# Semi-Supervised Disease Classification based on Limited Medical Image Data

Yan Zhang, Chun Li, Zhaoxia Liu, Ming Li This research was supported by
Guangdong Basic and Applied Basic Research Foundation (No.2021A1515220073).Y.
Zhang and Z. Liu are with School of Science, Minzu University of China,
Beijing, 100081, China. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Li is with Joint Research Center on Computational Mathematics and Control,
Shenzhen MSU-BIT University, Shenzhen, 518172, China.M. Li is with Institute
of Data Science, National University of Singapore, Singapore. E-mail:
<EMAIL_ADDRESS>*Corresponding author: Chun Li (E-mail:
lichun2020@smbu.edu.cn).

###### Abstract

In recent years, significant progress has been made in the field of learning
from positive and unlabeled examples (PU learning), particularly in the
context of advancing image and text classification tasks. However, applying PU
learning to semi-supervised disease classification remains a formidable
challenge, primarily due to the limited availability of labeled medical
images. In the realm of medical image-aided diagnosis algorithms, numerous
theoretical and practical obstacles persist. The research on PU learning for
medical image-assisted diagnosis holds substantial importance, as it aims to
reduce the time spent by professional experts in classifying images. Unlike
natural images, medical images are typically accompanied by a scarcity of
annotated data, while an abundance of unlabeled cases exists. Addressing these
challenges, this paper introduces a novel generative model inspired by Hölder
divergence, specifically designed for semi-supervised disease classification
using positive and unlabeled medical image data. In this paper, we present a
comprehensive formulation of the problem and establish its theoretical
feasibility through rigorous mathematical analysis. To evaluate the
effectiveness of our proposed approach, we conduct extensive experiments on
five benchmark datasets commonly used in PU medical learning: BreastMNIST,
PneumoniaMNIST, BloodMNIST, OCTMNIST, and AMD. The experimental results
clearly demonstrate the superiority of our method over existing approaches
based on KL divergence. Notably, our approach achieves state-of-the-art
performance on all five disease classification benchmarks. By addressing the
limitations imposed by limited labeled data and harnessing the untapped
potential of unlabeled medical images, our novel generative model presents a
promising direction for enhancing semi-supervised disease classification in
the field of medical image analysis. The code is available at the following
URL: https://github.com/wmh12138/Paper_Code/tree/master/train.

###### Index Terms:

PU learning, generative model, medical image classification, divergence
learning.

##  I Introduction

Positive and unlabeled learning (PU learning) is a learning approach that aims
to develop effective classifiers using only positive and unlabeled data, in
contrast to traditional supervised learning (TSL) methods [1, 2, 3, 4, 5, 6].
PU learning poses unique challenges due to the limited availability of labeled
data. To illustrate this point, let’s consider the field of medical image
processing, where the annotation of medical images requires the expertise of
well-trained clinical professionals. However, these experts often face time
constraints due to their busy schedules, making it challenging for them to
manually review a large number of image samples. Consequently, the scarcity of
labeled images hampers the development of effective classifiers.

Deep learning models rely heavily on the availability of extensive training
data and their corresponding expensive annotations. Unfortunately, the use of
medical image-aided diagnosis is restricted by several factors [7, 8, 9, 10]:

  * •

Labor-intensive annotation: The annotation of medical images is a labor-
intensive task that demands considerable time and effort from clinical
experts. Consequently, the lack of sufficient labeled data poses a fundamental
challenge for medical image-aided diagnosis.

  * •

Exploitation of unlabeled data: In addition to the limited number of labeled
images, the collection of medical data often includes a significant amount of
undiagnosed data, such as CT, MRI, and PET scans. These unlabeled samples
represent a challenging problem as they lack corresponding labels and exhibit
an uncertain illness state. Typically, these cases are considered
“suspicious”.

  * •

Expert involvement: Medical image annotations necessitate the expertise of
skilled clinicians, further impeding the availability of labeled data.

Addressing these challenges requires innovative approaches that can
effectively leverage both limited labeled data and the vast amount of
unlabeled data. By doing so, PU learning methods hold great potential for
enhancing the accuracy and efficiency of medical image analysis and aiding in
diagnosis.

PU learning is a valuable approach in the realm of semi-supervised binary
classification, specifically designed to handle ambiguous data scenarios [4,
11, 12]. It aims to recover labels from unknown samples by leveraging patterns
learned from positive cases. PU learning has gained significant attention as a
prominent area of research in semi-supervised learning, particularly
applicable to disease classification based on medical image analysis. Various
categories of PU learning methods have emerged, including:

  * •

Two-stage method [13, 14]: This approach involves two steps. Firstly, reliable
negative examples are extracted from the unlabeled sample set. In the second
step, a supervised or semi-supervised learning algorithm is employed to train
the classifier. However, this method requires the utilization of information
from the positive example set to compile an initial collection of trustworthy
negative examples from the unlabeled set.

  * •

Statistical learning based on positive samples: Denis et al. [15] proposed a
naive Bayesian classifier using the statistical query learning model, tailored
to PU learning. However, these models often necessitate the provision of prior
probability for positive examples.

  * •

One-sided noise learning: This approach treats unlabeled samples as noisy
negative examples, reducing the PU learning problem to an one-sided noise
learning problem. Examples include biased SVM [16] and biased PRTFIDF [17].
However, overfitting remains a challenge. To address this, Kiryo et al. [18]
proposed non-negative PU learning, improving the objective function based on
the unbiased learning model.

  * •

Predictive adversarial networks (PANs): Introduced by Hu et al. [19], this
method introduces a new objective function based on KL divergence and
generative adversarial networks (GANs). The generator $G$ of GANs is replaced
by a classifier $C$, and the optimization goal is to train a discriminator $D$
for classifying unlabeled data. The PAN learning approach has been shown, both
theoretically and experimentally, to train a binary classifier without relying
on prior probability. It outperforms current state-of-the-art PU learning
models such as NNPU [18] and UPU [20].

Previous works in this field can be categorized into three main groups [19,
21, 22, 23, 24, 25, 26]. However, each of these categories has its
limitations. For instance, the two-stage method requires utilizing information
from the positive example set in the initial stage to derive a reliable
negative example set from the unlabeled samples. In statistical learning based
on positive samples, users need to provide the prior probability of positive
examples, and overfitting issues often necessitate the application of
counterexample bias methods. In contrast, the proposed algorithm effectively
circumvents these aforementioned challenges.

