

the Bochner integral for $\mathcal{B}$. The right-hand side of this equality
is continuous. To shaw that the left-hand side of (1) is continuous, too, we
choose an arbitrary point $z_{0}\in\mathbb{D}$ and a compact neighborhood
$U\subset\mathbb{D}$ of $z_{0}$. By [13, Theorem 5.5] we have

$|f(w)|\leq|f(w)-f(0)|+|f(0)|\leq\|f\|_{\mathcal{B}}\beta(0,w)+|f(0)|,$

where $\beta(z,w)$ stands for the Bergman distance. Since $\beta$ is Möbius
invariant (see, e.g.,[13]), it follows that one can find such $z_{1},z_{2}\in
K$ that for all $z\in U$

$|f(\varphi_{w}(z))|\leq\|f\|_{\mathcal{B}}\beta(0,\varphi_{w}(z))+|f(-z)|=\|f\|_{\mathcal{B}}\beta(\varphi_{w}(w),\varphi_{w}(z))+|f(-z)|=$
$\|f\|_{\mathcal{B}}\beta(w,z)+|f(-z)|\leq\|f\|_{\mathcal{B}}(\beta(w,0)+\beta(0,z))+|f(-z)|\leq$
$\|f\|_{\mathcal{B}}\left(\frac{1}{2}\log\frac{1+|w|}{1-|w|}+\beta(0,z_{1})\right)+|f(-z_{2})|.$

Thus, if $z\in U$ we get a $\mu$ integrable majorant for the left-hand side of
(1) of the form

$K(w)\left(C_{1}\log\frac{1+|w|}{1-|w|}+C_{2}\right).$

Since both sides of the equality (1) are continuous, it is valid for all
$z\in\mathbb{D}$, i.e.,

$\mathcal{H}_{K,\mu}f=(B)\int_{\mathbb{D}}K(w)f\circ\varphi_{w}d\mu(w).$

Since $\|f\circ\varphi_{w}\|_{\mathcal{B}}=\|f\|_{\mathcal{B}}$ [13, p. 101],
and $f\circ\varphi_{w}(0)=f(w)$, it follows that

$\|\mathcal{H}_{K,\mu}f\|^{\prime}_{\mathcal{B}}\leq\int_{\mathbb{D}}|K(w)|\|f\circ\varphi_{w}\|^{\prime}_{\mathcal{B}}d\mu(w)=$
$\|f\|_{\mathcal{B}}\int_{\mathbb{D}}|K(w)|d\mu(w)+\int_{\mathbb{D}}|K(w)||f(w)|d\mu(w).$

As was mentioned above,

$|f(w)|\leq\|f\|_{\mathcal{B}}\beta(0,w)+|f(0)|.$

Therefore

$\|\mathcal{H}_{K,\mu}f\|^{\prime}_{\mathcal{B}}\leq\int_{\mathbb{D}}|K(w)|d\mu(w)\|f\|_{\mathcal{B}}+$
$\int_{\mathbb{D}}|K(w)|\beta(0,w)d\mu(w)\|f\|_{\mathcal{B}}+\int_{\mathbb{D}}|K(w)|d\mu(w)|f(0)|\leq$
$\int_{\mathbb{D}}|K(w)|(2+\beta(0,w))d\mu(w)\|f\|^{\prime}_{\mathcal{B}}.$

This completes the proof.

###  2.2 Bergman spaces

Let $\alpha>-1$ and $dA_{\alpha}(z):=(\alpha+1)(1-|z|^{2})^{\alpha}dA(z)$. For
$p>0$ the Bergman spaces with standard weights are defined by

$L^{p}_{a}(dA_{\alpha}):=H(\mathbb{D})\cap L^{p}(dA_{\alpha})$

where $H(\mathbb{D})$ is the space of analytic functions in $\mathbb{D}$ (see,
e.g., [13, Chapter 4]).

In this subsection we shall consider the case $d\mu=dA,\|\cdot\|_{p,\alpha}$
denotes the norm in $L^{p}_{a}(dA_{\alpha})$ induced from
$L^{p}(dA_{\alpha})$.

Theorem 2. Let $p\geq 1$ and the function $K(w)/(1-|w|)^{\frac{2+\alpha}{p}}$
belongs to $L^{1}(dA)$. Then the operator $\mathcal{H}_{K,A}$ is bounded on
$L^{p}_{a}(dA_{\alpha})$ and

$\|\mathcal{H}_{K,A_{\alpha}}\|_{L^{p}_{a}\to
L^{p}_{a}}\leq\int_{\mathbb{D}}|K(w)|\left(\frac{1+|w|}{1-|w|}\right)^{\frac{2+\alpha}{p}}dA(w).$

Proof. According to [13, Proposition 4.3],

$\int_{\mathbb{D}}g\circ\varphi_{w}(z)dA_{\alpha}(z)=\int_{\mathbb{D}}g(z)\frac{(1-|w|^{2})^{2+\alpha}}{(1-\overline{w}z)^{2(2+\alpha)}}dA_{\alpha}(z)$

for nonnegative $g$. Putting here $g=|f|^{p}$ where $f\in
L^{p}_{a}(dA_{\alpha})$ we get

$\|f\circ\varphi_{w}\|^{p}_{p,\alpha}=\int_{\mathbb{D}}|f\circ\varphi_{w}(z)|^{p}dA_{\alpha}(z)=\int_{\mathbb{D}}|f(z)|^{p}\frac{(1-|w|^{2})^{2+\alpha}}{|1-\overline{w}z|^{2(2+\alpha)}}dA_{\alpha}(z).$

Since $|1-\overline{w}z|\geq 1-|w|$, this implies that

$\|f\circ\varphi_{w}\|^{p}_{p,\alpha}\leq\int_{\mathbb{D}}|f(z)|^{p}\frac{(1-|w|^{2})^{2+\alpha}}{(1-|w|)^{2(2+\alpha)}}dA_{\alpha}(z)=\left(\frac{1+|w|}{1-|w|}\right)^{2+\alpha}\|f\|^{p}_{p,\alpha}.$
$None$

Note that all the conditions of Lemma 1 are satisfied for
$(X,m)=(\mathbb{D},dA_{\alpha})$, $(\Omega,\mu)=(\mathbb{D},dA)$,
$\mathcal{F}(X)=L^{p}_{a}(dA_{\alpha})$, and $F(w,z)=K(w)f(\varphi_{w}(z))$
(indeed, (b) and (c) follow from the estimate (2); (a) is a consequence of a
well known theorem of Riesz).

