# Inequalities for higher order differences of the logarithm of the
overpartition function and a problem of Wang-Xie-Zhang

Gargi Mukherjee  Institute for Algebra, Johannes Kepler University,
Altenberger Straße 69, A-4040 Linz, Austria.  gargi.mukherjee@dk-
compmath.jku.at

###### Abstract.

Let $\overline{p}(n)$ denote the overpartition function. In this paper, our
primary goal is to study the asymptotic behavior of the finite differences of
the logarithm of the overpartition function, i.e.,
$(-1)^{r-1}\Delta^{r}\log\overline{p}(n)$, by studying the inequality of the
following form

$\log\Bigl{(}1+\dfrac{C(r)}{n^{r-1/2}}-\dfrac{C_{1}(r)}{n^{r}}\Bigr{)}<(-1)^{r-1}\Delta^{r}\log\overline{p}(n)<\log\Bigl{(}1+\dfrac{C(r)}{n^{r-1/2}}\Bigr{)}\
\text{for}\ n\geq N(r),$

where $C(r),C_{1}(r),\text{and}\ N(r)$ are computable constants depending on
the positive integer $r$, determined explicitly. This solves a problem posed
by Wang, Xie and Zhang in the context of searching for a better lower bound of
$(-1)^{r-1}\Delta^{r}\log\overline{p}(n)$ than $0$. By settling the problem,
we are able to show that

$\lim_{n\rightarrow\infty}(-1)^{r-1}\Delta^{r}\log\overline{p}(n)=\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}n^{\frac{1}{2}-r}.$

Mathematics Subject Classifications. Primary 05A20; 11B68.
Keywords. Overpartition, $\log$-concavity, Finite difference.

##  1\. Introduction

An overpartition of a positive integer $n$ is a nonincreasing sequence of
positive integers whose sum is $n$ in which the first occurrence of a number
may be overlined, $\overline{p}(n)$ denotes the number of overpartitions of
$n$, and we define $\overline{p}(0)=1$. For example, there are $8$
overpartitions of $3$ enumerated by
$3,\overline{3},2+1,\overline{2}+1,2+\overline{1},\overline{2}+\overline{1},1+1+1,\overline{1}+1+1$.
A thorough study of the overpartition function started with the work of
Corteel and Lovejoy [4], although it has been studied under different
nomenclature that dates back to MacMahon. Similar to the Hardy-Ramanujan-
Rademacher formula for the partition function (cf. [9],[13]), Zuckerman’s [15]
formula for $\overline{p}(n)$ states that

$\overline{p}(n)=\frac{1}{2\pi}\underset{2\nmid
k}{\sum_{k=1}^{\infty}}\sqrt{k}\underset{(h,k)=1}{\sum_{h=0}^{k-1}}\dfrac{\omega(h,k)^{2}}{\omega(2h,k)}e^{-\frac{2\pi
inh}{k}}\dfrac{d}{dn}\biggl{(}\dfrac{\sinh\frac{\pi\sqrt{n}}{k}}{\sqrt{n}}\biggr{)},$
(1.1)

where

$\omega(h,k)=\text{exp}\Biggl{(}\pi
i\sum_{r=1}^{k-1}\dfrac{r}{k}\biggl{(}\dfrac{hr}{k}-\biggl{\lfloor}\dfrac{hr}{k}\biggr{\rfloor}-\dfrac{1}{2}\biggr{)}\Biggr{)}$

for positive integers $h$ and $k$. Similarly as Lehmer [10] obtained an error
bound for the partition function $p(n)$, Engel [6] determined an error term
for $\overline{p}(n)$ and found that

$\overline{p}(n)=\frac{1}{2\pi}\underset{2\nmid
k}{\sum_{k=1}^{N}}\sqrt{k}\underset{(h,k)=1}{\sum_{h=0}^{k-1}}\dfrac{\omega(h,k)^{2}}{\omega(2h,k)}e^{-\frac{2\pi
inh}{k}}\dfrac{d}{dn}\biggl{(}\dfrac{\sinh\frac{\pi\sqrt{n}}{k}}{\sqrt{n}}\biggr{)}+R_{2}(n,N),$
(1.2)

where

$\bigl{|}R_{2}(n,N)\bigr{|}<\dfrac{N^{5/2}}{\pi
n^{3/2}}\sinh\biggl{(}\dfrac{\pi\sqrt{n}}{N}\biggr{)}.$ (1.3)

A positive sequence $\\{a_{n}\\}_{n\geq 0}$ is $\log$-concave if for all
$n\geq 1$,

$a^{2}_{n}-a_{n-1}a_{n+1}\geq 0.$

Engel [6] proved that $\\{\overline{p}(n)\\}_{n\geq 2}$ is $\log$-concave by
using the asymptotic formula (1.2) with $N=3$ followed by (1.3). Prior to
Engel’s work on overpartitions, the $\log$-concavity of the partition function
$p(n)$ and its associated inequalities has been studied in a wider spectrum,
details can be found in [1], [2], and [5]. Liu and Zhang [11] proved a family
of inequalities for the overpartition function.
Chen, Guo and Wang [3] introduced the notion of ratio $\log$-convexity of a
sequence and established that ratio $\log$-convexity implies $\log$-convexity
under a certain initial condition. A sequence $\\{a_{n}\\}_{n\geq k}$ is
called ratio $\log$-convex if $\\{a_{n+1}/a_{n}\\}_{n\geq k}$ is $\log$-convex
or, equivalently, for $n\geq k+1$,

$\log a_{n+2}-3\log a_{n+1}+3\log a_{n}-\log a_{n-1}\geq 0.$

Let $\Delta$ be the difference operator defined by $\Delta f(n)=f(n+1)-f(n)$.
Similar to the work done by Chen et al. [2] for $p(n)$, Wang, Xie and Zhang
[14] proved the following two theorems.

######  Theorem 1.1.

[14, Theorem 3.1] For each $r\geq 1$, there exists a positive number $n(r)$
such that for all $n\geq n(r)$,

$(-1)^{r-1}\Delta^{r}\log\overline{p}(n)>0.$

######  Theorem 1.2.

[14, Theorem 4.1] For each $r\geq 1$, there exists a positive number $n(r)$
such that for all $n\geq n(r)$,

$(-1)^{r-1}\Delta^{r}\log\overline{p}(n)<1+\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}},$

where $\left(\alpha\right)_{r}:=\alpha\cdot(\alpha+1)\cdots(\alpha+n-1)$.

They raised the following question:

######  Problem 1.3.

[14, Problem 3.4] Does there exist a positive number A such that

$\dfrac{(-1)^{r-1}\Delta^{r}\log\overline{p}(n)}{n^{-(r-1/2)}}>A,$

for any $r$ and all sufficiently large $n$?

In other words, their statement reads “Moreover, we also wish to seek for a
sharp lower bound for $(-1)^{r-1}\Delta^{r}\log\overline{p}(n)$”.
The main motivation of this paper is to give an affirmative solution to the
Problem 1.3 in Theorems 1.4 and 1.6. This in turn clarifies the asymptotic
growth of $(-1)^{r-1}\Delta^{r}\log\overline{p}(n)$, see Corollary 1.7.
Moreover, we reprove the $\log$-concavity and its companion inequality in
Corollary 1.8.

######  Theorem 1.4.

For $n\geq 26$,

$\log\Bigl{(}1+\dfrac{\pi}{2\sqrt{n}}\Bigr{)}<\Delta\log\overline{p}(n)<\log\Bigl{(}1+\dfrac{\pi}{2\sqrt{n}}+\dfrac{\pi^{2}}{40n}\Bigr{)}.$
(1.4)

For $r\geq 2$, we define

######  Definition 1.5.

$\displaystyle N_{0}(m)$ $\displaystyle:=$
$\displaystyle\begin{cases}1,&\quad\text{if}\ m=1,\\\ 2m\log m-m\log\log
m,&\quad\text{if}\ m\geq 2,\end{cases}$ (1.5) $\displaystyle N_{1}(r)$
$\displaystyle:=$
$\displaystyle\max\Biggl{\\{}85,\Biggl{\lceil}\dfrac{4}{\pi^{2}}N^{2}_{0}(2r+2)\Biggr{\rceil}\Biggr{\\}},$
(1.6) $\displaystyle C(r)$ $\displaystyle:=$
$\displaystyle\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1},$ (1.7)
$\displaystyle C_{1}(r)$ $\displaystyle:=$ $\displaystyle(r-1)!+4r^{2}C(r),$
(1.8) $\displaystyle C_{2}(r)$ $\displaystyle:=$
$\displaystyle\sum_{k=0}^{2r-2}\dfrac{1}{(k+1)\pi^{k+1}}\Bigl{(}\dfrac{k+1}{2}\Bigr{)}_{r}\dfrac{1}{r^{k}}+\dfrac{r}{10^{r}},$
(1.9) $\displaystyle N_{2}(r)$ $\displaystyle:=$
$\displaystyle\Biggl{\lceil}\Biggl{(}\dfrac{1+C_{1}(r)}{C(r)}\Biggr{)}^{2}\Biggr{\rceil},$
(1.10) $\displaystyle N_{3}(r)$ $\displaystyle:=$
$\displaystyle\max\Biggl{\\{}N_{1}(r),2r^{2},\Biggl{\lceil}\Biggl{(}\dfrac{2^{r+1}\Bigl{(}C_{2}(r)+1\Bigr{)}}{(r-1)!}\Biggr{)}^{2}\Biggr{\rceil},\Biggl{\lceil}\sqrt[r-1]{\Biggl{(}\dfrac{2^{r}C^{2}(r)}{(r-1)!}\Biggr{)}}\Biggr{\rceil}\Biggr{\\}},$
(1.11) $\displaystyle N(r)$ $\displaystyle:=$
$\displaystyle\max\Bigl{\\{}N_{2}(r),N_{3}(r)\Bigr{\\}}.$ (1.12)

######  Theorem 1.6.

For $r\in\mathbb{Z}_{\geq 2}$ and $n\geq N(r)$,

$0<\log\Bigl{(}1+\dfrac{C(r)}{n^{r-1/2}}-\dfrac{C_{1}(r)}{n^{r}}\Bigr{)}<(-1)^{r-1}\Delta^{r}\log\overline{p}(n)<\log\Bigl{(}1+\dfrac{C(r)}{n^{r-1/2}}\Bigr{)},$
(1.13)

where $C(r)$ and $C_{1}(r)$ are given in (1.7)-(1.8).