To address the limitations of previous approaches, we propose and evaluate a
novel HD-PAN based on Hölder divergence, inspired by its appealing
mathematical properties. We investigate the influence of the objective
function, exploring different divergence formulas to enhance classification
results. Hölder divergence is found to play a crucial role in PU learning-
based classification. Our method is extensively evaluated on several medical
image datasets, demonstrating its effectiveness and robustness. We achieve
state-of-the-art performance on five disease classification benchmarks, namely
BreastMNIST, PneumoniaMNIST, BloodMNIST, OCTMNIST [27, 28], and AMD.

In summary, our contributions in this work can be outlined as follows:

  1. 1.

Proposal of the HD-PAN, leveraging the mathematical properties of Hölder
divergence.

  2. 2.

Investigation of the objective function, introducing improvements based on
different divergence formulas, highlighting the significance of Hölder
divergence in PU learning-based classification.

  3. 3.

Attainment of state-of-the-art performance on five disease classification
benchmarks, BreastMNIST, PneumoniaMNIST, BloodMNIST, OCTMNIST [27, 28], and
AMD.

In this work, we introduce several notations that are used throughout the
work. These notations are summarized in Table I.

TABLE I: Main Notations Used in This Work. Notation |  Definition
---|---
$p(z)$ |  noise distribution
$G(z)$ |  generated samples by $G(.)$
$x\sim{p_{data}}(x)$ |  real sample
$P^{p}$ |  generative distribution of labeled data in PU learning
$P^{u}$ |  generative distribution of unlabeled data in PU learning
$X^{p},X^{u}$ |  positive and unlabeled dataset, respectively
$X^{pu}$ |  given PU dataset
$P_{i}^{pu}$ |  given PU data $X^{pu}$ the true probability distribution of the positive and unlabeled two results of the $i$th sample

##  II Related Works

###  II-A Generative Adversarial Networks

GANs, initially proposed by Ian et al. in 2014 [29, 30], are adversarial
learners designed to train a generator that can produce data resembling real
samples. GANs consist of two main components: the discriminator $D$ and the
generator $G$. In essence, the generator $G$ aims to deceive the discriminator
$D$ by generating synthetic samples, while the discriminator $D$ strives to
differentiate between the generated samples $G(z)$ and the true data samples
from $p_{data}(x)$. The objective functions for training GANs are as follows:

$\left\\{{\begin{array}[]{*{20}{c}}{\mathop{\max
L(D)=}\limits_{D}\left(\begin{array}[]{l}{E_{x\sim{p_{data}}(x)}}\left[{\log
D(x)}\right]\\\
+{E_{z\sim{p_{z}}(z)}}\left[{\log(1-D(G(z)))}\right]\end{array}\right),}\\\
{\mathop{\min
L\left(G\right)}\limits_{G}={E_{z\sim{p_{z}}(z)}}\left[{\log(1-D(G(z)))}\right].\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil\end{array}}\end{array}}\right.$
(1)

GANs possess several advantages. Firstly, they require relatively low
computational resources compared to other generative models. Additionally,
GANs are capable of generating high-quality images. Moreover, GANs employ an
unsupervised learning approach, making them applicable in unsupervised and
semi-supervised learning domains. These advantages have led to the widespread
adoption of GANs in medical image analysis and computer vision. For instance,
Yu et al. [31] employed GANs to synthesize cross-modality MR images by
establishing global and local mappings. Upadhyay et al. [32] utilized
uncertainty-guided progressive GANs for medical image translation. Marouf et
al. [33] proposed a conditional single-cell CSCGAN to generate realistic
single-cell RNA-seq data. Cen and David [34] improved protein function
prediction by employing synthetic feature samples and GANs. Jiang et al. [35]
employed a conditional GAN to synthesize COVID-19 CT images. Yoon et al. [36]
introduced the ADS-GAN framework for data synthesis. Zhao et al. [37] utilized
a GAN for the prediction of Alzheimer’s disease progression. Hu et al. [38]
developed unsupervised learning methods based on GAN for cell-level visual
representation. Chen et al. [39] proposed JAS-GAN for the segmentation of
unbalanced data. Phattarapong et al. [40] developed EEGANet to remove ocular
artifacts from EEG signals. Zhou et al. [41] developed DR-GAN for fine-grained
lesion synthesis on diabetic retinopathy images. Luo et al. [42] employed a
GAN for dehazing cataractous retinal images. Yang et al. [43] developed a
semi-supervised approach based on GAN for bi-modality medical image synthesis.
For further exploration of GANs, refer to [44, 45, 46], which provide more
theoretical and applied research on GANs.

###  II-B Positive and Unlabeled Learning

Currently, PU learning has gained significant attention in the field of semi-
supervised learning, and researchers have made notable contributions in this
area. PU learning is a binary classification algorithm that assigns only two
values, positive and negative to the data. PU learning leverages a small
amount of labeled positive data in the training phase to infer the
characteristics of positive samples and then applies these learned
characteristics to classify unknown data and recover their labels.

For instance, Xu et al. [21] developed a one-step strategy to directly train a
multi-class classifier using given multi-class data. Gong et al. [22] employed
traditional decision margin-based methods to train classifiers, but the model
performance was subpar, particularly in image classification scenarios. Bekker
and Davis [23] explored random assumptions for PU learning, while Bao et al.
[24] investigated multi-component PU learning models. Sansone et al. [25]
focused on scalable PU learning models. Unfortunately, these models often
require prior probability estimation, which can be a challenge. Zhao et al.
[26] utilized entropy minimization principles and mixup regularization to
address the issue of trivial label distribution consistency. However, it has
been observed that incorrect estimation of the prior probability can lead to
poor results. Addressing these limitations, Hu et al. [19] proposed an
adversarial training network-based classification model, integrating the
adversarial principle into PU learning for the first time. Their approach does
not rely on prior knowledge and demonstrates excellent performance in terms of
classification accuracy, sensitivity, and robustness.