It follows in view of Lemma 1 that for $dA_{\alpha}$ a.e. $z\in\mathbb{D}$

$\mathcal{H}_{K,\mu}f(z)=(B)\int_{\mathbb{D}}K(w)f\circ\varphi_{w}d\mu(w)(z),$
$None$

the Bochner integral for $L^{p}_{a}(dA_{\alpha})$. As in the proof of Theorem
1 to show that the equality (3) is valid for all $z\in\mathbb{D}$ we shall
prove the continuity of its left-hand side. To this end we shall prove that
for every $z_{0}\in\mathbb{D}$ and for every compact neighborhood
$U\subset\mathbb{D}$ of $z_{0}$ there is a $dA$ integrable majorant for the
integrand of the left-hand side of (3). First note that by [13, Proposition
4.13] there is a constant $C>0$ such that for $f\in
L^{p}_{a}(dA_{\alpha}),z\in\mathbb{D}$

$|f(z)|^{p}\leq\frac{C}{(1-|z|^{2})^{2+\alpha}}\|f\|^{p}_{p,\alpha}.$

It follows that

$|f(\varphi_{w}(z))|\leq\frac{C^{1/p}}{(1-|\varphi_{w}(z)|^{2})^{\frac{2+\alpha}{p}}}\|f\|_{p,\alpha}.$

On the other hand,

$1-|\varphi_{w}(z)|^{2}=\frac{(1-|w|^{2})(1-|z|^{2})}{|1-\overline{w}z|^{2}}\geq\frac{(1-|w|^{2})(1-|z|^{2})}{(1+|w|)^{2}}=$
$\frac{(1-|w|)(1-|z|^{2})}{1+|w|}\geq\frac{1}{2}(1-|z|^{2})(1-|w|).$

Thus,

$|f(\varphi_{w}(z))|\leq\frac{2^{\frac{2+\alpha}{p}}C^{1/p}}{(1-|z|^{2})^{\frac{2+\alpha}{p}}(1-|w|)^{\frac{2+\alpha}{p}}}\|f\|_{p,\alpha}.$

This yields that for every compact neighborhood $U\subset\mathbb{D}$ of
$z_{0}$ there is a constant $C_{U}$ such that

$|K(w)||f(\varphi_{w}(z))|\leq\frac{C_{U}}{(1-|w|)^{\frac{2+\alpha}{p}}},$

and the right-hand side here belongs to $L^{1}(dA)$. Thus, the left-hand side
of (3) is continuous, as well.

Then since $\|\cdot\|_{p,\alpha}$ is a norm, we have in view of (3) and (2)
that

$\|\mathcal{H}_{K,A_{\alpha}}f\|_{p,\alpha}\leq\int_{\mathbb{D}}|K(w)|\|f\circ\varphi_{w}\|_{p,\alpha}dA_{\alpha}(w)\leq$
$\int_{\mathbb{D}}|K(w)|\left(\frac{1+|w|}{1-|w|}\right)^{\frac{2+\alpha}{p}}dA(w)\|f\|_{p,\alpha}.$

This proves the theorem.

###  2.3 Hardy spaces

As is well known, for $0<p<\infty$ the Hardy space $H^{p}=H^{p}(\mathbb{D})$
consists of analytic functions $f$ in the unit disc $\mathbb{D}$ such that

$\|f\|^{p}_{H^{p}}:=\sup_{0<r<1}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}d\theta<\infty.$

Then $\|\cdot\|_{H^{p}}$ is a norm for $p\in[1,\infty)$. It is easy to verify
also that this is a $p$-norm for $p\in(0,1)$, in particular for
$\|\cdot\|_{H^{p}}^{p}$ the triangle inequality holds. Every function $f(z)\in
H^{p}(\mathbb{D})$ has boundary values $f(e^{i\theta})\in
L^{p}(\partial\mathbb{D})$ and the map $f(z)\mapsto f(e^{i\theta})$ is an
isometrical isomorphism of $H^{p}(\mathbb{D})$ onto some closed subspace of
$L^{p}(\partial\mathbb{D},\frac{d\theta}{2\pi})$ (see, e. g., [12], [13]).

Denote by $\widetilde{\Lambda}_{p}$ the class of all holomorphic functions $g$
on $\mathbb{D}$ which are continuous on the closure of $\mathbb{D}$ and such
that if $1/p\notin\\{2,3,\dots\\}$ and $n=\lfloor 1/p\rfloor$ the derivative
$g^{(n-1)}$ of the $2\pi$ periodic counterpart of the boundary function
$g(e^{i\theta})$ belongs to the Lipschitz class $\Lambda_{\\{1/p\\}}$ on
$\mathbb{R}$ (here $\\{1/p\\}$ stands for the fractional part of $1/p$) and if
$1/p=n+1$, then $g^{(n-1)}\in\Lambda_{\ast}$ where $\Lambda_{\ast}$ denotes
the class of all continuous functions $\varphi$ on $\mathbb{R}$ such that
there is a constant $A$ with the property

$|\varphi(x+h)-2\varphi(x)+\varphi(x-h)|<Ah$

for all $x$ and for all $h>0$.

Theorem 3. Let the function $K(w)/(1-|w|)^{1/p}$ belongs to $L^{1}(\mu)$.

1) If $1\leq p<\infty$, then the operator $\mathcal{H}_{K,\mu}$ is bounded on
$H^{p}(\mathbb{D})$ and

$\|\mathcal{H}_{K,\mu}\|_{H^{p}\to
H^{p}}\leq\int_{\mathbb{D}}|K(w)|\left(\frac{1+|w|}{1-|w|}\right)^{1/p}d\mu(w).$

2) Let $0<p<1$. The operator $\mathcal{H}_{K,\mu}$ is bounded on
$H^{p}(\mathbb{D})$ if and only if for every $g\in\widetilde{\Lambda}_{p}$

$c_{n}=\int_{\mathbb{D}}K(w)\left(\frac{g(0)}{\overline{w}^{n}}+\underset{z=\overline{w}}{\mathrm{res}}\left(\left(\frac{wz-1}{z-\overline{w}}\right)^{n}\frac{g(z)}{z}\right)\right)d\mu(w),\quad
n\in\mathbb{Z}_{+}$ $None$

is a sequens of Fourier coefficients of some function
$h\in\widetilde{\Lambda}_{p}$.

Proof. 1) The inequality

$|f(z)|\leq 2^{\frac{1}{p}}\|f\|_{H^{p}}(1-|z|)^{-\frac{1}{p}}\quad(f\in
H^{p}(\mathbb{D}),z\in\mathbb{D})$

(see [12, p. 36]) shows that the condition (a) of Lemma 1 is satisfied for
$(X,m)=(\mathbb{D},dA)$ and $(\Omega,\mu)=(\mathbb{D},\mu)$. Conditions (b)
and (c) are the consequences of the inequality

$\|f\circ\varphi_{w}\|_{H^{p}}\leq\left(\frac{1+|w|}{1-|w|}\right)^{1/p}\|f\|_{H^{p}}.$

This inequality follows from the Littlewood’s subordination theorem (see,
e.g., [13, Theorem 11.12]), but we shall give a simple direct proof. If we as
usual identify the function $f(z)\in H^{p}$ with its boundary value
$f(e^{i\theta})$, then

$\|f\circ\varphi_{w}\|_{H^{p}}^{p}=\frac{1}{2\pi}\int_{0}^{2\pi}|f(\varphi_{w}(e^{i\theta}))|^{p}d\theta=\frac{1}{2\pi
i}\int_{0}^{2\pi}|f(\varphi_{w}(e^{i\theta}))|^{p}\frac{de^{i\theta}}{e^{i\theta}}.$

If we put in the last integral $e^{it}=\varphi_{w}(e^{i\theta})$, then
$e^{i\theta}=\varphi_{w}(e^{it})$, and
$de^{i\theta}=\frac{|w|^{2}-1}{(1-\overline{w}e^{it})^{2}}ie^{it}dt$. Thus,