######  Corollary 1.7.

For $r\in\mathbb{Z}_{\geq 1}$,

$\lim_{n\rightarrow\infty}n^{r-1/2}(-1)^{r-1}\Delta^{r}\log\overline{p}(n)=\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}.$
(1.14)

  * Proof.

Multiplying both sides of (1.4) (resp. (1.13)) by $\sqrt{n}$ (resp. by
$n^{r-1/2}$) and taking limit as $n$ tends to infinity, we obtain (1.14). ∎

######  Corollary 1.8.

[6, Theorem 1.2] For $n\geq 4$, $\overline{p}(n)$ is $\log$-concave.

  * Proof.

Observe that $N(2)=344$ and from the lower bound of (1.13), we observe that
$\\{\overline{p}(n)\\}_{n\geq 344}$ is $\log$-concave and for the rest $5\leq
n\leq 343$, we confirm by numerical checking in Mathematica. ∎

######  Corollary 1.9.

[11, Equation (1.6)] For $n\geq 2$,

$\dfrac{\overline{p}(n-1)}{\overline{p}(n)}\Bigl{(}1+\dfrac{\pi}{4n^{3/2}}\Bigr{)}>\dfrac{\overline{p}(n+1)}{\overline{p}(n)}.$
(1.15)

  * Proof.

Similar to the proof of Corollary 1.8, take $r=2$ and from the upper bound of
(1.13), we conclude the proof. ∎

######  Corollary 1.10.

For $n\geq 18$, $\overline{p}(n)$ is ratio $\log$-convex.

  * Proof.

Take $r=3$ and observe that $N(3)=1486$ and rest of the proof is similar to
the proof of Corollary 1.8.. ∎

As an immediate consequence of Corollary 1.10, we have

######  Corollary 1.11.

[7, Corollary 1.3] The sequence $\\{\sqrt[n]{\overline{p}(n)}\\}_{n\geq 4}$ is
$\log$-convex.

This paper is organized as follows. A preliminary setup for decomposing
$(-1)^{r-1}\Delta^{r}\log\overline{p}(n)$ $=H_{r}+G_{r}$ (cf. see (2.4) and
(2.5)), as done in [14] and consequently, estimations for both $H_{r}$ and
$G_{r}$ are given in Section 2. Proofs of Theorems 1.4 and 1.6 are given in
Section 3.

##  2\. preliminary lemmas

Following the notations given in Engel [6] and Wang, Xie and Zhang [14], split
$\overline{p}(n)$ as

$\overline{p}(n)=\widehat{T}(n)\Biggl{(}1+\dfrac{\widehat{R}(n)}{\widehat{T}(n)}\Biggr{)},$
(2.1)

where

$\displaystyle\widehat{T}(n)$ $\displaystyle=$
$\displaystyle\dfrac{1}{8n}\Bigl{(}1-\dfrac{1}{\widehat{\mu}(n)}\Bigr{)}e^{\widehat{\mu}(n)}$
(2.2) $\displaystyle\text{and}\ \ \widehat{R}(n)$ $\displaystyle=$
$\displaystyle\dfrac{1}{8n}\Bigl{(}1+\dfrac{1}{\widehat{\mu}(n)}\Bigr{)}e^{-\widehat{\mu}(n)}+R_{2}(n,3)$
(2.3)

with $\widehat{\mu}(n)=\pi\sqrt{n}$.
Taking the logarithm on both sides of (2.1) and plugging the definitions from
(2.2)-(2.3), we obtain

$\log\overline{p}(n)=\log\dfrac{\pi^{2}}{8}-3\log\widehat{\mu}(n)+\log(\widehat{\mu}(n)-1)+\widehat{\mu}(n)+\log\
\Biggl{(}1+\dfrac{\widehat{R}(n)}{\widehat{T}(n)}\Biggr{)}.$

Therefore,

$(-1)^{r-1}\Delta^{r}\log\overline{p}(n)=H_{r}+G_{r},$ (2.4)

where

$\displaystyle H_{r}$ $\displaystyle=$
$\displaystyle(-1)^{r-1}\Delta^{r}(-3\log\widehat{\mu}(n)+\log(\widehat{\mu}(n)-1)+\widehat{\mu}(n))$
(2.5) $\displaystyle G_{r}$ $\displaystyle=$
$\displaystyle(-1)^{r-1}\Delta^{r}\log\
\Biggl{(}1+\dfrac{\widehat{R}(n)}{\widehat{T}(n)}\Biggr{)}.$ (2.6)

Then we have that for $r\geq 1$,

$H_{r}-|G_{r}|\leq(-1)^{r-1}\Delta^{r}\log\overline{p}(n)\leq H_{r}+|G_{r}|.$
(2.7)

To estimate the bounds for $(-1)^{r-1}\Delta^{r}\log\overline{p}(n)$, we need
to establish bounds for $H_{r}$ and $|G_{r}|$. Our first goal is to determine
a bound for $|G_{r}|$ for $r\geq 1$ and then we further proceed with $H_{r}$
but splitting in two cases, i.e., for $r=1$ and $r\geq 2$.

######  Lemma 2.1.

[8, Lemma 2.1] For any integer $m\geq 1$ and $x\geq N_{0}(m)$,

$\displaystyle x^{m}e^{-x}<1,$

where $N_{0}(m)$ is defined in (1.5).

Recall that
$N_{1}(r)=\max\Biggl{\\{}85,\Biggl{\lceil}\dfrac{4}{\pi^{2}}N^{2}_{0}(2r+2)\Biggr{\rceil}\Biggr{\\}}$
(cf. (1.6)).

######  Lemma 2.2.

For all $n\geq N_{1}(r)$ and $r\geq 1$,

$|G_{r}|<\dfrac{1}{n^{r+1}}.$ (2.8)

  * Proof.

Define $\widehat{e}(n):=\frac{\widehat{R}(n)}{\widehat{T}(n)}$. From the
definition of $\widehat{R}(n)$ and $\widehat{T}(n)$ (cf. Equation
(2.2)-(2.3)), we have

$\displaystyle|\widehat{e}(n)|$ $\displaystyle=$
$\displaystyle\dfrac{|\widehat{R}(n)|}{|\widehat{T}(n)|}$ (2.9)
$\displaystyle=$
$\displaystyle\Biggl{|}\dfrac{\frac{1}{8n}\Bigl{(}1+\frac{1}{\widehat{\mu}(n)}\Bigr{)}e^{-\widehat{\mu}(n)}+R_{2}(n,3)}{\frac{1}{8n}\Bigl{(}1-\frac{1}{\widehat{\mu}(n)}\Bigr{)}e^{\widehat{\mu}(n)}}\Biggr{|}$
$\displaystyle<$
$\displaystyle\dfrac{\widehat{\mu}(n)+1}{\widehat{\mu}(n)-1}e^{-2\widehat{\mu}(n)}+\dfrac{36\sqrt{3}}{\widehat{\mu}(n)-1}e^{-2\widehat{\mu}(n)/3}$
$\displaystyle\hskip 113.81102pt\ \ \Bigl{(}\text{using}\ N=3\ \text{in}\
\eqref{Engel2}\ \text{and}\ \sinh(x)<\frac{e^{x}}{2}\ \text{for}\ x>0\Bigr{)}$
$\displaystyle=$
$\displaystyle\dfrac{1}{\widehat{\mu}(n)-1}e^{-\widehat{\mu}(n)/2}\Bigl{(}(\widehat{\mu}(n)+1)e^{-3\widehat{\mu}(n)/2}+36\sqrt{3}\
e^{-\widehat{\mu}(n)/6}\Bigr{)}.$

Since for all $n\geq 85$,

$(\widehat{\mu}(n)+1)e^{-3\widehat{\mu}(n)/2}+36\sqrt{3}\
e^{-\widehat{\mu}(n)/6}<\frac{1}{2}\ \text{and}\
\frac{1}{\widehat{\mu}(n)-1}<1,$

from (2.9), it follows that for all $n\geq 85$,

$|\widehat{e}(n)|<\dfrac{1}{2}\ e^{-\widehat{\mu}(n)/2}.$ (2.10)

Therefore, for all $n\geq 85$,

$\displaystyle|G_{r}|$ $\displaystyle=$
$\displaystyle\Bigl{|}\sum_{i=0}^{r}(-1)^{r-1}\Delta^{r}\log\
(1+\widehat{e}(n))\Bigr{|}\ \ (\text{by}\ \eqref{eqn6})$ (2.11)
$\displaystyle=$
$\displaystyle\Biggl{|}\sum_{i=0}^{r}(-1)^{r-i}\binom{r}{i}\log\
(1+\widehat{e}(n+i))\Biggr{|}$ $\displaystyle\leq$
$\displaystyle\sum_{i=0}^{r}\binom{r}{i}\Bigl{|}\log\
(1+\widehat{e}(n+i))\Bigr{|}$ $\displaystyle\leq$
$\displaystyle\sum_{i=0}^{r}\binom{r}{i}\dfrac{|\widehat{e}(n+i)|}{1-|\widehat{e}(n+i)|}\
\ \Bigl{(}\text{since},\ |\log(1+x)|\leq\dfrac{|x|}{1-|x|}\ \text{for}\
|x|<1\Bigr{)}$ $\displaystyle\leq$ $\displaystyle
2\sum_{i=0}^{r}\binom{r}{i}|\widehat{e}(n+i)|\ \Bigl{(}\text{as}\
\dfrac{x}{1-x}\leq 2x\ \text{for}\ 0<x\leq\frac{1}{2}\Bigr{)}$
$\displaystyle<$
$\displaystyle\sum_{i=0}^{r}\binom{r}{i}e^{-\widehat{\mu}(n+i)/2}\ (\text{by}\
\eqref{lem1eqn4})$ $\displaystyle\leq$
$\displaystyle\sum_{i=0}^{r}\binom{r}{i}e^{-\widehat{\mu}(n)/2}\
\Bigl{(}\text{since},\ \\{e^{-\widehat{\mu}(n)/2}\\}_{n\geq 1}\ \text{is a
decreasing sequence}\Bigr{)}$ $\displaystyle=$ $\displaystyle
2^{r}e^{-\widehat{\mu}(n)/2}.$

Now applying Lemma 2.1 with $m=2r+2$ and assigning
$x\mapsto\frac{\widehat{\mu}(n)}{2}$, it follows that for
$n\geq\Biggl{\lceil}\dfrac{4}{\pi^{2}}N^{2}_{0}(2r+2)\Biggr{\rceil}$,

$e^{-\widehat{\mu}(n)/2}<\Bigl{(}\dfrac{2}{\pi}\Bigr{)}^{2r+2}\dfrac{1}{n^{r+1}}\implies
2^{r}e^{-\widehat{\mu}(n)/2}<\Bigl{(}\dfrac{2\sqrt{2}}{\pi}\Bigr{)}^{2r+2}\dfrac{1}{n^{r+1}}<\dfrac{1}{n^{r+1}}.$
(2.12)

∎

Before we state the bounds for $H_{r}$, we recall the following result due to
Odlyzko [12] on the relation between the higher order differences of a smooth
function and its derivatives.