##  III Background

###  III-A Hölder Divergence

Kullback-Leibler (KL) divergence is a metric used to quantify the similarity
between two probability distributions. For a random variable $x$, the KL
divergence from distribution $P$ to distribution $Q$ is defined as:

${D_{KL}}(P||Q)=\int{p(x)\ln\left({\frac{{p(x)}}{{q(x)}}}\right)dx},$ (2)

In 2014, the concept of “Hölder divergence” was introduced, building upon the
definition of the Hölder score [47, 48, 49]. It utilizes the tightness of
inequalities to establish a dissimilarity measurement system. For an
inequality expressed as $lhs\leq rhs$, where $lhs$ and $rhs$ denote the left
and right sides of the inequality respectively, the tightness, denoted as
$\Delta=rhs-lhs$, is evaluated. When $lhs>0$, the following difference can be
defined:

$D=\log\frac{{rhs}}{{lhs}}=-\log\frac{{lhs}}{{rhs}}\geq 0,$ (3)

For a two-parameter inequality $lhs(p,q)\leq rhs(p,q)$ where $p\neq q$, if
$lhs(p,q)<rhs(p,q)$ is always true, the inequality is called proper. If $p=q$
and $lhs(p,q)=rhs(p,q)$ holds, then the inequality is considered tight.
Furthermore, a proper inequality can define a proper difference; that is,
$q=p$ and $D(p,q)=0$ represent a proper fix. Inequalities other than proper
inequalities are referred to as ill-posed inequalities. Therefore, pseudo-
divergence is established based on ill-posed inequalities. Hölder divergence
is defined using the Hölder inequality, which states that for two positive
real-valued functions $p(x)$ and $q(x)$ in the same probability space
$\Omega$, the following relationship holds:

$\int_{\Omega}{p(x)q(x)dx}\leq{\left({\int_{\Omega}{p{{(x)}^{\alpha}}dx}}\right)^{\frac{1}{\alpha}}}\left({\int_{\Omega}{q{{(x)}^{\beta}}dx}}\right)^{{}^{\frac{1}{\beta}}},$
(4)

where $\alpha$ and $\beta$ satisfy $\alpha\beta>0$ and
$\frac{1}{\alpha}+\frac{1}{\beta}=1$. They are referred to as a pair of Hölder
conjugate exponents. For the Hölder conjugate exponents $\alpha$ and $\beta$,
if the probability density functions $p(x)\in{L^{\alpha}}(\Omega,\mu)$ and
$q(x)\in{L^{\beta}}(\Omega,\mu)$, then the Hölder pseudo-divergence is defined
as:

$D_{\alpha}^{H}\left({p(x):q(x)}\right)=-\log\left({\frac{{\int_{\Omega}{p(x)q(x)dx}}}{{{{\left({\int_{\Omega}{p{{(x)}^{\alpha}}dx}}\right)}^{\frac{1}{\alpha}}}{{\left({\int_{\Omega}{q{{(x)}^{\beta}}dx}}\right)}^{\frac{1}{\beta}}}}}}\right),$
(5)

where $\mu$ and $L(\Omega,\mu)$ refer to the Lebesgue measure and Lebesgue
measure space, respectively.

Figure 1: The Framework of HD-PAN. HD-PAN presents a comparison between two
models, $D(\cdot)$ and $C(\cdot)$. The classifier $C(\cdot)$ aims to assess
the probability of the unlabeled data from set $U$ being positive data,
subsequently forwarding the data to the discriminator $D(\cdot)$ to determine
its authenticity as positive data.

##  IV Methodology

This section introduces a semi-supervised disease classification strategy
based on Hölder divergence, referred to as HD-PAN for brevity. PU learning is
employed as the underlying semi-supervised approach.

###  IV-A Motivation

The primary objective of this study is to classify diseases using medical
image databases. Therefore, it is necessary to learn the features present in
medical images and train a classifier accordingly. However, diseased features
in medical images are not limited to a single location, and images depicting
diseases often exhibit diseased characteristics in multiple regions. To
achieve the goal of accurately detecting lesion areas within images, the deep
learning algorithm must effectively identify these affected regions, making it
a multi-classification problem. In the domain of class distribution
recognition research, KL divergence demonstrates a significant increase,
reaching its maximum value at the boundary. Consequently, in multi-class
recognition, KL divergence tends to focus on a few categories with more
prominent properties, disregarding others. In contrast, Hölder divergence
exhibits distinct characteristics, presenting a gradually increasing curve and
the ability to recognize all groups, including those with evident traits. In
this experiment, Hölder divergence is employed as a substitute for KL
divergence.

Furthermore, Hölder divergences encompass both the Cauchy-Schwarz and the one-
parameter family of skew Bhattacharyya divergences, making them noteworthy. In
cases where the natural parameter space is a cone or an affine, Hölder
divergences allow for closed-form expressions between distributions belonging
to the same exponential families. This is particularly advantageous when
studying exponential families with conic or affine natural parameter spaces,
such as multinomials or multivariate normals. Both statistical Hölder distance
families enable the use of closed-form formulas and do not necessitate
distribution normalization, as they are inherently appropriate divergences
between probability densities [48, 49].

###  IV-B Problem Statement

It is well-known that GANs consist of a generator $G(\cdot)$ and a
discriminator $D(\cdot)$. However, in PU learning, the primary goal is to
train a predictor or classifier that accurately predicts data labels, denoted
as $C(\cdot)$. Instead of training a generator $G(\cdot)$, the classifier
$C(\cdot)$ aims to determine the probability of unlabeled data from set $U$
being positive data. Subsequently, the data is passed to the discriminator
$D(\cdot)$ to determine its authenticity as positive data. In our HD-PAN, we
retain the discriminator $D(\cdot)$ but replace the generator $G(\cdot)$ with
the classifier $C(\cdot)$ to achieve the classification objective.
Mathematically, this can be described by the following formulation:

$\begin{array}[]{l}\mathop{\min}\limits_{C}\mathop{\max}\limits_{D}V(D,C)\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\end{array}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil\end{array}=\left(\begin{array}[]{l}{E_{{x^{p}}\sim{P^{p}}({x^{p}})}}\left[{\log
D({x^{p}})}\right]\\\
+{E_{{x^{s}}={{\arg}_{{x^{u}}\sim{P^{u}}({x^{u}})}}C({x^{u}})=1}}\left[{\log(1-D({x^{s}}))}\right]\end{array}\right),\end{array}$
(6)

where $x_{s}$ represents data from the unlabeled data set $U$ that is
classified as positive by the classifier $C(\cdot)$. $P^{p}$ denotes the
generation distribution of labeled data in PU learning, and $P^{u}$ represents
the generative distribution of unlabeled data in PU learning. As the last term
in the equation is discrete, the model’s inventors trained the model using
policy gradients from reinforcement learning. The framework of HD-PAN is
illustrated in Figure 1.

###  IV-C Objective Function

The objective of PU learning is to train a classifier $C(\cdot)$ to classify
unlabeled data. Despite incorporating the idea of adversarial networks,
training a classifier $C(\cdot)$ remains essential. Therefore, we can consider
the classifier $C(\cdot)$ and the discriminator $D(\cdot)$ as two probability
distributions. The objective of the classifier $C(\cdot)$ is to fit the entire
probability distribution of the discriminator $D(\cdot)$, while the
discriminator $D(\cdot)$ aims to differentiate itself from the probability
distribution of the classifier $C(\cdot)$. The approach can be summarized as
follows:

  1. 1.