$\|f\circ\varphi_{w}\|_{H^{p}}^{p}=\frac{1-|w|^{2}}{2\pi}\int_{0}^{2\pi}\frac{|f(e^{it})|^{p}dt}{(\overline{w}-e^{-it})(w-e^{it})}=$
$(1-|w|^{2})\frac{1}{2\pi}\int_{0}^{2\pi}\frac{|f(e^{it})|^{p}dt}{1+|w|^{2}-2\mathrm{Re}(we^{-it})}\leq\frac{1+|w|}{1-|w|}\|f\|_{H^{p}}^{p}.$

So, by Lemma 1 for $dA$ a.e. $z\in\mathbb{D}$

$\mathcal{H}_{K,\mu}f(z)=(B)\int_{\mathbb{D}}K(w)f\circ\varphi_{w}d\mu(w)(z),$

the Bochner integral for $H^{p}(\mathbb{D})$. As in the proof of Theorem 1 to
show that this equality holds for all $z\in\mathbb{D}$ it suffices to prove
the continuity of its left-hand side. To this end first note that by [13,
Theorem 9.1]

$|f(\varphi_{w}(z))|\leq\frac{\|f\|_{H^{p}}}{(1-|\varphi_{w}(z)|^{2})^{1/p}}.$

On the other hand (see the proof of Theorem 2),

$1-|\varphi_{w}(z)|^{2}\geq\frac{1}{2}(1-|z|^{2})(1-|w|).$

Thus,

$|f(\varphi_{w}(z))|\leq\frac{2^{1/p}\|f\|_{H^{p}}}{(1-|z|^{2})^{1/p}(1-|w|)^{1/p}}.$

It follows, that for every compact neighborhood $U\subset\mathbb{D}$

$|K(w)||f(\varphi_{w}(z))|\leq C_{U}\frac{|K(w)|}{(1-|w|)^{1/p}}$

is a $\mu$ integrable majorant for the left-hand side. This proves the desired
continuity.

Since $\|\cdot\|_{H^{p}}$ is a norm, we have in view of the preceding
inequality that

$\|\mathcal{H}_{K,\mu}f\|_{H^{p}}\leq\int_{\mathbb{D}}|K(w)|\|f\circ\varphi_{w}\|_{H^{p}}d\mu(w)\leq$
$\int_{\mathbb{D}}|K(w)|\left(\frac{1+|w|}{1-|w|}\right)^{1/p}d\mu(w)\|f\|_{H^{p}}.$

This completes the proof of the first statement.

2) Let $0<p<1$. Note that $K\in L^{1}(\mu)$ Since $H^{p}(\mathbb{D})$ is an
$F$-space and the dual $H^{p}(\mathbb{D})^{*}$ separates the points of
$H^{p}(\mathbb{D})$ [12, p. 118], Theorem II.2.7 in [14] shows that
$\mathcal{H}_{K,\mu}$ is bounded on $H^{p}(\mathbb{D})$ if and only if a
linear functional $l\circ\mathcal{H}_{K,\mu}$ belongs to
$H^{p}(\mathbb{D})^{*}$ for each $l\in H^{p}(\mathbb{D})^{*}$.

We shall employ the general form of a linear functional on $H^{p}(\mathbb{D})$
[12, Theorem 7.5]. By this theorem every bounded linear functional on
$H^{p}(\mathbb{D})$ has a unique representation of the form

$l_{g}(f)=\lim_{r\to
1-0}\frac{1}{2\pi}\int_{0}^{2\pi}f(re^{i\theta})g(e^{-i\theta})d\theta,$

where $g\in\widetilde{\Lambda}_{p}$ (and vice versa).

Thus, $\mathcal{H}_{K,\mu}$ is bounded on $H^{p}(\mathbb{D})$ if and only if
for every $g\in\widetilde{\Lambda}_{p}$ there is such
$h\in\widetilde{\Lambda}_{p}$ that

$l_{g}\circ\mathcal{H}_{K,\mu}=l_{h}.$ $None$

In other wards, for every $f\in H^{p}(\mathbb{D})$

$\lim_{r\to
1-0}\frac{1}{2\pi}\int_{0}^{2\pi}\left(\int_{\mathbb{D}}K(w)f(\varphi_{w}(re^{i\theta}))d\mu(w)\right)g(e^{-i\theta})d\theta$
$=\lim_{r\to
1-0}\frac{1}{2\pi}\int_{0}^{2\pi}f(re^{i\theta})h(e^{-i\theta})d\theta.$
$None$

Let $\mathcal{H}_{K,\mu}$ is bounded on $H^{p}(\mathbb{D})$. Putting
$f(z)=z^{n}$, $n\in\mathbb{Z}_{+}$ in (6) we get in view of the Fubini theorem
that

$\lim_{r\to
1-0}\int_{\mathbb{D}}K(w)\left(\frac{1}{2\pi}\int_{0}^{2\pi}(\varphi_{w}(re^{i\theta}))^{n}g(e^{-i\theta})d\theta\right)d\mu(w)$
$=\lim_{r\to
1-0}\frac{1}{2\pi}\int_{0}^{2\pi}(re^{i\theta})^{n}h(e^{-i\theta})d\theta,$
$None$

or by the Lebesgue theorem ($\varphi_{w}$ and $g$ are bounded)

$\int_{\mathbb{D}}K(w)\frac{1}{2\pi}\int_{0}^{2\pi}((\varphi_{w}(e^{i\theta}))^{n}g(e^{-i\theta})d\theta
d\mu(w)$
$=\frac{1}{2\pi}\int_{0}^{2\pi}e^{in\theta}h(e^{-i\theta})d\theta,\quad
n\in\mathbb{Z}_{+}.$ $None$

Note that by the Cauchy theorem

$\frac{1}{2\pi}\int_{0}^{2\pi}\left((\varphi_{w}(e^{i\theta})\right)^{n}g(e^{-i\theta})d\theta=\frac{1}{2\pi}\int_{0}^{2\pi}\left(\frac{we^{it}-1}{e^{it}-\overline{w}}\right)^{n}g(e^{it})dt$
$=\frac{1}{2\pi
i}\int_{\\{|z|=1\\}}\left(\frac{wz-1}{z-\overline{w}}\right)^{n}\frac{g(z)}{z}dz=\frac{g(0)}{\overline{w}^{n}}+\underset{z=\overline{w}}{\mathrm{res}}\left(\left(\frac{wz-1}{z-\overline{w}}\right)^{n}\frac{g(z)}{z}\right).$
$None$

Therefore (8) is equivalent to

$\int_{\mathbb{D}}K(w)\left(\frac{g(0)}{\overline{w}^{n}}+\underset{z=\overline{w}}{\mathrm{res}}\left(\left(\frac{wz-1}{z-\overline{w}}\right)^{n}\frac{g(z)}{z}\right)\right)d\mu(w)$
$=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-in\theta}h(e^{i\theta})d\theta,\quad
n\in\mathbb{Z}_{+}$ $None$

and the necessity follows.