######  Proposition 2.3.

Let $r$ be a positive integer. Suppose that $f(x)$ is a function with infinite
continuous derivatives for $x\geq 1$, and $(-1)^{k-1}f^{(k)}(x)>0$ for $k\geq
1$. Then for $r\geq 1$,

$(-1)^{r-1}f^{(r)}(x+r)\leq(-1)^{r-1}\Delta^{r}f(x)\leq(-1)^{r-1}f^{(r)}(x).$

######  Lemma 2.4.

For all $n\geq 1$,

$L^{(1)}(n)\leq H_{1}\leq U^{(1)}(n),$ (2.13)

where

$\displaystyle U^{(1)}(n)$ $\displaystyle=$
$\displaystyle\dfrac{\pi}{2\sqrt{n}}-\dfrac{3}{2(n+1)}+\dfrac{\pi}{2\sqrt{n}(\widehat{\mu}(n)-1)}$
(2.14) $\displaystyle\text{and}\ L^{(1)}(n)$ $\displaystyle=$
$\displaystyle\dfrac{\pi}{2\sqrt{n+1}}-\dfrac{3}{2n}+\dfrac{\pi}{2\sqrt{n+1}(\widehat{\mu}(n+1)-1)}.$
(2.15)

  * Proof.

Equation (2.13) follows immediately by applying Proposition 2.3 on each of the
factors in $H_{r}$ being present in (2.5) for $r=1$. ∎

######  Lemma 2.5.

For $r\geq 2$ and $n\geq 2r^{2}$,

$\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{C_{1}(r)}{n^{r}}<H_{r}<\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{2^{r}\
n^{r}}+\dfrac{C_{2}(r)}{n^{r+\frac{1}{2}}},$ (2.16)

where $C(r)$, $C_{1}(r)$, and $C_{2}(r)$ are given by (1.7)-(1.9).

  * Proof.

Rewrite (2.5) as

$H_{r}=(-1)^{r-1}\Delta^{r}(\widehat{\mu}(n)-2\log\widehat{\mu}(n))-\sum_{k=1}^{\infty}(-1)^{r-1}\Delta^{r}\Bigl{(}\dfrac{1}{k\widehat{\mu}(n)^{k}}\Bigr{)}$
(2.17)

and applying Proposition 2.3, we get

$\begin{split}\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{(n+r)^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{n^{r}}&+\sum_{k=1}^{\infty}\dfrac{1}{k\pi^{k}}\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}\dfrac{1}{(n+r)^{r+\frac{k}{2}}}\leq
H_{r}\\\
&\leq\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{(n+r)^{r}}+\sum_{k=1}^{\infty}\dfrac{1}{k\pi^{k}}\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}\dfrac{1}{n^{r+\frac{k}{2}}}.\end{split}$
(2.18)

Since for all positive integers $n$, $r$ and $k$,

$\sum_{k=1}^{\infty}\dfrac{1}{k\pi^{k}}\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}\dfrac{1}{(n+r)^{r+\frac{k}{2}}}>0.$

Therefore,

$\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{(n+r)^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{n^{r}}<H_{r}\leq\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{(n+r)^{r}}+\sum_{k=1}^{\infty}\dfrac{1}{k\pi^{k}}\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}\dfrac{1}{n^{r+\frac{k}{2}}}.$
(2.19)

Now we further investigate the lower bound of $H_{r}$, given in (2.19).

$\displaystyle H_{r}$ $\displaystyle\geq$
$\displaystyle\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{(n+r)^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{n^{r}}$
(2.20) $\displaystyle=$
$\displaystyle\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}\Bigl{(}1+\dfrac{r}{n}\Bigr{)}^{-r+\frac{1}{2}}-\dfrac{(r-1)!}{n^{r}}$
$\displaystyle=$
$\displaystyle\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}+\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}\sum_{m=1}^{\infty}\binom{-\frac{2r-1}{2}}{m}\Bigl{(}\dfrac{r}{n}\Bigr{)}^{m}-\dfrac{(r-1)!}{n^{r}}.$

To bound the infinite series in (2.20), we proceed as follows

$\displaystyle\Biggl{|}\sum_{m=1}^{\infty}\binom{-\frac{2r-1}{2}}{m}\Bigl{(}\dfrac{r}{n}\Bigr{)}^{m}\Biggr{|}$
$\displaystyle=$
$\displaystyle\Biggl{|}\sum_{m=1}^{\infty}\dfrac{(-1)^{m}}{4^{m}}\dfrac{\binom{2r+2m-2}{r+m-1}\binom{r+m-1}{r-1}}{\binom{2r-2}{r-1}}\Bigl{(}\dfrac{r}{n}\Bigr{)}^{m}\Biggr{|}$
(2.21) $\displaystyle\leq$
$\displaystyle\sum_{m=1}^{\infty}\dfrac{1}{4^{m}}\dfrac{\binom{2r+2m-2}{r+m-1}\binom{r+m-1}{r-1}}{\binom{2r-2}{r-1}}\Bigl{(}\dfrac{r}{n}\Bigr{)}^{m}$
$\displaystyle\leq$
$\displaystyle\sum_{m=1}^{\infty}\dfrac{2\sqrt{r-1}}{\sqrt{\pi(r+m-1)}}\binom{r+m-1}{r-1}\Bigl{(}\dfrac{r}{n}\Bigr{)}^{m}$
$\displaystyle\hskip 85.35826pt\Biggl{(}\text{since},\
\dfrac{4^{k}}{2\sqrt{k}}\leq\binom{2k}{k}\leq\dfrac{4^{k}}{\sqrt{\pi k}}\
\forall\ k\geq 1\Biggr{)}$ $\displaystyle<$
$\displaystyle\dfrac{2r}{n}\sum_{m=0}^{\infty}\binom{r+m}{r-1}\Bigl{(}\dfrac{r}{n}\Bigr{)}^{m}$
$\displaystyle\leq$
$\displaystyle\dfrac{2r}{n}\sum_{m=0}^{\infty}r^{m+1}\Bigl{(}\dfrac{r}{n}\Bigr{)}^{m}\
\Biggl{(}\text{as},\ \binom{r+m}{r-1}\leq r^{m+1}\Biggr{)}$ $\displaystyle=$
$\displaystyle\dfrac{2r^{2}}{n}\sum_{m=0}^{\infty}\Bigl{(}\dfrac{r^{2}}{n}\Bigr{)}^{m}\leq\dfrac{4r^{2}}{n}\
\ \text{for all}\ \ n\geq 2r^{2}.$

From (2.20) and (2.21), it follows that for $n\geq 2r^{2}$,

$\displaystyle H_{r}$ $\displaystyle\geq$
$\displaystyle\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}-\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{4r^{2}}{n^{r+\frac{1}{2}}}-\dfrac{(r-1)!}{n^{r}}$
(2.22) $\displaystyle>$
$\displaystyle\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}-\Bigl{(}(r-1)!+2\pi
r^{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\Bigr{)}\dfrac{1}{n^{r}}.$

This finishes the estimation of the lower bound for $H_{r}$.
For the upper bound estimation of $H_{r}$, we start with (2.19) in the
following way

$\displaystyle H_{r}$ $\displaystyle\leq$
$\displaystyle\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{(n+r)^{r}}+\sum_{k=1}^{\infty}\dfrac{1}{k\pi^{k}}\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}\dfrac{1}{n^{r+\frac{k}{2}}}$
$\displaystyle<$
$\displaystyle\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{(2n)^{r}}+\sum_{k=1}^{\infty}\dfrac{1}{k\pi^{k}}\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}\dfrac{1}{n^{r+\frac{k}{2}}}\
\ \Bigl{(}\text{since},\ \dfrac{1}{(n+r)^{r}}>\dfrac{1}{(2n)^{r}}\ \forall\
n>r\Bigr{)}$ $\displaystyle=$
$\displaystyle\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{(2n)^{r}}+\dfrac{1}{n^{r+\frac{1}{2}}}\sum_{k=0}^{2r-2}\dfrac{1}{(k+1)\pi^{k+1}}\Bigl{(}\dfrac{k+1}{2}\Bigr{)}_{r}\dfrac{1}{\sqrt{n}^{k}}+\dfrac{1}{n^{r+\frac{1}{2}}}\sum_{k=2r}^{\infty}\dfrac{1}{k\pi^{k}}\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}\dfrac{1}{\sqrt{n}^{k-1}}$
$\displaystyle\leq$
$\displaystyle\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{(2n)^{r}}+\dfrac{1}{n^{r+\frac{1}{2}}}\underset{:=\widehat{C_{2}}(r)}{\underbrace{\sum_{k=0}^{2r-2}\dfrac{1}{(k+1)\pi^{k+1}}\Bigl{(}\dfrac{k+1}{2}\Bigr{)}_{r}\dfrac{1}{r^{k}}}}+\dfrac{r}{n^{r+\frac{1}{2}}}\underset{:=S(r)}{\underbrace{\sum_{k=2r}^{\infty}\dfrac{1}{k\pi^{k}}\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}\dfrac{1}{r^{k}}}}$
$\displaystyle\hskip 256.0748pt\Bigl{(}\text{since},\
\dfrac{1}{\sqrt{n}^{k}}\leq\dfrac{1}{r^{k}}\forall\ n\geq r^{2}\Bigr{)}.$

In order to estimate the infinite series $S(r)$, we need to give an upper
bound of $\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}$ by rewriting as

$\Bigl{(}\dfrac{k}{2}\Bigr{)}_{r}=\Bigl{(}\dfrac{k}{2}\Bigr{)}^{r}\prod_{i=0}^{r-1}\Bigl{(}1+\dfrac{2i}{k}\Bigr{)}:=\Bigl{(}\dfrac{k}{2}\Bigr{)}^{r}P(r,k).$

Now,

$\log
P(r,k)=\sum_{i=0}^{r-1}\log\Bigl{(}1+\dfrac{2i}{k}\Bigr{)}<\sum_{i=0}^{r-1}\dfrac{2i}{k}=\dfrac{r(r-1)}{k}\
\implies P(r,k)<e^{\frac{r(r-1)}{k}}.$ (2.24)