The classifier $C(\cdot)$ assigns high probabilities to data that is
challenging for the discriminator $D(\cdot)$ to distinguish and low
probabilities to data that the discriminator $D(\cdot)$ can easily
differentiate.

  2. 2.

The discriminator $D(\cdot)$ needs to discriminate between samples classified
as positive data by the classifier $C(\cdot)$ and the real positive data. The
probability here refers to the likelihood of the data being positive.

Additionally, the original PAN assumes that the output results of samples
after passing through $D(\cdot)$ and $C(\cdot)$ follow a 0-1 distribution.
Considering the analysis above and the mathematical properties of Hölder
divergence compared to KL divergence, it is more reasonable to use Hölder
divergence as a measure of the distance between two probability distributions.
This leads to the following objective function of HD-PAN:

$\begin{array}[]{l}\mathop{\min}\limits_{C}\mathop{\max}\limits_{D}V(D,C)\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\end{array}\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\end{array}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil\end{array}=-\sum\limits_{i=1}^{n}{D_{\alpha}^{H}(P_{i}^{pu}||D_{i}^{pu})}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil\end{array}+\lambda(\sum\limits_{i=1}^{{n_{0}}}{D_{\alpha}^{H}(D_{i}^{u}||C_{i}^{u})-}\sum\limits_{i=1}^{{n_{0}}}{D_{\alpha}^{H}(D_{i}^{u}||\hat{C}_{i}^{u})}),\end{array}$
(7)

where $P_{i}^{pu}$ represents the positive sample of the $i$-th sample in the
given PU data $X^{pu}$ (including the positive data set $X^{p}$ and the
unlabeled set $X^{u}$) and the true probability distribution of the two
unknown results. Here, $n$ is the total number of data in the sample set
$X^{pu}$ during training, and $n_{0}$ is the total number of data in the
unlabeled data set $X^{u}$ during training. $\hat{C}_{i}^{u}$ represents the
inverse distribution of $C_{i}^{u}$, and $\lambda$ is a hyperparameter for
balancing the distance. By replacing $D(x^{pu})$ and $C(x^{pu})$ with $D$ and
$C$, respectively, the objective function can be simplified as follows:

$\begin{array}[]{l}\mathop{\min}\limits_{C}\mathop{\max}\limits_{D}V(D,C)\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\end{array}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil\end{array}=\left(\begin{array}[]{l}{E_{{x^{p}}\sim{P^{p}}({x^{p}})}}[\log
D({x^{p}})]\\\
+{E_{{}_{{x^{u}}\sim{P^{u}}({x^{u}})}}}[\log(1-D({x^{u}}))]\end{array}\right)\\\
\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil\end{array}+\left(\begin{array}[]{l}\lambda{E_{{}_{{x^{u}}\sim{P^{u}}({x^{u}})}}}[(\log(1-C({x^{u}}))\\\
-\log(C({x^{u}})))(2D({x^{u}})-1)]\end{array}\right),\end{array}$ (8)

where $P^{p}$ represents the distribution of positive data. The algorithmic
flowchart is depicted in Algorithm 1.

Input: Dataset $\mathcal{D}$, positive training data $X^{p}$ and unlabeled
training data $X^{u}$; Randomly initialized HD-PAN; $t=0$,

$D(.)$, $C(.)$, $\leftarrow$ Randomly initialize network parameters

while  _number of training steps_ do

// Training $D(.)$ $k$ steps, we set $k=1$.

while  _k steps_ do

Sample a mini-batch of $m$ positive examples
$\left\\{{x_{1}^{p},\ldots,x_{m}^{p}}\right\\}$ from $X^{p}$

Sample a mini-batch of $m$ unlabeled examples
$\left\\{{x_{1}^{u},\ldots,x_{m}^{u}}\right\\}$ from $X^{u}$

Update $D(.)$ by ascending its stochastic gradient:
${\nabla_{{\Theta_{d}}}}\left(\begin{array}[]{l}-\sum\limits_{i=1}^{n}{D_{\alpha}^{H}(P_{i}^{pu}||D_{i}^{pu})}\\\
+\lambda(\sum\limits_{i=1}^{{n_{0}}}{D_{\alpha}^{H}(D_{i}^{u}||C_{i}^{u})}\\\
-\sum\limits_{i=1}^{{n_{0}}}{D_{\alpha}^{H}(D_{i}^{u}||\hat{C}_{i}^{u})})\end{array}\right)$

end while

Sample a mini-batch of $m$ unlabeled examples
$\left\\{{x_{1}^{u},\ldots,x_{m}^{u}}\right\\}$ from $X^{u}$

Update $C(.)$ by ascending its stochastic gradient:
${\nabla_{{\Theta_{c}}}}\left(\begin{array}[]{l}-\sum\limits_{i=1}^{n}{D_{\alpha}^{H}(P_{i}^{pu}||D_{i}^{pu})}\\\
+\lambda(\sum\limits_{i=1}^{{n_{0}}}{D_{\alpha}^{H}(D_{i}^{u}||C_{i}^{u})}\\\
-\sum\limits_{i=1}^{{n_{0}}}{D_{\alpha}^{H}(D_{i}^{u}||\hat{C}_{i}^{u})})\end{array}\right)$

end while

Algorithm 1 Training HD-PAN.

##  V Experiments and Analysis

###  V-A Datasets

To enhance the model’s logical coherence, result reliability, and algorithm
robustness for disease classification, this study employs the MedMNIST [27,
28], a widely recognized medical imaging dataset, for testing purposes, and
Age-related Macular degeneration (AMD) (https://amd.grand-challenge.org/).
MedMNIST, developed and made available by researchers from Shanghai Jiaotong
University, encompasses ten sub-datasets consisting of diverse medical imaging
modalities. In this work, the focus is on five specific sub-datasets:
BreastMNIST, PneumoniaMNIST, OCTMNIST, BloodMNIST, and AMD. A brief overview
of these datasets is provided below:

  * •

BreastMNIST [27, 28]: This dataset comprises two classes and consists of 546
training images, 78 validation images, and 156 test images, all labeled
accordingly. Each image has a size of $28\times 28$ pixels.

  * •

PneumoniaMNIST [27, 28]: With a binary classification task, the PneumoniaMNIST
dataset contains 4708 training images, 524 validation images, and 624 test
images, accompanied by respective labels. Each image in this dataset is of
size $28\times 28$ pixels.

  * •

OCTMNIST [27, 28]: This dataset encompasses five classes and includes 97,477
training images, 10,832 validation images, and 1000 test images, each with
corresponding labels. The images in OCTMNIST are of size $28\times 28$ pixels.