To prove the sufficiency, for every $g\in\widetilde{\Lambda}_{p}$ let
$h\in\widetilde{\Lambda}_{p}$ be such that for its Fourier coefficients
$c_{n}$ (4) holds. Then formula (10) is valid. In view of (9) this implies
(8). In turn, (8) implies (7) and then for every algebraic polynomial $q_{n}$
we get

$\int_{\mathbb{D}}K(w)\left(\lim_{r\to
1-0}\frac{1}{2\pi}\int_{0}^{2\pi}q_{n}(\varphi_{w}(re^{i\theta}))g(e^{-i\theta})d\theta\right)d\mu(w)$
$=\lim_{r\to
1-0}\frac{1}{2\pi}\int_{0}^{2\pi}q_{n}(re^{i\theta})h(e^{-i\theta})d\theta.$

In other words,

$\int_{\mathbb{D}}K(w)l_{g}(q_{n}\circ\varphi_{w})d\mu(w)=l_{h}(q_{n}).$
$None$

Taking into account that polynomials are dense in $H^{p}$ (see, e. g., [13,
Corollary 9.5]), for every $f\in H^{p}$ one can choose a sequence $q_{n}$ of
polynomials that converges to $f$ in $H^{p}$. Since

$l_{h}(q_{n})\to l_{h}(f),\ l_{g}(q_{n}\circ\varphi_{w})\to
l_{g}(f\circ\varphi_{w})\mbox{ as }n\to\infty,$

and by Lemma 1

$|l_{g}(q_{n}\circ\varphi_{w})|\leq\|l_{g}\|\|q_{n}\|_{p}\left(\frac{1+|w|}{1-|w|}\right)^{1/p}\leq\mathrm{const}\left(\frac{1+|w|}{1-|w|}\right)^{1/p},$

formula (11) implies in view of the Lebesgue theorem that the property (5) is
valid. This proves the sufficiency.$\Box$

Putting $g(z)=z$ in the previous theorem we get the following

Corollary 1. Let $0<p<1$ and let the function $K(w)/(1-|w|)^{1/p}$ belongs to
$L^{1}(\mu)$. If the operator $\mathcal{H}_{K,\mu}$ is bounded on
$H^{p}(\mathbb{D})$, then

$c_{n}=n\int_{\mathbb{D}}K(w)w^{n-1}(|w|^{2}-1)d\mu(w),\quad
n\in\mathbb{Z}_{+}$

is a sequens of Fourier coefficients of some function
$h\in\widetilde{\Lambda}_{p}$.

##  3 Identity Approximation by Hausdorff Operators

Although the class of operators we are considering differs from the class
considered in [4], there are analogs of Theorem 14 from [4] that was devoted
to identity approximation by HausdorffBerezin operators.

In the following for the function $f$ on $\mathbb{D}$ we put
$f^{\vartriangle}(z)=f(-z)$ and for $0<\varepsilon<1$ consider operators of
the form

$\mathcal{H}_{\varepsilon}f(z):=\int_{\mathbb{D}}K(w)(f\circ\varphi_{\varepsilon
w})^{\vartriangle}(z)d\mu(w).$ $None$

As in [4, Section 5] for the goals of approximation it is naturally to assume
that

$K\in L^{1}(d\mu),\ \mbox{ and }\ \int_{\mathbb{D}}Kd\mu=1.$ $None$

Theorem 4. Let $1\leq p<\infty$. Under the assumption (13) with $dA_{\alpha}$
instead of $d\mu$ the operators (12) with $dA_{\alpha}$ instead of $d\mu$ are
identity approximations in $L^{p}(dA_{\alpha})$, namely,

$\lim\limits_{\varepsilon\to 0}\|\mathcal{H}_{\varepsilon}f-f\|_{p,\alpha}=0.$
$None$

Proof. As in the proof of Theorem 2 we have

$\mathcal{H}_{\varepsilon}f=(B)\int_{\mathbb{D}}K(w)(f\circ\varphi_{\varepsilon
w})^{\vartriangle}dA_{\alpha}(w),$ $None$

the Bochner integral in $L^{p}(dA_{\alpha})$. It follows in view of (13) that

$\|\mathcal{H}_{\varepsilon}f-f\|_{p,\alpha}\leq\int_{\mathbb{D}}|K(w)|\|(f\circ\varphi_{\varepsilon
w})^{\vartriangle}-f\|_{p,\alpha}dA_{\alpha}(w).$ $None$

We shall show that the operators $\mathcal{H}_{\varepsilon}$ are uniformly
bounded when $\varepsilon\in(0,1/2)$, specifically,

$\|\mathcal{H}_{\varepsilon}\|\leq
2^{\frac{2(2+\alpha)}{p}}\int_{\mathbb{D}}|K(w)|dA_{\alpha}(w).$ $None$

Indeed, using [13, Proposition 4.3] with $a=\varepsilon w$ we have

$\|(f\circ\varphi_{\varepsilon
w})^{\vartriangle})\|_{p,\alpha}^{p}=\|f\circ\varphi_{\varepsilon
w}\|_{p,\alpha}^{p}=$ $\int_{\mathbb{D}}|f(z)|^{p}\frac{(1-|\varepsilon
w|^{2})^{(2+\alpha)}}{|1-\varepsilon\overline{w}z|^{2(2+\alpha)}}dA_{\alpha}(z)\leq$
$\int_{\mathbb{D}}|f(z)|^{p}\frac{1}{(1-\varepsilon)^{2(2+\alpha)}}dA_{\alpha}(z)\leq
2^{2(2+\alpha)}\|f\|_{p,\alpha}^{p}.$ $None$

Thus, (17) follows from (15) and (18).

In view of (17) to prove (14) it remains to check this equality on a dense
subset of $L^{p}(dA_{\alpha})$. We shall use (16) to check (14) on the subset
of bounded functions. Let $|f(z)|\leq C$. Then

$\|(f\circ\varphi_{\varepsilon
w})^{\vartriangle}-f\|_{p,\alpha}^{p}=\int_{\mathbb{D}}|(f\circ\varphi_{\varepsilon
w})(-z)-f(z)|^{p}dA_{\alpha}(z)\to 0$

as $\varepsilon\to 0$ by the Lebesgue’s dominated convergence theorem
($|(f\circ\varphi_{\varepsilon w})(-z)-f(z)|\leq 2C$). But formula (18)
implies that

$\|(f\circ\varphi_{\varepsilon
w})^{\vartriangle}-f\|_{p,\alpha}\leq(2^{\frac{2(2+\alpha)}{p}}+1)\|f\|_{p,\alpha}.$

It follows (again by the Lebesgue’s dominated convergence theorem) that the
right-hand side of (16) vanishes as $\varepsilon\to 0$. This completes the
proof.