Using (2.24), we obtain

$S(r)<\sum_{k=2r}^{\infty}\dfrac{1}{k\pi^{k}}\Bigl{(}\dfrac{k}{2}\Bigr{)}^{r}e^{\frac{r(r-1)}{k}}\dfrac{1}{r^{k}}\leq\dfrac{e^{\frac{r-1}{2}}}{2^{r}}\sum_{k=2r}^{\infty}\dfrac{k^{r-1}}{(\pi
r)^{k}}\ \ \Bigl{(}\text{since},e^{\frac{r(r-1)}{k}}\leq
e^{\frac{r-1}{2}}\forall\ k\geq 2r\Bigr{)}.$ (2.25)

Moreover, $k^{r-1}<r^{k}$ for all $r\geq 2$ and $k\geq 2r$. To observe this
fact, we first note that to prove $k^{r-1}<r^{k}$, it is equivalent to show

$\dfrac{r-1}{\log r}<\dfrac{k}{\log k}.$ (2.26)

Define $f(x):=\dfrac{x}{\log x}$ and observe that $f(x)$ is strictly
increasing for all $x>e$. As $k\geq 2r\geq 4>e$, it follows that $f(k)>f(2r)$
and the fact that $f(2r)>\dfrac{r-1}{\log r}$ for $r\geq 2$, we conclude
(2.26).
Applying (2.26) in (2.25), we get

$S(r)<\dfrac{e^{\frac{r-1}{2}}}{2^{r}}\sum_{k=2r}^{\infty}\dfrac{1}{\pi^{k}}=\dfrac{\pi}{\sqrt{e}(\pi-1)}\Bigl{(}\dfrac{\sqrt{e}}{2\
r^{2}}\Bigr{)}^{r}<\dfrac{1}{10^{r}}.$ (2.27)

Hence, by (2.27) and (Proof.), we obtain for all $n\geq r^{2}$,

$H_{r}<\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{2^{r}\
n^{r}}+\dfrac{\widehat{C_{2}}(r)}{n^{r+\frac{1}{2}}}+\dfrac{r}{10^{r}\
n^{r+\frac{1}{2}}}=\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{2^{r}\
n^{r}}+\underset{=C_{2}(r)}{\underbrace{\Bigl{(}\widehat{C_{2}}(r)+\dfrac{r}{10^{r}}\Bigr{)}}}\dfrac{1}{n^{r+\frac{1}{2}}}.$
(2.28)

∎

##  3\. Proof of Theorem 1.4 and 1.6

_Proof of Theorem 1.4_: Applying (2.13) and (2.8) in (2.7), we have for $n\geq
85$,

$L^{(1)}(n)-\dfrac{1}{n^{2}}<\Delta\log\overline{p}(n)<U^{(1)}(n)+\dfrac{1}{n^{2}}.$
(3.1)

It is straightforward to show that for $n\geq 457$,

$-\dfrac{3}{2(n+1)}+\dfrac{\pi}{2\sqrt{n}(\widehat{\mu}(n)-1)}+\dfrac{1}{n^{2}}<-\dfrac{\pi^{2}}{10n}$
(3.2)

and therefore

$U^{(1)}(n)+\dfrac{1}{n^{2}}<\dfrac{\pi}{2\sqrt{n}}-\dfrac{\pi^{2}}{10n}.$
(3.3)

Define $c_{n}:=\dfrac{\pi}{2\sqrt{n}}-\dfrac{\pi^{2}}{10n}$ and
$d_{n}:=\dfrac{\pi}{2\sqrt{n}}+\dfrac{\pi^{2}}{40n}$. Observe that $c_{n}<1$
for $n\geq 1$ and $d_{n}<1$ for $n\geq 3$ and consequently for $n\geq 3$,

$c_{n}<d_{n}-\dfrac{d^{2}_{n}}{2}+\dfrac{d^{3}_{n}}{3}-\dfrac{d^{4}_{n}}{4}<\log(1+d_{n})$
(3.4)

since, $\log(1+x)>x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}-\dfrac{x^{4}}{4}\
\text{for}\ x>0$. Invoking (3.3) and (3.4) in (3.1), we get for $n\geq 457$,

$\Delta\log\overline{p}(n)<\log\Bigl{(}1+\dfrac{\pi}{2\sqrt{n}}+\dfrac{\pi^{2}}{40n}\Bigr{)}.$
(3.5)

Similarly as before, it can be readily shown that for $n\geq 79$,

$L^{(1)}(n)-\dfrac{1}{n^{2}}>\dfrac{\pi}{2\sqrt{n}}-\dfrac{\pi^{2}}{8n}+\dfrac{\pi^{3}}{24n^{3/2}}$
(3.6)

and

$\dfrac{\pi}{2\sqrt{n}}-\dfrac{\pi^{2}}{8n}+\dfrac{\pi^{3}}{24n^{3/2}}>\log\Bigl{(}1+\dfrac{\pi}{2\sqrt{n}}\Bigr{)}$
(3.7)

as $\log(1+x)<x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}$ for $x>0$. Applying (3.6)
and (3.7) into (3.1), it follows that for $n\geq 85$,

$\Delta\log\overline{p}(n)>\log\Bigl{(}1+\dfrac{\pi}{2\sqrt{n}}\Bigr{)}.$
(3.8)

Equations (3.5) and (3.8) conclude the proof of Theorem 1.4 except for $26\leq
n\leq 456$, which we confirm by numerical checking in Mathematica. ∎

_Proof of Theorem 1.6_: Applying (2.8) and (2.16) to the lower bound of (2.7),
it follows that for $n\geq\max\\{N_{1}(r),2r^{2}\\}$,

$\displaystyle(-1)^{r-1}\Delta^{r}\log\overline{p}(n)$ $\displaystyle>$
$\displaystyle\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}-\Bigl{(}(r-1)!+2\pi
r^{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\Bigr{)}\dfrac{1}{n^{r}}-\dfrac{1}{n^{r+1}}$
(3.9) $\displaystyle>$
$\displaystyle\dfrac{\pi}{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}-\Bigl{(}1+(r-1)!+2\pi
r^{2}\Bigl{(}\dfrac{1}{2}\Bigr{)}_{r-1}\Bigr{)}\dfrac{1}{n^{r}}$
$\displaystyle=$
$\displaystyle\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{1+C_{1}(r)}{n^{r}}.$

Following (1.10),
$N_{2}(r)=\Biggl{\lceil}\Biggl{(}\dfrac{1+C_{1}(r)}{C(r)}\Biggr{)}^{2}\Biggr{\rceil}$.
Then for all $n\geq\max\\{N_{1}(r),2r^{2},N_{2}(r)\\}$, it follows that

$(-1)^{r-1}\Delta^{r}\log\overline{p}(n)>\log\Biggl{(}1+\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{1+C_{1}(r)}{n^{r}}\Biggr{)}>0.$
(3.10)

For the upper bound estimation, putting (2.8) and (2.16) together into the
upper bound of (2.7), it follows that for $n\geq\max\\{N_{1}(r),2r^{2}\\}$,

$\displaystyle(-1)^{r-1}\Delta^{r}\log\overline{p}(n)$ $\displaystyle<$
$\displaystyle\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{2^{r}\
n^{r}}+\dfrac{C_{2}(r)}{n^{r+\frac{1}{2}}}+\dfrac{1}{n^{r+1}}$ (3.11)
$\displaystyle<$
$\displaystyle\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{2^{r}\
n^{r}}+\dfrac{C_{2}(r)+1}{n^{r+\frac{1}{2}}}.$

Next, our goal is to show

$-\dfrac{(r-1)!}{2^{r}\
n^{r}}+\dfrac{C_{2}(r)+1}{n^{r+\frac{1}{2}}}<-\dfrac{C^{2}(r)}{2\ n^{2r-1}}$

which is equivalent to

$\dfrac{C^{2}(r)}{2}<n^{r-1}\Biggl{[}\dfrac{(r-1)!}{2^{r}}-\dfrac{C_{2}(r)+1}{\sqrt{n}}\Biggr{]}.$
(3.12)

Note that for all
$n\geq\Biggl{\lceil}\Biggl{(}\dfrac{2^{r+1}\Bigl{(}C_{2}(r)+1\Bigr{)}}{(r-1)!}\Biggr{)}^{2}\Biggr{\rceil}$,
$\dfrac{(r-1)!}{2^{r+1}}-\dfrac{C_{2}(r)+1}{\sqrt{n}}>0$ and therefore

$n^{r-1}\Biggl{[}\dfrac{(r-1)!}{2^{r}}-\dfrac{C_{2}(r)+1}{\sqrt{n}}\Biggr{]}=n^{r-1}\Biggl{[}\dfrac{(r-1)!}{2^{r+1}}+\dfrac{(r-1)!}{2^{r+1}}-\dfrac{C_{2}(r)+1}{\sqrt{n}}\Biggr{]}>n^{r-1}\dfrac{(r-1)!}{2^{r+1}}.$
(3.13)

Hence, to prove (3.12), it is sufficient to prove

$n^{r-1}\dfrac{(r-1)!}{2^{r+1}}>\dfrac{C^{2}(r)}{2}\ \text{which holds for
all}\ \
n\geq\Biggl{\lceil}\sqrt[r-1]{\Biggl{(}\dfrac{2^{r}C^{2}(r)}{(r-1)!}\Biggr{)}}\Biggr{\rceil}.$
(3.14)

Recall that

$N_{3}(r)=\max\Biggl{\\{}N_{1}(r),2r^{2},\Biggl{\lceil}\Biggl{(}\dfrac{2^{r+1}\Bigl{(}C_{2}(r)+1\Bigr{)}}{(r-1)!}\Biggr{)}^{2}\Biggr{\rceil},\Biggl{\lceil}\sqrt[r-1]{\Biggl{(}\dfrac{2^{r}C^{2}(r)}{(r-1)!}\Biggr{)}}\Biggr{\rceil}\Biggr{\\}}\
\ (\text{cf.}\ \eqref{def7})$

. From (3.11) and (3.12), it follows that for $n\geq N_{3}(r)$,

$(-1)^{r-1}\Delta^{r}\log\overline{p}(n)<\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{C^{2}(r)}{2\
n^{2r-1}}<\log\Bigl{(}1+\dfrac{C(r)}{n^{r-1/2}}\Bigr{)}.$ (3.15)

Equation (3.10) and (3.15) together imply that for
$n\geq\max\\{N_{2}(r),N_{3}(r)\\}=N(r)$, (1.13) holds. ∎

Acknowledgements

The author would like to express sincere gratitude to her advisor Prof. Manuel
Kauers for his valuable suggestions on the paper. The author was supported by
the Austrian Science Fund (FWF) grant W1214-N15, project DK13.