  * •

BloodMNIST [27, 28]: With eight classes, the BloodMNIST dataset consists of
11,959 training images, 1712 validation images, and 3421 test images, all
labeled accordingly. The images in this dataset are also sized $28\times 28$
pixels.

  * •

Age-related Macular degeneration (AMD)[50]: The Age-related Macular
Degeneration (AMD) dataset serves as the foundation for evaluating
sophisticated algorithms designed for the diagnosis of AMD, along with the
segmentation and classification of lesions in fundus photos obtained from AMD
patients. The primary aim is to assess and compare automated techniques for
detecting AMD, leveraging a standardized dataset of retinal fundus images.
This classification task focuses on a binary dataset. The training set
encompasses 240 images, each paired with corresponding labels, while the test
set includes 160 labeled images. The images are formatted in colored RGB, with
dimensions measuring $2124\times 2056$ pixels. AMD and Non-AMD examples are
shown in Figure 5.

By utilizing these diverse and representative sub-datasets from MedMNIST and
AMD, this research aims to achieve improved disease classification performance
while ensuring the validity and generalizability of the findings.

###  V-B Training Details

####  V-B1 Evaluation Metrics

To effectively evaluate the performance of our disease classification method,
we choose accuracy, recall, precision, and $F_{1}$-score as our evaluation
metrics [51]. These metrics provide comprehensive insights into the model’s
performance and are defined as follows:

  * •

Accuracy: Accuracy measures the overall correctness of the classification and
is computed as the ratio of the sum of true positives (TP) and true negatives
(TN) to the total number of samples: $Accuracy=\frac{{TP+TN}}{{TP+TN+FP+FN}}$.

  * •

Recall: Recall, also known as sensitivity or true positive rate, measures the
ability of the classifier to correctly identify positive instances. It is
calculated as the ratio of TP to the sum of TP and false negatives (FN):
${\rm{Recall}}=\frac{{TP}}{{TP+FN}}$.

  * •

Precision: Precision quantifies the accuracy of positive predictions made by
the classifier. It is calculated as the ratio of TP to the sum of TP and false
positives (FP): ${\rm{Precision}}=\frac{{TP}}{{TP+FP}}$.

  * •

$F_{1}$-score: The $F_{1}$-score is the harmonic mean of precision and recall,
providing a balanced measure that combines both metrics. It is calculated as:
${F_{1}}\text{-score}=2\times\frac{{\text{{Precision}}\times\text{{Recall}}}}{{\text{{Precision}}+\text{{Recall}}}}$.

By considering these evaluation metrics, including accuracy, recall,
precision, and $F_{1}$-score, we can comprehensively assess the performance of
our disease classification method. These metrics capture different aspects of
the model’s performance, such as overall accuracy, the ability to detect
positive instances, and the balance between precision and recall.

####  V-B2 The Experiment’s Purpose

Our experimental study aims to achieve the following three objectives:

  * •

Improving the objective function of PAN: In this work, we propose an enhanced
objective function for PAN, known as HD-PAN. By refining the objective
function and adjusting the Hölder parameters, our goal is to investigate
whether HD-PAN can outperform the original PAN in terms of disease
classification accuracy and recall. We aim to demonstrate the effectiveness of
the improved objective function in enhancing the performance of disease
classification.

  * •

Comparing HD-PAN with state-of-the-art models: We compare HD-PAN with other
state-of-the-art models. To ensure a fair comparison, we utilize the same
batch of datasets, network structures, and training parameters for all models.
By training these models on the identical datasets, we can evaluate their
performance based on accuracy and $F_{1}$-score. This comparative analysis
will help determine which model achieves better classification results and
more accurate predictions.

  * •

Evaluating HD-PAN for disease classification: We aim to explore the accuracy
and effectiveness of HD-PAN in disease classification tasks. By conducting
experiments on five medical imaging datasets, we utilize accuracy and
$F_{1}$-score as evaluation metrics to assess the performance of HD-PAN. These
metrics will enable us to measure the accuracy of disease classification and
gauge the overall effectiveness of HD-PAN in comparison to other models.

By addressing these three goals, our experiment aims to contribute to the
advancement of disease classification algorithms, enhance the understanding of
HD-PAN’s capabilities, and provide insights into its potential applications in
medical image processing.

###  V-C Preparation for Experiment

####  V-C1 Data Preprocessing

To conduct PU learning, it is essential to have a dataset comprising both
labeled and unlabeled samples. However, the number of labeled samples is often
limited compared to the total dataset size. Furthermore, the algorithm’s
ultimate objective is to achieve binary classification results, distinguishing
between positive and negative classes. Consequently, preprocessing of the five
medical imaging datasets is carried out prior to the experiment.

For BreastMNIST and PneumoniaMNIST, the datasets already consist of two
distinct types and therefore require no additional processing. For the AMD
dataset, we randomly select 50 images to form the positive dataset.

OCTMNIST, on the other hand, is a four-category dataset with category labels
ranging from 1 to 4. In order to convert it into a binary classification task,
the data is divided into positive and negative samples based on the parity of
the category labels.

BloodMNIST, an eight-category dataset with category labels from 1 to 8, also
required preprocessing. The data is divided into positive and negative samples
based on the parity of the category labels, designating images with even-
numbered categories as positive samples and those with odd-numbered categories
as negative samples.

By performing this preprocessing step, we ensure that all five medical imaging
datasets are suitably prepared for the PU learning experiment. The resulting
datasets allowed for effective training and evaluation of the proposed
method’s performance in disease classification tasks.

####  V-C2 The Establishment of the PU Learning Data Set

To construct the training dataset for PU learning, we obtain positive and
negative training samples for each dataset. For datasets with a limited number
of samples, namely BreastMNIST and PneumoniaMNIST, we randomly select 100
images as the positive data set. This selection ensure a representative subset
of positive samples for training.

In contrast, for datasets with a larger volume of data, namely OCTMNIST and
BloodMNIST, we aim to maintain a suitable balance between positive and
negative samples. We randomly select 2000 images as the positive data set for
OCTMNIST and 500 images as the positive data set for BloodMNIST. These
selections are made to ensure an appropriate distribution of positive samples
within the overall training dataset.

By constructing the PU learning training dataset in this manner, we ensure a
combination of labeled and unlabeled data from each dataset. This dataset
composition facilitates effective training of the proposed method and allows
for accurate disease classification in subsequent experiments.