Theorem 5. Let $1\leq p<\infty$. Under the assumption (13) the operators (12)
are identity approximations in $H^{p}(\mathbb{D})$, namely,

$\lim\limits_{\varepsilon\to 0}\|\mathcal{H}_{\varepsilon}f-f\|_{H^{p}}=0.$
$None$

The proof is similar with the proof of Theorem 4. Indeed, as in the proof of
Theorem 3 we have

$\mathcal{H}_{\varepsilon}f=(B)\int_{\mathbb{D}}K(w)(f\circ\varphi_{\varepsilon
w})^{\vartriangle}d\mu(w),$ $None$

the Bochner integral in $H^{p}(\mathbb{D})$. It follows in view of (13) that

$\|\mathcal{H}_{\varepsilon}f-f\|_{H^{p}}\leq\int_{\mathbb{D}}|K(w)|\|(f\circ\varphi_{\varepsilon
w})^{\vartriangle}-f\|_{H^{p}}d\mu(w).$ $None$

We shall show that the operators $\mathcal{H}_{\varepsilon}$ are uniformly
bounded when $\varepsilon\in(0,1/3)$, specifically,

$\|\mathcal{H}_{\varepsilon}\|\leq
2^{\frac{1}{p}}\int_{\mathbb{D}}|K(w)|d\mu(w).$ $None$

Indeed, in the proof of Theorem 3 it was shown that

$\|(f\circ\varphi_{\varepsilon
w})^{\vartriangle})\|_{H^{p}}=\left(\frac{1+\varepsilon|w|}{1-\varepsilon|w|}\right)^{\frac{1}{p}}\|f\|_{H^{p}}.$

Thus,

$\|(f\circ\varphi_{\varepsilon w})^{\vartriangle})\|_{H^{p}}\leq
2^{\frac{1}{p}}\|f\|_{H^{p}}$ $None$

as $\varepsilon\in(0,1/3)$, and (22) follows from (20) and (23).

In view of (22) to prove (19) it remains to check this equality on a dense
subset of $H^{p}(\mathbb{D})$. We shall use (21) to check (19) on the subset
of polynomials. First note that since
$|z_{1}^{k}-z_{2}^{k}|\leq(k+1)|z_{1}-z_{2}|$ for $z_{1},z_{2}\in\mathbb{D}$,
$k\in\mathbb{N}$, we have

$|((\varphi_{\varepsilon
w}(z))^{k})^{\vartriangle})-z^{k}|=\left|\left(\frac{\varepsilon
w+z}{1+\varepsilon\overline{w}z}\right)^{k}-z^{k}\right|\leq(k+1)\left|\frac{\varepsilon
w-\varepsilon\overline{w}z^{2}}{1+\varepsilon\overline{w}z}\right|\leq\frac{2(k+1)\varepsilon}{1-\varepsilon}.$

and therefore

$\|((\varphi_{\varepsilon
w}(z))^{k})^{\vartriangle})-z^{k}\|_{H^{p}}\leq\frac{2(k+1)\varepsilon}{1-\varepsilon}.$

It follows that

$\|(f\circ\varphi_{\varepsilon w})^{\vartriangle}-f\|_{H^{p}}\to 0$

as $\varepsilon\to 0$ for every polynomial $f$. But formula (23) implies that

$\|(f\circ\varphi_{\varepsilon
w})^{\vartriangle}-f\|_{H^{p}}\leq(2^{\frac{1}{p}}+1)\|f\|_{H^{p}}.$

Then the Lebesgue’s dominated convergence theorem shows that the right-hand
side of (21) vanishes as $\varepsilon\to 0$. This completes the proof.

## References

  * [1] Mirotin, A. R. : Boundedness of Hausdorff operators on Hardy spaces $H^{1}$ over locally compact groups. J. Math. Anal. Appl. 473, 519 – 533 (2019) doi: 10.1016/j.jmaa.2018.12.065
  * [2] Galanopoulos, P., Papadimitrakis, M.: Hausdorff and Quasi-Hausdorff Matrices on Spaces of Analytic Functions. Canadian Journal of Mathematics. 58, Issue 3 , 548 – 579 (2006) DOI: https://doi.org/10.4153/CJM-2006-023-5
  * [3] Antonevich, A., Lebedev, A.: Functional differential equetions: I. $C^{*}$-theory. Longman Scientific and Technical, Harlow (1994)
  * [4] Karapetyants, A., Samko, S., Zhu, K.: A Class of Hausdorff-Berezin Operators on the Unit Disc. Complex Anal. Oper. Theory, 13, no. 8, 3853–3870 (2019) doi.org/10.1007/s11785-019-00934-x
  * [5] Karapetyants, A., Samko, S.: HadamardBergman Convolution Operators. Complex Anal. Oper. Theory. 14, 77 (2020) https://doi.org/10.1007/s11785-020-01035-w
  * [6] Stylogiannis, G.: Hausdorff operators on Bergman spaces of the upper half plane. Concr. Oper. 7, 69–80 (2020)
  * [7] Arvantidis, A.G., Siskakis, A.G.: Cesaro operators on the Hardy Spaces of the Half-Plane. Canadian Mathematical Bulletin. 56, no 2, 229 – 240 (2013) DOI: https://doi.org/10.4153/CMB-2011-153-7
  * [8] Hung, H. D.,Ky, L. D. , Quang, T. T.: Hausdorff operators on holomorphic Hardy spaces and applications. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 150, no 3, 1095 – 1112 (2020) DOI: https://doi.org/10.1017/prm.2018.74
  * [9] Ballamoole, S., Bonyo, J. O., Miller, T. L. and Miller, V. G.: Cesaro-like operators on the Hardy and Bergman spaces of the half plane. Complex Anal. Oper. Theory 10, no. 1, 187–203 (2016) doi.org/10.1007/s11785-015-0481-8
  * [10] Aizenberg, L., Liflyand, E.: Hardy spaces in Reinhardt domains, and Hausdorff operators. Illinois J. Math. 53, 10331049 (2009)
  * [11] Hardy, G. H.: Divergent series. Clarendon Press, Oxford (1949)
  * [12] Duren, P. L.: Theory of $H^{p}$ spaces. Academic Press, New York and London (1970)
  * [13] Zhu, K.: Operator theory in function spaces. Second edition. American Mathematical Society (2007)
  * [14] Dunford N., Schwartz, J. T.: Linear Operators. Part I. Pure and Applied Mathematics. Vol. 7. Interscience Publishers Inc., New York (1958)
  * [15] Liflyand, E., Móricz, F.: The Hausdorff operator is bounded on the real Hardy space $H^{1}(\mathbb{R})$. Proc. Am. Math. Soc. 128, 1391 – 1396 (2000)
  * [16] Liflyand, E.: Hausdorff operators on Hardy spaces. Eurasian Math. J., 4, 101 – 141 (2013)
  * [17] Chen, J., Fan, D., Wang, S.: Hausdorff operators on Euclidean space (a survey article). Appl. Math. J. Chinese Univ. Ser. B (4), 28, 548–564 (2014)


# A hybrid deep learning approach for medical relation extraction

Veera Raghavendra Chikka  International Institute of Information
TechnologyGachibowliHyderabadIndia<EMAIL_ADDRESS>and
Kamalakar Karlapalem  International Institute of Information
TechnologyGachibowliHyderabadIndia<EMAIL_ADDRESS>
(2018)

###### Abstract.

Mining relationships between treatment(s) and medical problem(s) is vital in
the biomedical domain. This helps in various applications, such as decision
support system, safety surveillance, and new treatment discovery. We propose a
deep learning approach that utilizes both word level and sentence-level
representations to extract the relationships between treatment and problem.
While deep learning techniques demand a large amount of data for training, we
make use of a rule-based system particularly for relationship classes with
fewer samples. Our final relations are derived by jointly combining the
results from deep learning and rule-based models. Our system achieved a
promising performance on the relationship classes of I2b2 2010 relation
extraction task.