## References

  * [1] W. Y. C. Chen.  Recent developments on log-concavity and $q$-log-concavity of combinatorial polynomials.  22nd International Conference on Formal Power Series and Algebraic Combinatorics, 2010, http://www.billchen.org/talks/2010-FPSAC.pdf.
  * [2] W. Y. C. Chen, L. X. W. Wang and G. Y. B. Xie.  Finite differences of the logarithm of the partition function.  Math. Comput., 85:825–847, 2016.
  * [3] W. Y. C. Chen, J. J. F. Guo and L. X. W. Wang.  Infinitely log-monotonic combinatorial sequences.  Adv. Appl. Math., 52: 99–120, 2014.
  * [4] S. Corteel and J. Lovejoy.  Overpartitions.  Trans. Am. Math. Soc. 356, 1623–1635, 2004.
  * [5] S. DeSalvo and I. Pak.  Log-concavity of the partition function.  Ramanujan J., 38(1):61–73, 2015.
  * [6] B. Engel.  Log-concavity of the overpartition function.  Ramanujan J. 43(2), 229–241, 2017.
  * [7] G. Mukherjee.  Log-convexity and the overpartition function, Ramanujan J., to appear.
  * [8] G. Mukherjee, H. W. J. Zhang and Y. Zhong.  Higher order log-concavity of the overpartition function and its consequences, in preparation.
  * [9] G. H. Hardy, S. Ramanujan.  Asymptotic Formulae in Combinatory Analysis.  Proc. London Math. Soc. 17: 75–115, 1918.
  * [10] D. H. Lehmer.  On the remainders and convergence of the series for the partition function.  Trans. Amer. Math. Soc., 46:362–373, 1939.
  * [11] E. Y. S. Liu and H. W. J. Zhang.  Inequalities for the overpartition function.  Ramanujan J., 54(3): 485–509, 2021.
  * [12] A. M. Odlyzko.  Differences of the partition function.  Acta Arith., 49:237–254, 1988.
  * [13] H. Rademacher.  A convergent series for the partition function $p(n)$.  Proc. Nat. Acad. Sci. 23, 78–84, 1937.
  * [14] L. X. W. Wang, G. Y. B. Xie, A. Q. Zhang.  Finite difference of the overpartition function.  Adv. Appl. Math. 92, 51–72, 2018.
  * [15] H. S. Zuckerman.  On the coefficients of certain modular forms belonging to subgroups of the modular group.  Trans. Am. Math. Soc. 45(2), 298–321, 1939.


# Medical Knowledge-Guided Deep Learning for Imbalanced Medical Image
Classification ††thanks: Citation:

Long Gao
Department of Radiology
School of Medicine
University of Pittsburgh
Pittsburgh PA, USA, 15260
<EMAIL_ADDRESS>
&Chang Liu
Department of Bioengineering
Swanson School of Engineering
University of Pittsburgh
Pittsburgh PA, USA, 15260
<EMAIL_ADDRESS>
&Dooman Arefan
Department of Radiology
School of Medicine
University of Pittsburgh
Pittsburgh PA, USA, 15260
<EMAIL_ADDRESS>
&Ashok Panigrahy
Department of Radiology
University of Pittsburgh Medical Center
Children’s Hospital of Pittsburgh
Pittsburgh PA, USA, 15260
&Margarita L. Zuley
Department of Radiology
School of Medicine
University of Pittsburgh
Pittsburgh PA, USA, 15260
&Shandong Wu
Intelligent Systems Program
University of Pittsburgh
Pittsburgh PA, USA, 15260
<EMAIL_ADDRESS>

###### Abstract

Deep learning models have gained remarkable performance on a variety of image
classification tasks. However, many models suffer from limited performance in
clinical or medical settings when data are imbalanced. To address this
challenge, we propose a medical-knowledge-guided one-class classification
approach that leverages domain-specific knowledge of classification tasks to
boost the model’s performance. The rationale behind our approach is that some
existing prior medical knowledge can be incorporated into data-driven deep
learning to facilitate model learning. We design a deep learning-based one-
class classification pipeline for imbalanced image classification, and
demonstrate in three use cases how we take advantage of medical knowledge of
each specific classification task by generating additional “middle” classes to
achieve higher classification performances. We evaluate our approach on three
different clinical image classification tasks (a total of 8459 images) and
show superior model performance when compared to six state-of-the-art methods.
All codes of this work will be publicly available upon acceptance of the
paper.

_K_ eywords Deep learning $\cdot$ Image classification $\cdot$ Data imbalance

##  1 Introduction

Medical imaging classification tasks have achieved remarkable performance with
the development of deep learning models. Usually, the training of most deep
learning models requires a large number of samples from balanced classes.
However, in many medical or clinical applications, samples belonging to
certain classes are hard to collect or rare due to low prevalence, resulting
in data being imbalanced, which poses a great challenge for building machine
learning models [1].

To address the data imbalance problem, One-Class Classification (OCC) is
proposed to learn a model from the class that has a majority of samples, while
other classes that have minority of samples are considered as the anomaly to
be detected or separated [2, 3]. This is also similar to the problem of
anomaly detection [3, 4]. For the OCC, the key point is to build a model that
performs well on the majority class samples, while under-performs on other
class samples. Classical methods, such as One-Class Support Vector Machine
(OCSVM) [5] and Support Vector Data Description (SVDD) focus on mapping
features [6], they are hard to handle large-scale datasets [3]. On the other
hand, deep learning-based methods, such as autoencoder [7, 8] and generative
adversarial network [9], focus on modeling the distribution of the majority
class samples. However, these methods are purely data-driven, leading to
highly varying effects in different medical image classification tasks. For
deep learning-based medical image analysis, there is emerging but limited work
that attempts to integrate prior knowledge to improve machine learning [10].
Some studies [11, 12] introduce color and texture knowledge to obtain regions
of interest for image segmentation. To date, there is little to none previous
work that incorporates specific prior medical knowledge into OCC for
classifying imbalanced medical images.

In this paper, we propose a novel end-to-end method, namely, $M$edical
$K$nowledge-guided $D$eep $L$earning (MKDL), for classifying imbalanced
medical images by one-class classification. The principle behind MKDL lies in
the fact that many medical image classification tasks are partly or largely
based on certain pre-known image features, such as image contrast and
brightness. These pre-known features can be computationally represented and
then leveraged as prior knowledge to facilitate more effective model learning
and therefore boost the classification performance. We formulate this idea in
the self-supervised learning [13] framework, where we construct a balanced
classification task with newly generated "middle" classes through transforming
the original samples of the “majority” class into additional image sample
counterparts. After the classifier is trained on a mixture of the original and
transformed image samples, it learns to distinguish features on samples from
the “majority” class while under-performs on samples from the “minority”
class, making the two classes of the data distinguishable.

The contributions of this work are summarized as follows:

$\cdot$ We propose a new deep learning approach that can incorporate domain-
specific prior medical knowledge to improve the classification tasks on
imbalanced imaging data.

$\cdot$ We propose a simple but effective transformation method to utilize
task/data-specific knowledge into deep learning process.

$\cdot$ We demonstrate our approach on three different use cases where
specific knowledge is reflected by the proposed transformations and
outperforming classification performance are shown in comparison to six
related methods.

##  2 Related Work

Our work focuses on integrating medical knowledge into deep learning models in
OCC to deal with imbalanced medical image data classification. So here we
mainly review some of the major related work on imbalanced data classification
and prior-knowledge-guided machine learning.

Imbalanced Data Classification  Imbalanced data classification can be dealt
with two schemes: one-class classification or binary classification with
special strategies [2]. The first scheme is also known as anomaly detection
[14, 15]. In this scheme, a model is trained with samples from the majority
class, while samples from the minority class are regarded as abnormalities to
be detected by the trained model. The classical methods usually rely on
mapping features into new feature space [5] by functions or fitting the
distribution of majority samples [16]. For example, OCSVM and SVDD employ
kernel functions to map input features into new hyperplane/hypersphere. These
methods are limited for large scale data because of the curse of
dimensionality [3]. With the development of deep learning, convolutional
neural networks (CNN) also have been widely applied in anomaly detection. A
straightforward method is to train an autoencoder with normal samples, and
classify testing samples by analyzing the reconstruction loss [7, 8]. Some
methods also integrate adversarial learning [9] and other strategies [17, 16]
to enhance the performance [18, 4, 19]. For example, AnoGAN [4] learns the
distribution of normal samples by training a GAN. DAOL optimizes autoencoder
by a GAN model to fit the normal chest X ray samples. Similar structure is
also adopted by ADOCC [18] for CT and PET images. These reconstruction-loss-
based methods are relatively difficult to train, and the computational cost
increases when dealing with large scale datasets.

When using binary classifiers, the key is to balance the influence of
different class samples in the binary classification models [20, 21]. A
straightforward method is to balance the sample numbers by over-sampling or
under-sampling [22, 21]. Another method is to generate more minority class
samples by sampling from the borderline area [23, 24]. Some previous studies
also use cost-sensitive methods [25] such as focal loss [26] and CosNet [27]
to manipulate weights on the samples based on the cost metrics.

Prior-knowledge-guided learning Integrating prior knowledge is a broad concept
which has been intensively studied in different contexts. Some previous work
using prior knowledge focuses on image segmentation, detection, and
classification. For instance, PCA-CNN [10] proposes a PCA-based layer to
enforce the model to focus on specific regions for segmentation tasks. [28]
proposes a method to integrate prior knowledge to minimize the distribution
distance of different domains for segmentation. [29] combines the information
about important regions to improve the performance of glaucoma diagnosis. The
work of [30] introduces gene features into lung cancer binary classification
task to improve the performance. [31] proposes a CNN model to integrate the
nodule size and shape distribution as a prior knowledge for thyroid nodule
detection in ultrasound images. As far as we know, little work attempts to use
prior medical knowledge in the context of the OCC task on medical images. A
previous work that is somewhat related to using knowledge for OCC is [32].
This paper generates negative samples by shifting edge points along the
negative direction of estimated data density gradient, thus selects optimal
hyper-parameters for OCSVM.