Figure 2: Visualization of Quantitative Evaluation: The figure presents a
comprehensive quantitative evaluation based on inter-class and intra-class
experimental results conducted on two medical image datasets, namely
BreastMNIST and PneumoniaMNIST. Subfigures (a) and (b) showcase the inter-
class experimental results of BreastMNIST, while (c) and (d) illustrate the
intra-class experimental outcomes for the same dataset. Additionally,
subfigures (e) and (f) depict the inter-class experimental results for
PneumoniaMNIST, and (g) and (h) present the intra-class experimental results
for PneumoniaMNIST. Figure 3: Visualization of Quantitative Evaluation: The
figure presents a detailed quantitative evaluation through inter-class and
intra-class experimental results conducted on two medical image datasets,
OCTMNIST and BloodMNIST. Subfigures (a) and (b) illustrate the inter-class
experimental outcomes for OCTMNIST, while (c) and (d) showcase the intra-class
experimental results for the same dataset. Furthermore, subfigures (e) and (f)
depict the inter-class experimental results for BloodMNIST, and (g) and (h)
showcase the intra-class experimental outcomes for BloodMNIST, respectively.
Figure 4: Quantitative Evaluation Visualization: The figure illustrates the
intra-class and inter-class experimental outcomes for the AMD dataset.
Subfigures (a) and (b) depict the inter-class experimental results, while
subfigures (c) and (d) showcase the intra-class experimental outcomes. Figure
5: AMD and Non-AMD Examples: Panels (a)-(c) showcase examples of AMD, while
panels (d)-(f) depict examples of Non-AMD.

####  V-C3 Compare Experimental Models

The comparative experimental models employ in this study are NPU [20], UUPU
[18], PAN [19], and Dist-PU [26]. Each model brings unique contributions to
the field of PU learning, as outlined below:

  * •

UPU (Du Plessis, 2015) [20]: UPU introduces a convex formulation with double
hinge loss for PU learning, effectively addressing bias issues. The authors
not only prove the convergence rate of the estimators but also demonstrate the
computational efficiency of their non-convex method in experimental settings.

  * •

NNPU (Ryuichi Kiryo, 2017) [18]: NNPU proposes a non-negative risk estimator
for PU learning, offering robustness against overfitting. This model allows
for greater flexibility in utilizing limited positive data, making it
advantageous in scenarios where positive data is scarce.

  * •

PAN (Wenpeng Hu, 2019) [19]: PAN leverages adversarial training to accomplish
PU learning classification tasks. By avoiding the need for class prior
probabilities, PAN exhibits superior performance compared to UPU and NNPU,
particularly in real-world operations where class prior probabilities are
often unavailable.

  * •

Dist-PU (Yunrui Zhao, 2022) [26]: Dist-PU proposes a model that emphasizes the
consistency of label distributions between predicted and true labels. This
approach enhances the accuracy and reliability of PU learning by incorporating
the principle of label distribution consistency.

It is important to note that both UPU and NNPU rely on class prior
probabilities as inputs, which may not be readily available in real-world
scenarios. In our experiments, we provide the correct class prior
probabilities in advance. Despite this favorable condition, UPU and NNPU still
exhibit inferior performance compare to PAN, which does not rely on class
prior probability inputs. For consistency, we adopt the same network
structure, parameters, and optimization methods as the original models when
using UPU and NNPU.

Figure 6: The figure presents a comparison of attention maps between PAN
(second row) and HD-PAN (third row), with the first row depicting the original
images.

####  V-C4 Network Structure and Hyperparameter

To ensure a fair comparison of experimental results, we adopt the same network
architecture as the comparison models to construct our classifier.
Specifically, for the BreastMNIST, PneumoniaMNIST, and OCTMNIST datasets,
which consist of grayscale images with a size of $28\times 28$ pixels, we
utilize a three-layer Multilayer Perceptron (MLP). As for the BloodMNIST
dataset, which comprises RGB-format images with the same size, we employ a
Convolutional Neural Network (CNN) for validation. The CNN architecture is
defined as follows:
$\left({{\rm{28}}\times{\rm{28}}\times{\rm{}}3}\right)-\left[{C\left({3{\rm{}}\times{\rm{}}3,{\rm{84}}}\right)}\right]{\rm{}}-{\rm{}}C\left({3{\rm{}}\times{\rm{}}3,{\rm{84}},{\rm{}}2}\right)$
$-{\rm{}}\left[{C\left({1{\rm{}}\times{\rm{}}1,{\rm{168}}}\right)}\right]$
$-{\rm{}}C\left({1{\rm{}}\times{\rm{}}1,{\rm{8}}}\right)$
$-{\rm{}}1000{\rm{}}-{\rm{}}1000{\rm{}}-{\rm{}}1$. In the architecture,
$C\left({3{\rm{}}\times{\rm{}}3,{\rm{84}}}\right)$ represents a convolutional
layer with a $3\times 3$ kernel and 84 channels.

Regarding the hyperparameters in PU learning, three key factors include the
learning rate, the number of labeled data points, and the batch size. However,
in our HD-PAN approach, we construct a novel objective function based on
Hölder divergence. This function requires the computation of Hölder exponents
$\alpha$ and $\beta$. Since $\alpha$ and $\beta$ are Hölder conjugate
exponents, we assign a value to $\alpha$, and $\beta$ is computed accordingly.
Moreover, the batch size is set to 64, and we experiment with learning rates
of 0.4, 0.5, 0.6, 0.7, and 0.8, respectively. Additionally, we discuss the
Hölder exponent $\alpha$ as different divergences, such as Cauchy-Schwarz
divergence [49], Bregman divergence [49], and $\gamma$-divergence [49], are
formed by different values of $\alpha$ in Hölder divergence. Therefore, it is
necessary to investigate the value of $\alpha$, considering values of 1.5,
1.6, 1.7, 1.8, 1.9, and 2.0, respectively.

####  V-C5 Experimental Steps

During the experiment, we sequentially execute UPU, NNPU, Dist-PU, and PAN on
five medical image datasets. Throughout the training process, we modify the
parameters of these models to examine the impact of different learning rates
on the training outcomes. Additionally, we investigate the influence of the
Hölder parameter $\alpha$ on the training results by varying its value during
the training process. Subsequently, for each learning rate, we evaluate the
optimal training impact.

####  V-C6 Experimental Results and Analysis

For binary classification tasks, accuracy (Acc) and the $F_{1}$-score are
important metrics. In the context of PU learning, the primary objective is to
identify all positive data points through the algorithm for further analysis.
If the model overlooks some positive samples while detecting others, its
performance will be significantly compromised, failing to meet the
requirements of robustness and interpretability. In the realm of medical
image-aided diagnosis, misclassifying several diseased samples as disease-free
patients can lead to numerous misdiagnoses and missed treatment opportunities,
which is undesirable. In PU learning, the emphasis is placed on the
identification and classification of positive samples, and the $F_{1}$-score,
as the harmonic mean of precision and recall, provides a comprehensive measure
for positive samples. Consequently, when evaluating PU learning algorithms,
the $F_{1}$-score is more suitable as the primary metric.