Medical Relation Extraction, Bi-directional LSTM, Deep Learning

††conference: KDD Workshop on Machine Learning for Medicine and Healthcare;
Aug 2018; London, United Kingdom††journalyear: 2018

##  1\. Introduction

Understanding the relationships between treatment and problem is crucial in
various biomedical applications. The extensive generation of electronic health
records prompt immense knowledge of these relations in the biomedical
literature. Manually extracting these relations is difficult and time-
consuming. This necessitates the automatic techniques with minimal human
intervention. Various challenges have been organized in the past decade to
identify relationships in the biomedical literature (uzuner20112010, ).
Relation extraction usually includes two steps: (1) identifying named
entities, (2) identifying relations between them. Unlike named entities,
relation extraction is a complex task which aims to extract context and
structure from the text (ananiadou2010event, ). A simple bag-of-words and co-
occurrence modelling cannot detect the relationship among these entities.
Studies (chun2006extraction, ) report that only 30% of co-occurring protein-
protein entities have actual interactions. Also, relationships have internal
structures which are very difficult to detect (ananiadou2010event, ).

In order to promote the research in relation extraction, I2b2 2010 community
organized a challenge (uzuner20112010, ) on identifying relationship types
between medical problems, and treatments. The relationships between treatment
and problem addressed in this task are:

  1. (1)

Treatment is administered for a medical problem (TrAP), suggests that this
treatment is an accepted procedure for the medical problem. For example, “She
did have some pain and was treated with Percocet”.

  2. (2)

Treatment improves the medical problem(TrIP). For example, “He had occasional
episodes of rapid atrial fibrillation that were controlled with IV
metoprolol”.

  3. (3)

Treatment worsens the medical problem (TrWP). For example, “The stroke
occurred while taking aspirin and plavix”.

  4. (4)

Treatment causes medical problem(TrCP). For example, “The patient had
pancytopenia and vomiting on DDI”.

  5. (5)

Treatment is not administered because of a medical problem (TrNAP), suggests
that this treatment should not be used for this medical problem. For example,
“His coumadin was held during his stay given his acute bleed”.

Most of the participants in I2b2 2010 challenge have used machine learning
techniques like Support vector machines (SVM) for the identification of
relationship type among the entities (de2010nrc, )(jiang2010hybrid,
)(patrick2010i2b2, )(roberts2010extraction, ). Among the participant systems,
hybrid approach (roberts2010extraction, ) with SVM and rule-based approach
reported top performance in the relation extraction task. Our goal is to
develop a hybrid method to improve the performance of deep learning technique.
The contributions of our work are:

  * •

Usually CNN is the popular neural network architecture used in relation
extraction task. In this work, we have experimented with Bi-directional Long
Short Term Memory (LSTM) to capture sequential dependency between the relation
arguments.

  * •

Comparison and analysis of SVM and deep learning for relation extraction task
have been reported.

  * •

For the task of relation classification between the medical entities, the
state of art approaches consider only the word-level and three word window
proximity. We experiment by considering the complete sentence as a matrix
comprising word vectors of all the words along with position indexes.

##  2\. Bi-directional LSTM

Long-Short Term Memory (LSTM) is an improvement over Recurrent Neural Networks
(RNNs) designed to overcome the problem of exponential decay of back
propagation error (hochreiter1997long, ). Thus enabling LSTMs to handle long-
term dependencies. LSTM layer is a chain-like structure of components, known
as memory blocks. Each block contains one or more memory cells and three
regulatory structures - input, output and forget gates.These gates allow LSTM
to add, delete, and reset the information flowing through each block and
stores the information in the form of cell state (yu2015using, ).

LSTMs flows the data in one directional, where as Bidirectional-LSTM processes
the information from both directions. Figure 1 shows the LSTM hidden layer
comprising of LSTM block passing information from either side of the layer.
This allows LSTM to learn the contextual dependency of current word on either
sides. Figure 1 shows our neural network architecture for relation extraction
task. Input sentence is passed for each sample pair of treatment and problem
present in the sentence. The sentence is represented with both word-level and
sentence level features. Word level features used are: word token, Part of
speech tags, and Chunk phrase tags. Lookup table computes the representations
for each of the features along with position vectors. Position vector contains
the distance of the word from the treatment and problem for which relationship
type has to be identified (collobert2011natural, ). For example, if $w_{2}$
and $w_{5}$ are the medical entities, then the position vectors of the
sentence for entities $w_{2}$ and $w_{5}$ are (-1, 0, 1, 2, 3) and (-4, -3,
-2, -1, 0), respectively. The features are initialized with the random
representations. These representation are fine-tuned while training through
back propagation.

###  2.1. Sentence level features for relation extraction

  1. (1)

POS tag sequence: The part-of-speech sequence of the phrase between the
entities. Frequent 100 POS tag sequence are pre-computed from the training
dataset. The vector of dimension 100 is used as feature.

  2. (2)

Point wise mutual information: A statistical measure between two entities,
based on their co-occurrences. This value is used as a feature.

  3. (3)

Added feature words used to find the assertion of the sentence. These are the
dictionary keywords used in Apache cTAKES (savova2010mayo, ) (namely,
allergy.txt, cause.txt, fail.txt, certainty.txt, history.txt,
hypothetical_cue_list.txt and uncertainty.txt). These words give the context
of the sentence. The index of the word tokens in each of the dictionary is
used as feature.

Sentence level representations are computed by a separate module. These
representations are appended Bi-LSTM outputs in the Merge Layer. Finally, a
fully connected Linear layer is used to get the output size of number of
relationships (ntags).

Figure 1. Bi-directional LSTM

##  3\. Rule based approach

Rule-based approaches are widely used in medical applications which need
critical quality for high precision outcomes. Relationship between pair of
medical terms depends on the context in which they occur in the sentence. For
example, a sentence $s=w_{1},...,m_{1},...,w_{i},...,m_{2},...,w_{n}$
containing medical terms $m_{1}$, and $m_{2}$ with relation $r$ can be
described as $s=s_{b}m_{1}s_{m}m_{2}s_{a}$, where $s_{b}$, $s_{m}$ and $s_{a}$
are before, middle and after word-context portions of the medical terms
($m_{1}$, and $m_{2}$) respectively.

Given a test sentence $t$ containing medical terms $m_{01}$ and $m_{02}$, if
the before, middle and after portions of $t$ are similar to that of sentence
$s$, it implies that the relation $r$ of $s$ exists in between $m_{01}$ and
$m_{02}$. Two approaches were used to identify the relation ship between
medical terms:

  * •

Pattern of sentence/phrase: The relation of two entities in a sentence depends
on the phrase in between the entities. Such phrase patterns were manually
framed by checking the training dataset samples. For example, a pattern for
relation “TrAP” (Treatment Administered Problem) can be defined as
“$<problem>$ is diagnosed with $<treatment>$”.