##  3 Proposed methods

###  3.1 Task formulation

We focus on the classification problems on imbalanced data by the One-Class
Classification approach. In OCC, the class with the majority samples is
considered as the normal class, while the samples in the minority class are
considered abnormalities or outliers. It can be formulated as follows: for a
set of samples $\mathcal{X}$ belonging to the majority class, we aim to learn
a scoring function $F(\mathcal{X})$: $\mathcal{X}$ $\rightarrow$ $R$. A higher
value of $F(X)$ indicates that the given sample $X$ has a higher likelihood of
belonging to the majority class.

Figure 1:  The pipeline of method. First, a given sample $X$ is transformed to
generate n (here n=4 as an example) images based on medical knowledge. Then a
classifier is trained to classify the transformed images into the n classes .
Finally, we calculate the classification score ($s$) that represents the
probability of sample $X$ belonging to the majority class. Figure 2:
Demonstration images of the different classes (the first two rows) and
transformed images (the third row, generated from the right-top sample marked
with the yellow box) in the three use cases: (a) Breast density; (b) Kidney
tumor; (c) Skin cancer.

###  3.2 Rationale

With OCC, our goal is to construct a task, where the model can learn
effectively on samples belonging to the majority class, while exhibiting a low
recognition performance on samples belonging to the minority class. To this
end, inspired by self-supervised learning [13], we construct a classification
task by training the model to classify a set of artificially transformed
images into corresponding classes which represent the transformations. In this
process, selecting appropriate image transformations is crucial. A series of
transformations (e.g., rotation, shift [33, 13], etc) have been studied on
natural images. However, different from natural images where the classes are
usually relevant to different objects/scenes, the classification of
medical/clinical images are often related to different image properties such
as the brightness and contrast features. Based on this observation, a novel
idea is to implement image transformations through manipulating the
classification-related contrast and brightness features, guided by pre-known
medical knowledge that is specific to an image classification task or dataset.
These transformations generate additional samples (we denote them as of newly
constructed “middle” classes, since these samples have a different
distribution from the original majority class). Then, based on the original
and the “middle” classes, we build a classifier to effectively learn the
intrinsic image features of the majority class.

###  3.3 Proposed pipeline

The pipeline with proposed methods ( Figure 1 )includes three steps as
outlined in the following paragraphs.

First, generate n image counterparts from an original sample $X$ according to
specific prior knowledge. See details in Section 3.4 for demonstration on
three use cases. Details of using prior medical knowledge to generate these
images are demonstrated in Section 3.4 on three use cases. Specifically, the
generation of images is defined as:
$\mathcal{T}=\\{\mathcal{T}_{1},\cdots,\mathcal{T}_{n}\\}$ with the
corresponding class labels {1, $\cdots$, n}.
$\mathcal{T}_{i}$=$\mathcal{T}_{i}(X)$ represents the $i$th transformation
operation applied to an original sample $X$. A variety of operations may be
used to implement the transformations. Considering that in many cases
different classes of medical images are usually more relevant to the image
contrast, brightness, and/or color features, in this work we use a simple
linear transformation operation, namely, Linear Magnification (LM), as defined
in Formula 1, to utilize potential medical knowledge for classification.

$\mathcal{T}_{i}=c\cdot\mathcal{T}_{1}+b;\ \ \ \mathcal{T}_{1}=X.$ (1)

where $\mathcal{T}_{1}$ is the original sample $X$; $c$ and $b$ denote the two
coefficients adjusting the contrast and brightness of the sample,
respectively.

It is important to point out that even though the operation of the proposed
transformation is similar to data augmentation, the aim of the transformation
operation in our approach is fundamentally different from data augmentation
[34]. Data augmentation is to modify class-invariant features and preserve
class-relevant features to keep the generated images in the same class as the
original images. In contrast, the aim of the transformation operations in our
approach is to modify the class-relevant features so that the generated images
will have a different distribution from the original class.

Second, train a classifier with the image-label pairs ($\mathcal{T}_{i}$, $i$)
generated from the majority class samples through the first step. Here we
employ the Wide Residual Network (WRN) [35] as the classifier, and use cross-
entropy loss as the loss function.

Third, calculate the classification score for a given sample. The classifier
will output a 1$\times$n vector for each input image. For the n images
generated from $X$, we can obtain a matrix Pn×n. P($i$, $j$) represents the
probability of image $\mathcal{T}_{i}$ ($i\in${1, …, n} belonging to class $j$
($j\in${1, …, n}). The classification score $s$ is calculated by the summation
of the diagonal elements of matrix P (see Formula 2). Since the classification
model is trained on the generated images from the samples belonging to the
“majority” class, a given test sample from the samples of the majority class
will be well-recognized by the classifier. Hence P will be more like a unit
diagonal matrix in this situation. On the other hand, when a test sample
coming from the “minority” class is tested, the classifier will under-perform.
Hence P will be less like a unit diagonal matrix. Based on this difference, a
classification score $s$ is calculated as the summation of the diagonal
elements of matrix P by Formula 2; then by assessing the value of score $s$, a
testing sample can be classified into either the majority or the minority
class.

$s=\sum_{i=1}^{i=n}P(i,i)$ (2)

###  3.4 Incorporating prior medical knowledge: three use cases

The transformation operations are a key to generate transformed images in
order to build an effective classification model. Different transformation
operations can be used or customized according to the characteristics of
specific tasks/datasets. In medical fields, while there is more generic
knowledge that can be useful across different fields, knowledge often is
specific to a concrete task/dataset and therefore, it is vital to represent
and implement specific knowledge for a specific use case. In this section, we
demonstrate how prior medical knowledge is utilized to generate the additional
samples of middle classes on three different use cases for image
classification tasks. For each task, we specify the knowledge that is related
to contrast and brightness of images, and then implement the knowledge through
the proposed transformations.

Mammographic breast density category classification: This task focuses on
distinguishing the four Breast Imaging Reporting and Data System (BI-
RADS)-based breast density categories (see Figure 2(a)) based on digital
mammograms: A) almost entire fatty, B) scattered areas of fibroglandular
density, C) heterogeneously dense, D) extremely dense. Category A and B are
considered a lower risk of developing breast cancer, while C and D indicate a
higher risk of developing breast cancer. Statistically, 10% patients in breast
cancer screening fall into category A and D, respectively, and 40% patients
fall into B and C, respectively. Thus, it forms a typical four-class
classification problem with an imbalanced number of samples across the
classes.

Medical knowledge: The classification of BI-RADS density categories is
generally based on the amount of dense/fibroglanduar tissue in the breast.
Images belonging to Category C and D have more dense pixels/regions, resulting
in a higher level of contrast and brightness. Therefore, this knowledge can be
reflected by simply magnifying the pixel intensity values via implementing the
LM transformation operation. An example is shown in the third row of Figure
2(a), where we generate two middle classes B2 and B3 by multiplying a constant
on the pixel intensity of the image B: B2 =1.2$\times$B, B3=1.4$\times$B. By
training with image-label pairs: (B, 1), (B2, 2) and (B3, 3), the classifier
can effectively learn the features related to the density category
classification.

Kidney tumor diagnosis: This task focuses on distinguishing the benign lesions
(including benign and cyst) from malignant lesions (renal cell carcinoma)
based on CT images. Because of the low prevalence of kidney carcinoma (i.e.,
malignancy is less often is general population) or benign (for example, in
certain special circumstances such as in cancer specialty hospitals, there are
many more carcinoma cases than benign), the classification of the two classes
can be imbalanced.

Medical knowledge: As shown in the first two rows of Figure 2(b), the
malignancy cases have higher contrast and more amounts of bright regions
because they have more artery blood vessels revealed by the contrast agent.
Thus, if a classifier is trained on images generated from malignancy cases (an
example is shown in the third row of Figure 2(b)), it will have low accuracy
when distinguishing images generated from benign cases that have less amounts
of contrast and brightness.

Melanocytic nevus diagnosis in skin cancer: This task focuses on
distinguishing different skin cancer types based on dermatoscopic images. The
classification is often imbalanced because each type of skin cancer has a
different prevalence rate. Melanocytic nevus (MN) is the most common skin
cancer among other types of skin cancers such as melanoma, basal cell
carcinoma, actinic keratosis, benign keratosis, dermatofibroma, and vascular
lesion [36]. Here we aim to classify images as MN or other skin cancers.

Medical knowledge: Color is an important distinguishing factor for the
clinical diagnosis of skin cancer, especially for MN [37]. As shown in the
first two rows of Figure 2(c), MN lesions are usually manifested brown in
images. Thus, by adjusting contrast and brightness, we can generate new images
to train a classifier to learn the color characteristics associated with MN
diagnosis.

In summary, these three clinical image classification tasks are usually
challenging due to imbalanced data. Using contrast and brightness
transformation operations guided by specific medical knowledge, we can
generate additional images that have different distributions from the given
class to facilitate self-supervised learning.

##  4 Experiments

In this section, we evaluate MKDL from four aspects: 1) we show the
classification performance of MKDL with a fixed transformation operation and
compare MKDL to other related methods (Section 4.3); 2) we evaluate the
performance of MKDL under different transformation operations with a varying
combination of transformation parameters (Section 4.4); 3) we analyse the
influence of training sample number to show that MKDL is robust to the small
scale training set (Section 4.5).

###  4.1 Datasets

Table 1: Datasets statistics. Dataset | Dimension | Total | Training | Testing
---|---|---|---|---
Majority | Minority | Majority | Minority
Breast Density | 128$\times$128$\times$1 | B: 2058 | A,C,D: 1690 | B: 1500 | 558 | 1690
C: 2261 | A,B,D: 1487 | C: 1500 | 761 | 1478
Kidney Tumor | 64$\times$64$\times$1 | 147 | 32 | 115 | 32 | 32
Skin Cancer | 128$\times$128$\times$3 | 2022 | 978 | 1500 | 522 | 978

Corresponding to the three use cases discussed in Section 3.4, our experiments
include three different clinical imaging datasets with different modalities,
as briefly summarized in Table 1. All images within a single dataset are
resized to the same dimension, and all pixel intensity values are scaled to
reside in [0, 1]. For the Breast Density dataset, identifying Category B and C
is the most demanded and challenging task for radiologists because these two
categories are similar to each other but indicate different clinical actions
(i.e., category B for regular surveillance but category C for enhanced
screening). Thus, we design two clinically-demanded experiments by 1) taking
category B as the majority class and all other categories as the "minority"
class, 2) taking category C as the majority class and all other categories as
the "minority" class. In the Kidney Tumor dataset, the minority class has much
fewer samples than the majority class. The Skin Cancer dataset is publicly
available [36], we randomly select 3000 images for our experiments due to
considerations on computational limitation.