TABLE II: Quantitative evaluation based on intra-class experimental results (accuracy and $F_{1}$-score) for breastMNIST, pneumoniaMNIST, OCTMNIST, bloodMNIST, and AMD datasets. Dataset |  Models |  Accuracy |  $F_{1}$-Score
---|---|---|---
BreastMNIST |  UPU [20] |  0.7885 |  0.8649
|  PAN [19] |  0.7500 |  0.8367
|  NNPU [18] |  0.7821 |  0.8448
|  Dist-PU[26] |  0.7737 |  0.9879
|  HD-PAN ($\alpha=1.8$) |  0.8141 |  0.8755
PneumoniaMNIST |  UPU [20] |  0.8542 |  0.8853
|  PAN [19] |  0.8221 |  0.8703
|  NNPU [18] |  0.8061 |  0.8530
|  Dist-PU [26] |  0.6378 |  0.9826
|  HD-PAN ($\alpha=1.9$) |  0.8782 |  0.9003
OCTMNIST |  UPU [20] |  0.6870 |  0.6756
|  PAN [19] |  0.6370 |  0.7151
|  NNPU [18] |  0.6860 |  0.6990
|  HD-PAN ($\alpha=1.5$) |  0.6910 |  0.7295
BloodMNIST |  UPU [20] |  0.8340 |  0.8010
|  PAN [19] |  0.8100 |  0.7587
|  NNPU [18] |  0.8360 |  0.8173
|  HD-PAN ($\alpha=2.0$) |  0.8385 |  0.8196
AMD |  UPU [20] |  0.7391 |  0.5510
|  PAN [19] |  0.6708 |  0.5378
|  NNPU [18] |  0.6584 |  0.5210
|  HD-PAN ($\alpha=1.8$) |  0.7143 |  0.5717
TABLE III: Quantitative Evaluation of inter-class Experimental Results (Accuracy and $F_{1}$-Score) on BreastMNIST, PneumoniaMNIST, OCTMNIST, BloodMNIST, and AMD Datasets. Dataset |  Models |  Accuracy |  $F_{1}$-Score
---|---|---|---
BreastMNIST |  HD-PAN ($\alpha=1.5,lr=0.6$) |  0.7692 |  0.8429
|  HD-PAN ($\alpha=1.6,lr=0.6$) |  0.7949 |  0.8547
|  HD-PAN ($\alpha=1.7,lr=0.8$) |  0.8141 |  0.8724
|  HD-PAN ($\alpha=1.8,lr=0.7$) |  0.8141 |  0.8755
|  HD-PAN ($\alpha=1.9,lr=0.4$) |  0.7885 |  0.8445
|  HD-PAN ($\alpha=2.0,lr=0.4$) |  0.7885 |  0.8560
PneumoniaMNIST |  HD-PAN ($\alpha=1.5,lr=0.6$) |  0.8638 |  0.8922
|  HD-PAN ($\alpha=1.6,lr=0.5$) |  0.8638 |  0.8927
|  HD-PAN ($\alpha=1.7,lr=0.7$) |  0.8670 |  0.8920
|  HD-PAN ($\alpha=1.8,lr=0.7$) |  0.8510 |  0.8774
|  HD-PAN ($\alpha=1.9,lr=0.8$) |  0.8782 |  0.9003
|  HD-PAN ($\alpha=2.0,lr=0.5$) |  0.8574 |  0.8894
OCTMNIST |  HD-PAN ($\alpha=1.5,lr=0.4$) |  0.6910 |  0.7295
|  HD-PAN ($\alpha=1.6,lr=0.6$) |  0.6550 |  0.6588
|  HD-PAN ($\alpha=1.7,lr=0.7$) |  0.6610 |  0.6903
|  HD-PAN ($\alpha=1.8,lr=0.6$) |  0.6630 |  0.6847
|  HD-PAN ($\alpha=1.9,lr=0.4$) |  0.6620 |  0.6920
|  HD-PAN ($\alpha=2.0,lr=0.4$) |  0.6680 |  0.7049
BloodMNIST |  HD-PAN ($\alpha=1.5,lr=0.7$) |  0.8194 |  0.7962
|  HD-PAN ($\alpha=1.6,lr=0.7$) |  0.8264 |  0.7913
|  HD-PAN ($\alpha=1.7,lr=0.8$) |  0.8278 |  0.7861
|  HD-PAN ($\alpha=1.8,lr=0.8$) |  0.8185 |  0.7761
|  HD-PAN ($\alpha=1.9,lr=0.7$) |  0.8173 |  0.7870
|  HD-PAN ($\alpha=2.0,lr=0.7$) |  0.8385 |  0.8196
AMD |  HD-PAN ($\alpha=1.8,lr=0.8$) |  0.7143 |  0.5714
|  HD-PAN ($\alpha=1.3,lr=0.6$) |  0.7143 |  0.5636
|  HD-PAN ($\alpha=1.5,lr=0.4$) |  0.7019 |  0.5455
|  HD-PAN ($\alpha=1.4,lr=0.6$) |  0.6894 |  0.5517
|  HD-PAN ($\alpha=1.7,lr=0.6$) |  0.6894 |  0.5517
|  HD-PAN ($\alpha=1.1,lr=0.8$) |  0.7143 |  0.5714

####  V-C7 Performance of HD-PAN on Different Hölder Parameters $\alpha$

Using the BreastMNIST dataset as an example, we conducted experiments to
investigate the effects of different learning rates and Hölder parameters
($\alpha$) on the performance of HD-PAN. The aim is to identify the best
combination of parameters based on accuracy and the $F_{1}$-score. For
instance, when $\alpha$ is set to 1.8 and the learning rate to 0.7, HD-PAN
achieves the best training results on the BreastMNIST dataset, with an
accuracy of 0.8141 and an $F_{1}$-score of 0.8755.

It is important to note that the termination condition for training was set at
15 epochs, with the criterion that the $F_{1}$-score should not increase by
less than 0.01.

Table III and Figures 2–4 illustrate that different values of $\alpha$ can
yield varying optimal training results for the same dataset. Taking the
BloodMNIST dataset as an example, when $\alpha$ is set to 2.0 and the learning
rate to 0.7, the accuracy rate reached 0.8384, while the $F_{1}$-score reach
0.8196. However, changing the value of $\alpha$ lead to unsuccessful learning
outcomes, regardless of variations in the learning rate. Therefore, adjusting
the Hölder parameter $\alpha$ can improve the classification performance of
the model for different types of medical imaging datasets.