  * •

Shortest Dependency path: The verbs that are present in the shortest
dependency path of pair of the medical terms can be used to identify the
relation. For example,
Sentence: Given her fever the patient was treated with Ceptaz and Levaquin.
Medical entities: fever (problem), Ceptaz (drug), and Levaquin (drug)
Relationships: (Ceptaz, fever, TrAP), (Levaquin, fever, TrAP).
In the above sentence, the verb “treated” infers that the medicines “Ceptaz”
and “Levaquin” were administered for problem “fever”. Having verb “treated” in
the dependency path of pair of entities ‘Ceptaz’ and ‘fever’, infers the
relationship TrAP. Similarly, for the pair ‘Levaquin’ and ‘fever’. If there is
no dependency path exists in between the medical terms, the string of words
between the medical terms is used as the path. The verbs for each of the
relation are collected from the training dataset. For the relationship ’TrCP’
(Treatment controls problem), has verbs like control, regulate, modulate,
restrict, etc., are used to identify the relationship. We check whether the
shortest dependency path between the entities contains such terms.

##  4\. Results and Discussion

###  4.1. Dataset statistics:

We used a subset of i2b22010 challenge dataset which is publicly available.
The organizers of I2b2 have restricted the distribution of remaining dataset.
The original training dataset consists of 349 documents, 27,837 concepts, and
5264 relations, and the test set consists of 477 documents, 45,009 concepts,
and 9069 relations. Statistics of our i2b2 2010 subset data is given in Table
1. Out of the available 426 documents, 10% (42 documents) were used for
testing, and the remaining 384 documents were used for training. This was done
in order to maximize the training set to provide as large as possible dataset
for deep learning purposes. All the experiments are carried out using this
dataset.

Table 1. Relationship type instances Relationship types | Number of instances
---|---
Total | Train | Test
TrAP | 2617 | 2342 | 275
TrCP | 526 | 455 | 71
TrIP | 203 | 167 | 36
TrNAP | 174 | 161 | 13
TrWP | 133 | 119 | 14
All | 3653 | 3244 | 409

###  4.2. Relation Extraction Experimentation

Experiments are evaluated using commonly used Precision (P), Recall (R) and
F-score (F). These metrics are computed based on the True Positives (TP),
False Positives (FP), and False Negatives (FN).
$Precision(P)=\frac{TP}{TP+FP}$

$Recall(R)=\frac{TP}{TP+FN}$

$F_{score}(F)=\frac{2\times Recall\times Precision}{Recall+Precision}$

For the relation extraction task, we have used keras (chollet2015keras, ) for
the implementation of Bi-LSTM neural network architecture. As done in previous
works, we train the relation extraction model by giving the gold relations and
test the model by checking for relations between gold medical entities. The
parameters of LSTM architecture that are restored to default are
$LSTM\\_output\\_size$ as 64 and $number~{}of~{}epochs$ as 20.

####  4.2.1. Effect of negative samples:

The total number of samples in our dataset with respect to relations is around
4000 as given in Table 1. In order to balance these positive samples, we have
experimented with varied number of negative samples ($null$ relationships) as
5000, 10000, 20000 and 30000, where we have obtained higher F-score for 20000
negative samples.

####  4.2.2. Effect of Varying Embedding size:

Word embedding is the numerical representation of words in the form of
vectors. We used Glove tool (pennington2014glove, ) to obtain vector
representations (embeddings) for words. We have empirically tried various word
embedding sizes. Results on embedding sizes 40, 100 and 200 are shown in Table
2. Based on Table 2, we chose to use embedding size $40$ for further
experimentation.

Table 2. Relation Extraction Results with varying embedding sizes | Embedding sizes
---|---
| 40 | 100 | 200
| TrAP | TrCP | TrIP | TrNAP | TrWP | Total | TrAP | TrCP | TrIP | TrNAP | TrWP | Total | TrAP | TrCP | TrIP | TrNAP | TrWP | Total
P | 0.53 | 0.44 | 0.5 | 0.45 | 0.08 | 0.51 | 0.563 | 0.48 | 0.26 | 0.17 | 0.08 | 0.45 | 0.49 | 0.42 | 0.37 | 0.23 | 0.07 | 0.44
R | 0.60 | 0.30 | 0.25 | 0.38 | 0.07 | 0.49 | 0.6 | 0.35 | 0.38 | 0.38 | 0.21 | 0.51 | 0.61 | 0.42 | 0.27 | 0.46 | 0.14 | 0.51
F | 0.56 | 0.36 | 0.33 | 0.41 | 0.0 | 0.50 | 0.58 | 0.40 | 0.31 | 0.24 | 0.12 | 0.48 | 0.54 | 0.35 | 0.31 | 0.30 | 0.09 | 0.47
Table 3. Relation Extraction Results using SVM (baseline), Bi-LSTM and Rule-Based approaches |  $SVM$ (baseline) | $Bi-LSTM$ | $RuleBased$
---|---|---|---
| TrAP | TrCP | TrIP | TrNAP | TrWP | Total | TrAP | TrCP | TrIP | TrNAP | TrWP | Total | TrAP | TrCP | TrIP | TrNAP | TrWP | Total
P | 0.36 | 0.65 | 0.61 | 0.26 | 0.33 | 0.38 | 0.61 | 0.47 | 0.41 | 0.14 | 0.15 | 0.50 | 0.56 | 1.0 | 1.0 | 1.0 | 1.0 | 0.61
R | 0.71 | 0.36 | 0.36 | 0.30 | 0.07 | 0.58 | 0.56 | 0.47 | 0.36 | 0.30 | 0.21 | 0.52 | 0.29 | 0.09 | 0.22 | 0.23 | 0.07 | 0.24
F | 0.48 | 0.46 | 0.45 | 0.28 | 0.11 | 0.46 | 0.58 | 0.47 | 0.38 | 0.20 | 0.17 | 0.51 | 0.38 | 0.17 | 0.36 | 0.37 | 0.13 | 0.35
Table 4. Results of Hybrid Bi-LSTM and Rule-based system | Bi-LSTM + RuleBased
---|---
| TrAP | TrCP | TrIP | TrNAP | TrWP | Total
Precision | 0.61 | 0.47 | 0.44 | 0.22 | 0.15 | 0.51
Recall | 0.56 | 0.47 | 0.41 | 0.46 | 0.21 | 0.53
F-score | 0.58 | 0.47 | 0.42 | 0.30 | 0.17 | 0.52

####  4.2.3. Comparison of SVM, Bi-LSTM and Rule-based approaches

Table 3 shows the results of relation extraction with SVM, Bi-directional LSTM
and Rule based approaches. From Table 3, we can infer that Bi-LSTM has shown
relatively better performance on identifying TrAP relation, implying the
importance of the large number of data samples required to train a neural
network model.

As the part of the I2b2 dataset is retricted to use, we have built a SVM model
on remaining dataset. We have used libsvm (chang2011libsvm, ) for the
implementation of SVM based on the top performing system (rink2011automatic, )
of the i2b2 challenge. The feature vector for SVM is given as a one-hot
encoded vector of word vocabulary, part of speech tags, chunk tags and
position indexes. SVM has shown a consistent results on identifying TrAP,
TrCP, and TrIP which are around 0.45 F-score. This implies that the SVM model
can be trained with comparatively less number of samples and it also balances
the class weights of the relationship types.