###  4.2 Baseline methods and experimental settings

The MKDL is compared to six previous OCC models: OCSVM [5], C-OCSVM, DSEBM
[38], DAGMM [16], D-SVDD [3], and AnoGAN [4]. The details are as follows:

OCSVM: This method focuses on mapping the sample into a new feature hyperplane
by a kernel function. Here the Radial Basis Function (RBF) kernel is used. RBF
needs the hyper-parameter $\nu$ (anomaly ratio), so we grid search
$\nu\in\\{0.01,\cdots,0.09\\}$ to select the best performing values based on
the ground truth labels. Therefore, what we report here is the upper bound of
OCSVM’s performance. Each image is reshaped into a vector as the input
feature.

C-OCSVM This is a combination of CAE and OCSVM. We first train a CAE to
compress images, then use the bottleneck features to train OCSVM. Similar to
previous one, we also use grid search to find the best performance.

DSEBM Deep Structure Energy-based Model uses an energy-based model to enforce
the encoder to map samples from the majority class into a low-energy feature
space.

DAGMM Deep Autoencoder Gaussian Mixture Model uses a Gaussian Mixture Model to
help the autoencoder to escape from local optima and fit the distribution of
training samples better.

D-SVDD Similar to SVDD, Deep SVDD uses an encoder network to map the samples
from the majority class into a new hypersphere. Here we use the default hyper-
parameter settings for this method.

AnoGAN This is a GAN-based method for medical images. After training a GAN
with samples from the majority class, it finds a latent code from the
distribution to generate the most similar image to a given testing image, and
use the difference between generated and original images as the loss. The data
augmentation is also used in AnoGAN because the GAN needs a large number of
samples to train.

We repeat each algorithm three times and report the average values of the
model performance. Keras [39] is employed on an NVIDIA TITAN GPU. Adam [40] is
adopted as the optimizer, with a learning rate of 0.0002 and a batch size of
128. These parameters are fixed across all the experiments. Two metrics are
used to evaluate the classification performance: i) Area Under Receiver
Operating Characteristic Curve (AUC), and ii) Area Under the Precision-Recall
curve for majority class (AUPR-maj) and minority class (AUPR-min).

Table 2: The AUC$\pm$std of different methods. The highest AUCs are marked in bold. Dataset | Evaluate | OCSVM | C-OCSVM | DAGMM | DSEBM | D-SVDD | AnoGAN | MKDL
---|---|---|---|---|---|---|---|---
Breast Density | B |  AUC | 74.4$\pm$1.1 | 69.5$\pm$6.2 | 58.7$\pm$8.7 | 70.0$\pm$8.7 | 59.2$\pm$9.5 | 69.8$\pm$3.7 | 78.4$\pm$0.9
AUPR-maj | 40.5$\pm$0.5 | 40.1$\pm$7.0 | 38.7$\pm$15.8 | 40.9$\pm$7.2 | 24.4$\pm$5.0 | 33.1$\pm$2.9 | 56.3$\pm$4.9
AUPR-min | 88.5$\pm$0.8 | 85.9$\pm$3.6 | 87.3$\pm$1.5 | 87.2$\pm$5.4 | 74.8$\pm$8.1 | 82.9$\pm$0.8 | 90.5$\pm$0.2
C |  AUC | 56.4$\pm$1.4 | 65.6$\pm$3.8 | 49.8$\pm$17.9 | 54.2$\pm$3.3 | 59.2$\pm$4.5 | 60.7$\pm$8.5 | 78.3$\pm$5.3
AUPR-maj | 29.4$\pm$0.6 | 44.2$\pm$2.5 | 36.2$\pm$10.4 | 50.8$\pm$13.7 | 35.6$\pm$7.6 | 39.2$\pm$5.2 | 58.1$\pm$8.5
AUPR-min | 63.1$\pm$0.9 | 79.5$\pm$4.0 | 40.0$\pm$1.0 | 69.3$\pm$4.5 | 68.9$\pm$8.1 | 72.1$\pm$4.7 | 89.2$\pm$2.4
Kidney Tumor |  |  AUC | 64.2$\pm$0.6 | 64.4$\pm$3.0 | 56.1$\pm$5.3 | 54.2$\pm$0.3 | 58.2$\pm$7.1 | 54.4$\pm$0.1 | 67.0$\pm$3.1
|  AUPR-maj | 63.2$\pm$0.4 | 63.2$\pm$2.1 | 50.6$\pm$16.8 | 50.1$\pm$0.2 | 43.7$\pm$3.8 | 50.3$\pm$0.7 | 63.0$\pm$3.3
|  AUPR-min | 70.0$\pm$0.4 | 67.5$\pm$4.8 | 61.7$\pm$11.5 | 61.6$\pm$0.5 | 49.9$\pm$5.9 | 61.7$\pm$0.1 | 71.3$\pm$2.8
Skin Cancer |  |  AUC | 67.4$\pm$1.3 | 62.9$\pm$5.3 | 55.1$\pm$5.7 | 52.0$\pm$4.4 | 61.7$\pm$1.5 | 55.2$\pm$5.3 | 72.8$\pm$1.8
AUPR-maj | 75.3$\pm$1.7 | 70.0$\pm$4.6 | 72.9$\pm$5.6 | 52.2$\pm$3.9 | 50.8$\pm$0.7 | 30.4$\pm$2.8 | 78.5$\pm$1.7
AUPR-min | 52.2$\pm$1.5 | 61.9$\pm$5.0 | 38.9$\pm$7.6 | 50.9$\pm$3.1 | 33.3$\pm$1.0 | 64.2$\pm$2.1 | 67.9$\pm$1.9

###  4.3 Comparison with previous related methods

In this experiment, we use a fixed transformation operation for all three
datasets to show the “general” performance of our approach on the three
different tasks. Note that there are four classes in the Breast Density
dataset, categories C and D having higher contrast than B, and categories A
and B having lower contrast than C. Taking this into consideration, we set n=5
for this task and we keep n=5 for the other two tasks too for consistency.
Specifically, according to step 1 in Section 3.3, we transform each original
sample into five images by setting the contrast coefficient $c$={0.2, 0.6, 1,
1.4, 1.8} and the brightness coefficient $b$={0, 0, 0, 0, 0} in Formula 1,
where c=1, b=0 represents the original sample. As shown in Table 2, the
proposed method outperforms the previous work in most of the experiments.

###  4.4 Comparison of different transformation operations

Table 3: The AUCs under different transformation operations. The highest values are marked in bold. Note that LM(5,0) is the result reported in Table 2 (the MKDL column). Dataset |  LM${(5,0)}$ |  S${(4,0)}$ |  R${(4,0)}$ |  LM${(5,1)}$ |  LM${(5,2)}$ |  LM${(3,0)}$ |  LM${(7,0)}$
---|---|---|---|---|---|---|---
Breast Density |  B |  78.44 |  76.48 |  67.02 |  78.50 |  81.28 |  57.50 |  65.50
C |  78.30 |  67.20 |  70.39 |  82.61 |  86.13 |  66.99 |  77.89
Kidney Tumor |  66.97 |  63.57 |  50.29 |  74.67 |  66.70 |  68.42 |  64.55
Skin Cancer |  72.81 |  69.23 |  67.47 |  77.37 |  76.41 |  71.31 |  76.45

In this part, we conduct the experiments using a different set of parameters
for the LM transformation specifically customized for each task/dataset. To
further reveal the benefits of using the medical knowledge guided
transformations, we also compare MKDL with two straightforward transformation
operations: shift (denoted by S(4,0)) and rotation (denoted by R(4,0)). In
addition, to investigate the influence of the different number of
transformation operations, we also test the performance when using n=3 and n=7
transformations (denoted by LM$(3,0)$ and LM$(7,0)$). The details of these
different transformation operations are summarized as follows:

$\textbf{S}(4,0)$: n=4, shift {0, $\frac{h}{3}$} pixels in each x and y
directions for an image of size h$\times$h.

$\textbf{R}(4,0)$ : n=4, rotate {0, 90∘, 180∘, 270∘}.

$\textbf{LM}(5,1):$ n=5, c=$\\{$0.6, 0.8, 1, 1.2, 1.4$\\}$, b=$\\{$0.2, -0.2,
0, 0.2, -0.2$\\}$.

$\textbf{LM}(5,2):$ n=5, c=$\\{$0.6, 0.8, 1, 1.2, 1.4$\\}$, b=$\\{$0.4, -0.4,
0, 0.4, -0.4$\\}$.

$\textbf{LM}(3,0):$ n=3, c=$\\{$0.8, 1, 1.2$\\}$, b=$\\{$-0.2, 0, 0.2$\\}$.

$\textbf{LM}(7,0):$ n=7, c=$\\{$0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6$\\}$,
b=$\\{$0.2, -0.2, 0, 0.2, -0.2$\\}$.

As shown in Table 3, the shift (S${(4,0)}$) and rotation (R${(4,0)}$)
transformation operations obtain relatively good performance in some tasks,
validating the proposed pipeline is a powerful method for AD. However, they
have lower performances than the proposed transformation operations (LM) that
leverage prior medical knowledge. Particularly, the model with the rotation
transformation collapses on the Kidney Tumor dataset (AUC=50.29) because the
lesions have no specific direction property, indicating that transformations
without the guidance of related knowledge can lead to failures. In contrast,
with the prior medical knowledge as we specified, the models exhibit superior
performances in the three given datasets. Furthermore, compared to
LM${(5,0)}$, the optimized transformation parameter values customized for each
task/dataset (LM${(5,1)}$ and LM${(5,2)}$) lead to increased performance.

###  4.5 Comparison with varying size of training samples

Table 4: The AUCs under a varying size of training sample. Dataset |  1500 |  500 |  250 |  100
---|---|---|---|---
Breast Density | MKDL |  81.28 |  80.63 |  80.23 |  76.83
Category B | OCSVM |  74.39 |  74.13 |  74.27 |  73.62
Breast Density | MKDL |  86.13 |  74.20 |  67.58 |  66.56
Category C | C-OCSVM |  65.60 |  65.83 |  56.71 |  54.65
Skin Cancer | MKDL |  76.41 |  73.16 |  73.35 |  71.96
OCSVM |  67.40 |  66.85 |  65.94 |  66.87

In this experiment we examine the effects of the MKDL method when training the
models with a varying size of training samples. We report the model
performance when using 100, 250, 500 samples for training. We perform the
experiment on breast density and skin cancer dataset and skip this experiment
on the kidney tumor dataset because it is already a quite small dataset. We
use LM(5,2) as the transformation operation in implementing MKDL. We compare
MKDL to other methods that obtain the best performance in the previous
experiment (see Table 2). As shown in Table 4, the performance of MKDL is more
stable than other methods with respect to the change of training sample size,
and MKDL consistently outperforms other methods. In particular, the AUCs when
using the smallest training dataset (100 samples) for MKDL are still higher
than the highest AUCs of OCSVM and C-OCSVM when they are trained using 1500
samples.