Following the experiments, we obtain results for the HD-PAN model under
various Hölder parameters ($\alpha$) and diverse learning rates (lr). Through
a comprehensive comparison of Accuracy and $F_{1}$-score, we identify the
optimal training outcomes and their corresponding learning rates for different
Hölder parameters. The training set comprises 240 training samples and 160
validation samples from the AMD, while the test set includes 156 test samples
from the same dataset.

To illustrate, let’s consider the row corresponding to HD-PAN
($\alpha=1.1,lr=0.8$). In this case, the HD-PAN achieves its peak training
performance with a Hölder exponent of $\alpha=1.1$ and a learning rate of 0.8.
Under this configuration, the Accuracy reaches 0.7143, and the $F_{1}$-score
is recorded at 0.5714.

####  V-C8 Comparison of HD-PAN and Other PU Learning Models

This study focuses on the exploration of medical imaging datasets and proposes
a novel PU learning algorithm by building upon previous research. The
experiment involves the selection of five distinct types of medical imaging
datasets, on which different disease classification models are trained. The
experimental findings, present in Table II and Figures 2–4, demonstrate that
HD-PAN can achieve optimal classification results for these diverse medical
imaging datasets by adjusting its settings. Specifically, HD-PAN surpasses
other models in terms of classification accuracy and recall.

In addition to identifying the most suitable model for PU learning, this work
also investigates the impact of objective functions based on divergence. By
analyzing the experimental results, the performance of HD-PAN is evaluated and
compared with other models, providing insights into the effectiveness of
divergence-based objective functions in disease classification tasks.

###  V-D Ablation Study

Attention map play a crucial role in machine learning by providing insights
into the neural network’s focus areas that contribute to the final
categorization decision. In our ablation study, we generate attention maps
using two different models and analyze the saliency changes in HD-PAN apply to
the BloodMNIST dataset [27, 28]. This allows us to compare the accuracy and
sensitivity of each model in capturing the information distribution within the
images. It is important to note that unless specified otherwise, our training
setups remain identical except for the ablated alterations. Figure 6
illustrates the results obtained from the ablation procedures.

Through our analyses, we find that incorporating HD divergence as a measure
significantly enhances the performance of PU learning image classification
tasks. The results show minimal improvement when solely utilizing KL
divergence within PAN, indicating that the key component of our approach lies
in the integration of HD divergence. Additionally, by examining the attention
map comparisons presented above, we observe that HD-PAN can more accurately
identify the regions of the image that contain crucial information, while
PAN’s output demonstrates lower sensitivity in this regard.

Overall, our findings underscore the efficacy of HD-PAN and the importance of
HD divergence in achieving improved performance in PU learning image
classification tasks. Furthermore, the attention map comparisons highlight the
superior precision of HD-PAN in pinpointing areas with significant
information, distinguishing it from the relatively less sensitive output of
PAN.

###  V-E Discussion

This research focuses on essential semi-supervised classification learning
technologies based on positive and unlabeled samples. An adversarial training
classification model, HD-PAN, is established using a generative model, and a
novel objective function is proposed. The training results on a medical
imaging dataset demonstrate that HD-PAN outperforms other common PU learning
models in terms of classification accuracy and $F_{1}$-score. However, PU
learning remains an evolving field facing new challenges. Although this work
has produced valuable research outcomes, several challenges and directions for
further exploration need to be addressed. Based on existing work and ongoing
research in this field, several possible study directions are outlined below:

  * •

Increase in the number of positive samples: In PU learning, positive samples
are typically significantly fewer in number compare to unlabeled samples.
However, learning the characteristics of positive samples is crucial for
accurately identifying positive and unlabeled data, thus directly impacting
classification accuracy. One approach to address this issue is to leverage
diffusion models to expand the number of samples based on a limited amount of
data. Therefore, combining diffusion models with PU learning can be pursued as
a novel research direction.

  * •

Exploration of new objective functions: Although this work improves the
objective function based on KL divergence, it is important to note that this
is just one way to measure the gap between two probability distributions.
There is still room for further exploration and enhancement in this regard.
Moreover, different types of datasets may require different objective
functions, potentially influencing the final classification performance.
Therefore, it is worth investigating which objective function is more robust
for most datasets. Additionally, apart from using divergence formulas,
exploring alternative approaches to explain the difference between probability
distributions can be a promising new direction for research.

  * •

Investigation of restricted sample size and class imbalance issues in
datasets: In this research, the PU learning classification problem is
investigated on several datasets, performing a simple 0-1 classification.
However, it is important to acknowledge that different datasets have diverse
characteristics, and some datasets, including those analyzed in this research,
cannot be classified solely based on image label parity. Hence, it is crucial
to thoroughly investigate the specific challenges posed by each dataset and
develop a unified PU learning framework that can capture commonalities and
characteristics across different types of datasets. By enhancing the model’s
robustness to various data types, this approach can effectively reduce the
costs associated with handling new classification tasks.

In summary, while this research has made significant advancements in PU
learning using positive and unlabeled samples, there remain important avenues
for further exploration. The suggested study directions include increasing the
number of positive samples through diffusion models, exploring alternative
objective functions, and addressing restricted sample size and class imbalance
issues in datasets. By delving into these areas, researchers can advance the
field of medical image processing and contribute to the development of more
effective and robust PU learning models.

##  VI Conclusion

PU learning is a highly prominent research area in machine learning, computer
vision, and natural language processing. While advancements in deep learning
technology have allowed for the successful resolution of some basic PU
learning problems, the expansion of its applicability has given rise to new
challenges across various domains. These challenges range from natural
language dataset classification to fake review detection, with medical imaging
datasets posing increasingly complex problems in disease recognition and
classification.

Although researchers have proposed viable alternatives, these methods are not
without limitations. To address these limitations, this work proposes an
adversarial learning approach based on generative adversarial neural networks.
Additionally, two different divergence formulas are introduced, and specific
formulations are derived to address diverse scenarios using mathematical
principles. This research culminates in the development of a novel PU learning
model, HD-PAN, which is thoroughly evaluated and validated using multiple
medical imaging datasets.

By leveraging the power of adversarial learning and incorporating specific
divergence formulas, this work contributes to advancing PU learning in the
field of medical image processing. The proposed HD-PAN demonstrates its
efficacy in tackling the intricate challenges associated with disease
recognition and classification within medical imaging datasets.

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