Framing rules for relation extraction is a very challenging task, since mere
part-of-speech tag sequences or chunk phrase sequences does not work. Rules
for relations should be at the word level. We have five relationship types
between treatment and problem, the semantics of the relations are very close
that it becomes difficult in framing rules. The sentences with TrIP (improves)
relationship type have verbs like improvement, relieved, resolved, controlled,
etc., which are directly attributed to TrIP. Where as the relationship types
TrCP (causes) and TrWP (worsens) are bit similar as both signifies the
negative impact of Treatment on the Problem. For example, the pattern
“$<problem>resistant~{}to<treatment>$” was treated as TrCP, where as,
“$<problem>intermittently~{}resistant~{}to<treatment>$” was tagged as TrWP.
Few sentences does not even have direct verbs to know which relationship type
they belong to, for example, ”$<problem>of~{}the<treatment>$”,
“$<problem>as~{}he~{}was~{}on<treatment>$”,
“$<problem>since~{}the<treatment>$”, etc. Also we have found few ambiguous
relationship annotations in the original dataset. In few sentences like “Known
allergies to Drugs” are sometimes tagged as TrAP and TrCP in our dataset. Few
of the ambiguous patterns in gold standard dataset are listed below:

  * •

“$<treatment>concern~{}for<problem>$” has given any of TrAP, TrIP or TrCP
relationship types.

  * •

“$<treatment>treated~{}for<problem>$” can take both TrAP and TrIP

  * •

“$<problem>after<treatment>$” has both TrAP and TrIP relationship types.

  * •

Similarly, for other patterns like “$<problem>due~{}to<treatment>$”, etc.

####  4.2.4. Performance of Hybrid Bi-LSTM and Rule-based system

We merge the rule-based approach relationships with high precision to Bi-LSTM
outcomes. That is, from Table 3, except “’TrAP’, all other relationship
samples of rule-based approach are combined with Bi-LSTM samples. Table 4
shows that this combination gave better performance for TrIP and TrNAP
relationships. The numbers of TrCP and TrWP are unchanged since the samples of
those relationship types were already identified by Bi-LSTM. The source code
of our system is available at:
https://github.com/RaghavendraCh/RelationExtraction_keras.

##  5\. Conclusion

In this paper, we have shown the important problem of extracting relationships
between treatment and medical problem. We have described Bi-directional LSTM
approach to automatically extract the relationships. The neural network is
trained by considering both word-level and sentence-level representations. We
have experimented Bi-LSTM by varying the number of negative samples, embedding
vector sizes and features, by keeping the parameters “lstm_output_size” and
“number_of_epochs” as constant. We have used support vector machines as the
baseline system for relation extraction. Bi-LSTM has shown relatively better
performance on identifying TrAP relation, implying the importance of a large
number of data samples required to train a neural network model. Also, the
rule-based approach has been used for relationships with less number of
samples available in our dataset. Finally, we have shown that rule-based
system with high-precision leverages the performance of the neural network. As
a future work, we plan to extend our work to different kinds of relationships,
such as test-problem, drug-problem, and drug-drug relations, that exists in
the biomedical literature.

## References

  * [1] S. Ananiadou, S. Pyysalo, J. Tsujii, and D. B. Kell.  Event extraction for systems biology by text mining the literature.  Trends in biotechnology, 28(7):381–390, 2010.
  * [2] C.-C. Chang and C.-J. Lin.  Libsvm: a library for support vector machines.  ACM Transactions on Intelligent Systems and Technology (TIST), 2(3):27, 2011.
  * [3] F. Chollet.  keras.  https://github.com/fchollet/keras, 2015.
  * [4] H.-W. Chun, Y. Tsuruoka, J.-D. Kim, R. Shiba, N. Nagata, T. Hishiki, and J. Tsujii.  Extraction of gene-disease relations from medline using domain dictionaries and machine learning.  In Pacific Symposium on Biocomputing, 2006.
  * [5] R. Collobert, J. Weston, L. Bottou, M. Karlen, K. Kavukcuoglu, and P. Kuksa.  Natural language processing (almost) from scratch.  Journal of Machine Learning Research, 12(Aug):2493–2537, 2011.
  * [6] B. de Bruijn, C. Cherry, S. Kiritchenko, J. Martin, and X. Zhu.  Nrc at i2b2: one challenge, three practical tasks, nine statistical systems, hundreds of clinical records, millions of useful features.  Proc. of i2b2/VA, 2010.
  * [7] S. Hochreiter and J. Schmidhuber.  Long short-term memory.  Neural computation, 9(8):1735–1780, 1997.
  * [8] M. Jiang, Y. Chen, M. Liu, T. Rosenbloom, S. Mani, J. C. Denny, and H. Xu.  Hybrid approaches to concept extraction and assertion classification-vanderbilt’s systems for 2010 i2b2 nlp challenge.  In Proceedings of the 2010 i2b2/VA Workshop on Challenges in Natural Language Processing for Clinical Data. Boston, MA, USA: i2b2, 2010.
  * [9] J. Patrick, D. Nguyen, Y. Wang, et al.  I2b2 challenges in clinical natural language processing 2010.  In Proceedings of the 2010 i2b2/VA Workshop on Challenges in Natural Language Processing for Clinical Data. Boston, MA, USA: i2b2, 2010.
  * [10] J. Pennington, R. Socher, and C. D. Manning.  Glove: Global vectors for word representation.  In EMNLP, volume 14, pages 1532–43, 2014.
  * [11] B. Rink, S. Harabagiu, and K. Roberts.  Automatic extraction of relations between medical concepts in clinical texts.  Journal of the American Medical Informatics Association, 18(5):594–600, 2011.
  * [12] K. Roberts, B. Rink, and S. Harabagiu.  Extraction of medical concepts, assertions, and relations from discharge summaries for the fourth i2b2/va shared task.  In Proceedings of the 2010 i2b2/VA Workshop on Challenges in Natural Language Processing for Clinical Data. Boston, MA, USA: i2b2, 2010.
  * [13] G. K. Savova, J. J. Masanz, P. V. Ogren, J. Zheng, S. Sohn, K. C. Kipper-Schuler, and C. G. Chute.  Mayo clinical text analysis and knowledge extraction system (ctakes): architecture, component evaluation and applications.  Journal of the American Medical Informatics Association, 17(5):507–513, 2010.
  * [14] Ö. Uzuner, B. R. South, S. Shen, and S. L. DuVall.  2010 i2b2/va challenge on concepts, assertions, and relations in clinical text.  Journal of the American Medical Informatics Association, 18(5):552–556, 2011.
  * [15] Z. Yu, V. Ramanarayanan, D. Suendermann-Oeft, X. Wang, K. Zechner, L. Chen, J. Tao, A. Ivanou, and Y. Qian.  Using bidirectional lstm recurrent neural networks to learn high-level abstractions of sequential features for automated scoring of non-native spontaneous speech.  In Automatic Speech Recognition and Understanding (ASRU), 2015 IEEE Workshop on, pages 338–345. IEEE, 2015.