##  5 Conclusion

In this work, we aim to address classification of imbalanced medical image
data and we propose a novel one-class classifier that leverages medical
knowledge to guide the deep learning of image features. By transforming the
class-relevant features of a given sample into different images, we construct
a classification task where the model can effectively learn essential
information about the majority class. Our method achieves superior performance
compared to previous one-class methods and other transformations, indicating
that integrating medical knowledge into deep learning models may be an
effective approach to improve data-driven machine learning. In future work, we
will further explore the generalizability of our methods on diverse datasets
from different machines and sites. While the medical knowledge is generic,
different datasets may demand adaptive implementation of our methods.

## Acknowledgement

This work was supported by National Institutes of Health (NIH)National Cancer
Institute (NCI) grants (1R01CA193603, 3R01CA193603-03S1, and 1R01CA218405), a
Radiological Society of North America (RSNA) Research Scholar Grant
(#RSCH1530), an Amazon Machine Learning Research Award, and the UPMC Hillman
Cancer Center Developmental Pilot Program. We gratefully acknowledge the
support of NVIDIA Corporation with the donation of the Titan X Pascal GPU for
our research. This research was also partly supported by the NIH through
resources provided by the National Resource for Biomedical Supercomputing (P41
RR06009), which is part of the Pittsburgh Supercomputing Center. Specifically,
it used the Bridges system, which is supported by NSF award number
ACI-1445606, at the Pittsburgh Supercomputing Center.

## References

  * [1] Geert Litjens, Thijs Kooi, Babak Ehteshami Bejnordi, Arnaud Arindra Adiyoso Setio, Francesco Ciompi, Mohsen Ghafoorian, Jeroen Awm Van Der Laak, Bram Van Ginneken, and Clara I Sánchez.  A survey on deep learning in medical image analysis.  Medical image analysis, 42:60–88, 2017.
  * [2] Justin M Johnson and Taghi M Khoshgoftaar.  Survey on deep learning with class imbalance.  Journal of Big Data, 6(1):27, 2019.
  * [3] Lukas Ruff, Robert A. Vandermeulen, Nico Görnitz, Lucas Deecke, Shoaib A. Siddiqui, Alexander Binder, Emmanuel Müller, and Marius Kloft.  Deep one-class classification.  In Proceedings of the 35th International Conference on Machine Learning, volume 80, pages 4393–4402, 2018.
  * [4] Thomas Schlegl, Philipp Seeböck, Sebastian M Waldstein, Ursula Schmidt-Erfurth, and Georg Langs.  Unsupervised anomaly detection with generative adversarial networks to guide marker discovery.  In International conference on information processing in medical imaging, pages 146–157. Springer, 2017.
  * [5] Bernhard Schölkopf, Robert C Williamson, and Alex J Smola.  Support vector method for novelty detection.  In Advances in neural information processing systems, pages 582–588, 2000.
  * [6] David MJ Tax and Robert PW Duin.  Support vector data description.  Machine learning, 54(1):45–66, 2004.
  * [7] Christoph Baur, Benedikt Wiestler, Shadi Albarqouni, and Nassir Navab.  Deep autoencoding models for unsupervised anomaly segmentation in brain mr images.  In International MICCAI Brainlesion Workshop, pages 161–169. Springer, 2018.
  * [8] Xiaoran Chen and Ender Konukoglu.  Unsupervised detection of lesions in brain mri using constrained adversarial auto-encoders.  arXiv preprint arXiv:1806.04972, 2018.
  * [9] Ian J Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.  Generative adversarial networks.  Advances in Neural Information Processing Systems, 3:2672–2680, 2014.
  * [10] Fausto Milletari, Alex Rothberg, Jimmy Jia, and Michal Sofka.  Integrating statistical prior knowledge into convolutional neural networks.  In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 161–168. Springer, 2017.
  * [11] Scott Konishi and Alan L Yuille.  Statistical cues for domain specific image segmentation with performance analysis.  In Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No. PR00662), volume 1, pages 125–132. IEEE, 2000\.
  * [12] Anant Madabhushi and Dimitris N Metaxas.  Combining low-, high-level and empirical domain knowledge for automated segmentation of ultrasonic breast lesions.  IEEE transactions on medical imaging, 22(2):155–169, 2003.
  * [13] Alexander Kolesnikov, Xiaohua Zhai, and Lucas Beyer.  Revisiting self-supervised visual representation learning.  In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 1920–1929, 2019.
  * [14] Arwa Aldweesh, Abdelouahid Derhab, and Ahmed Z Emam.  Deep learning approaches for anomaly-based intrusion detection systems: A survey, taxonomy, and open issues.  Knowledge-Based Systems, 189:105124, 2020.
  * [15] Peng Lv, Yanwei Yu, Yangyang Fan, Xianfeng Tang, and Xiangrong Tong.  Layer-constrained variational autoencoding kernel density estimation model for anomaly detection.  Knowledge-Based Systems, page 105753, 2020.
  * [16] Bo Zong, Qi Song, Martin Renqiang Min, Wei Cheng, Cristian Lumezanu, Daeki Cho, and Haifeng Chen.  Deep autoencoding gaussian mixture model for unsupervised anomaly detection.  ICLR, 2018.
  * [17] J Zhou, KL Chan, VFH Chong, and Shankar M Krishnan.  Extraction of brain tumor from mr images using one-class support vector machine.  In 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference, pages 6411–6414. IEEE, 2006.
  * [18] Qi Wei, Yinhao Ren, Rui Hou, Bibo Shi, Joseph Y Lo, and Lawrence Carin.  Anomaly detection for medical images based on a one-class classification.  In Medical Imaging 2018: Computer-Aided Diagnosis, volume 10575, page 105751M. International Society for Optics and Photonics, 2018.
  * [19] YuXing Tang.  Deep adversarial one-class learning for normal and abnormal chest radiograph classification.  In SPIE, volume 10950, 2019.
  * [20] Samuel Rota Bulo, Gerhard Neuhold, and Peter Kontschieder.  Loss max-pooling for semantic image segmentation.  In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 7082–7091. IEEE, 2017.
  * [21] Paul Viola and Michael Jones.  Rapid object detection using a boosted cascade of simple features.  In Proceedings of the 2001 IEEE computer society conference on computer vision and pattern recognition. CVPR 2001, volume 1, pages I–I. IEEE, 2001.
  * [22] Romero FAB De Morais and Germano C Vasconcelos.  Boosting the performance of over-sampling algorithms through under-sampling the minority class.  Neurocomputing, 343:3–18, 2019.
  * [23] Hui Han, Wen-Yuan Wang, and Bing-Huan Mao.  Borderline-smote: a new over-sampling method in imbalanced data sets learning.  In International conference on intelligent computing, pages 878–887. Springer, 2005.
  * [24] Hien M Nguyen, Eric W Cooper, and Katsuari Kamei.  Borderline over-sampling for imbalanced data classification.  International Journal of Knowledge Engineering and Soft Data Paradigms, 3(1):4–21, 2011.
  * [25] Bianca Zadrozny, John Langford, and Naoki Abe.  Cost-sensitive learning by cost-proportionate example weighting.  In Third IEEE international conference on data mining, pages 435–442. IEEE, 2003.
  * [26] Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollár.  Focal loss for dense object detection.  In Proceedings of the IEEE international conference on computer vision, pages 2980–2988, 2017.
  * [27] Alberto Bertoni, Marco Frasca, and Giorgio Valentini.  Cosnet: a cost sensitive neural network for semi-supervised learning in graphs.  In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 219–234. Springer, 2011.
  * [28] Yuxiao Zhang, Mao Ye, Yan Gan, and Wencong Zhang.  Knowledge based domain adaptation for semantic segmentation.  Knowledge-Based Systems, page 105444, 2019.
  * [29] Yidong Chai, Hongyan Liu, and Jie Xu.  Glaucoma diagnosis based on both hidden features and domain knowledge through deep learning models.  Knowledge-Based Systems, 161:147–156, 2018.
  * [30] Peng Guan, Desheng Huang, Miao He, and Baosen Zhou.  Lung cancer gene expression database analysis incorporating prior knowledge with support vector machine-based classification method.  Journal of experimental & clinical cancer research, 28(1):103, 2009\.
  * [31] Tianjiao Liu, Qianqian Guo, Chunfeng Lian, Xuhua Ren, Shujun Liang, Jing Yu, Lijuan Niu, Weidong Sun, and Dinggang Shen.  Automated detection and classification of thyroid nodules in ultrasound images using clinical-knowledge-guided convolutional neural networks.  Medical image analysis, 58:101555, 2019.
  * [32] Siqi Wang, Qiang Liu, En Zhu, Fatih Porikli, and Jianping Yin.  Hyperparameter selection of one-class support vector machine by self-adaptive data shifting.  Pattern Recognition, 74:198–211, 2018.
  * [33] Izhak Golan and Ran El-Yaniv.  Deep anomaly detection using geometric transformations.  In NeurIPS, 2018.
  * [34] Connor Shorten and Taghi M Khoshgoftaar.  A survey on image data augmentation for deep learning.  Journal of Big Data, 6(1):60, 2019.
  * [35] Sergey Zagoruyko and Nikos Komodakis.  Wide residual networks.  arXiv:1605.07146, 2016.
  * [36] Philipp Tschandl, Cliff Rosendahl, and Harald Kittler.  The ham10000 dataset, a large collection of multi-source dermatoscopic images of common pigmented skin lesions.  Scientific data, 2018.
  * [37] Alon Scope, Stephen W Dusza, Allan C Halpern, Harold Rabinovitz, Ralph P Braun, Iris Zalaudek, Giuseppe Argenziano, and Ashfaq A Marghoob.  The “ugly duckling” sign: agreement between observers.  Archives of dermatology, 144(1):58–64, 2008.
  * [38] Shuangfei Zhai, Yu Cheng, Weining Lu, and Zhongfei Zhang.  Deep structured energy based models for anomaly detection.  arXiv preprint arXiv:1605.07717, 2016.
  * [39] Franois Chollet.  Keras: https://github.com/fchollet/keras, 2015.
  * [40] Diederik P. Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  CoRR abs/1412.6980, 2014.