Abstract
To detect breast cancer in mammography screening practice, we modify the inertial relaxed CQ algorithm with Mann’s iteration for solving split feasibility problems in real Hilbert spaces to apply in an extreme learning machine as an optimizer. Weak convergence of the proposed algorithm is proved under certain mild conditions. Moreover, we present the advantage of our algorithm by comparing it with existing machine learning methods. The highest performance value of 85.03% accuracy, 82.56% precision, 87.65% recall, and 85.03% F1-score show that our algorithm performs better than the other machine learning models.
1. Introduction and Preliminaries
Breast cancer is common in women, with approximately 2 million diagnosed women worldwide each year [1]. Its mortality rate has increased over decades due to the change in risk factors, aging society, and better cancer registration and detection [2]. Many trials conclude that mammography screening at age 40s has mortality reduction, many years of life saved, and improved treatment, including evaluation of the extent of the disease [3]. However, there are some risks of this modality: overdiagnosis, false-positives, anxiety, and radiation injury [4]. Recently, imaging options combined with artificial intelligence are believed to be enhanced by integrating new screening protocols directed toward more personalized and precision medicine [1].
Many optimization algorithms were used to solve medical classification in machine learning; see [5, 6]. In this paper, we focus on the split feasibility problem (SFP) applying to mammography classification. Let and be two nonempty closed and convex subsets of real Hilbert space such that and , and let be a bounded linear operator. The problem SFP is to if such exist. The solution set of the problem SFP (1) is denoted by .
The first algorithm to solve the problem SFP (1) was presented by Censor and Elfving [7]. After that, many mathematicians (see [8–10]) applied the problem SFP (1) to solve many real-world problems such as machine learning, signal processing, image restoration, and many more. To find a solution of the problem SFP (1), Xu [11] proved that the point is a solution of the problem SFP (1) if and only if the point is a fixed point of the following mapping:
Later on, Byrne [12] was the first to propose a popular algorithm for solving the problem SFP (1). The algorithm of Byrne [12] was generated as follows:
where the parameter belongs in the interval which makes the mapping is nonexpansive where is an identity, i.e.,
and denotes for the adjoint operator of , while and are the orthogonal projections onto and , respectively. The overall cost of calculation is not high if the metric projections onto and are simple to calculate. However, precisely computing the metric projection is difficult or requires too much effort in some circumstances when and are complex constructs.
Later on, Yang [13] introduced a relaxed algorithm by modifying the and sets of the algorithm (3) to reveal sets as follows:
where and are two proper convex functions such that and . Since the projections and are easier to use, many mathematicians use them to modify numerous algorithms for solving the problem SFP (1); see [14–16].
One of the techniques to speed up the convergence of the algorithms is the inertial technique which Polyak first introduced [17] in 1964. Polyak’s algorithm was called the heavy ball method, and it was improved by Nesterov [18]. Later on, it has been widely used to solve a wide variety of problems in the optimization field, as seen in [9, 19–22].
In 2017, Dang et al. [9] modified the inertial technique which was introduced by Alvarez and Attouch [23] with the algorithm of the Byrne algorithm (3) for the problem SFP (1) in a real Hilbert space. This algorithm was defined as follows:
where the parameter is in the interval involving the norm of operator , and are the revel set introduced by Yang [13], and the extrapolation factor and such that
The weak convergence of algorithm generated by (6) was proved under the conditions of the extrapolation factor (8) and the stepsize parameter .
Very recently, Wang and Yu [32] generalized an inertial relaxed of Yang [13] by modifying as follows:
and of Wang and Yu [24] can be reduced to and of Yang [13] when and are set to 0. The inertial generalized relaxed algorithm (IGRCQ) of Wang and Yu [24] was introduced by , , and
where , and
For each , the functions are defined as follows:
It was shown that, under the conditions and , the sequence created by (10) weakly converges to a solution of the problem SFP (1).
In this paper, we modify the inertial technique with relaxed algorithms and Mann’s algorithm to solve the split feasibility problems in Hilbert spaces. We establish weak convergence theorems under suitable conditions. We apply our main result to solve a data classification problem by using extreme learning machine with mammographic mass data set from UCI, and then compare the performance of our algorithm with other existing machine learning algorithms.
2. Main Results
In this section, we introduce a new modified inertial projective algorithm by combining Mann algorithms with inertial technique and inertial relaxed algorithm. Assume that and are two nonempty closed and convex subsets of real Hilbert space such that and such that
where is a bounded linear operator, and are lower semicontinuous convex functions. We next assume that and are bounded operators. For a sequence in , we modify the half-spaces and by using the idea of Wang and Yu [24] as follows:
where , , and
where and . We see that and for each . Define and as in (13).
We now introduce a modified inertial projective algorithms as follows:
| ||||||||||||||||||||||||
Assume that the following condition hold:
(C1)
(C2) ;
(C3) .
Theorem 1. Let and be two real Hilbert spaces, and let and be nonempty closed convex subsets such that and . Let be a bounded linear operator. Assume that the solution set of the problem SFP (1) is nonempty, the condition (C1)-(C2) hold. Then the sequence generated by Algorithm 1:. converges weakly to a point
Proof. Let . Since is firmly nonexpansive, then for each , we have In the other hand, we set , From (18), we have Replacing (20) into (19), we have This implies that Since and are nonexpansive, we have
By Lemma 1 in [25] and (C1), we have that is convergence sequence for any . Therefore, is bounded. From the definition of , is also bounded. It follows from (20), (C1)-(C3) that
Again by and are nonexpansive, then we have which implies that
From exists, (C1) and (C3), we obtain
It follows from (C2) and (24) that
And it is clearly from (C1), we have
Finally, let be a weak sequential cluster point of . There exists a subsequence of which converges weakly to . From (30), we also have that converges weakly to and hence as . By the definition of , we have that . This implies that
where . By our assumption as is bounded (24) and (31), we have , this shows that . Again, by the definition of , we have that . This implies that
where . By our assumption as is bounded (27) and (32), we have , this shows that . By Opial’s lemma in [26], we can conclude that converges weakly to a solution in . This completes the proof.
3. Application to Data Classification Problem
Nowadays, many cancer patients are reported around the world each year. In the population survey in Global Cancer Statistics 2020, information was found that breast cancer was the most severe disease with 258 new cases per hour [27]. Breast cancer is more common in developed countries than in developing countries, and the number of cases varies with per capita income (GDP per capita). In addition, people’s way of life in the city (urbanization) and those environments result in more risk behaviors for breast cancer. Importantly, it also found that the number of patients tends to increase significantly each year as well. In Thailand, breast cancer is the 1st most common cancer among females and the 3rd most common among both males and females. There are 8,266 deaths from breast cancer per year or about 1 person per hour. The above data shows that breast cancer is rapidly increasing and directly threatening the female population globally, including Thailand. In addition, the number of doctors specializing in such diseases is limited, not enough to provide services to the patients. Therefore, the use of technology-based knowledge related to artificial intelligence or machine learning including deep learning is the basis for creating tools or innovations that are efficient and accurate in assisting medical personnel in screening and diagnosing breast cancer. As a result, patients will be screened quickly and accurately, cured in the early stages and could reduce mortality. It is also an indicator of the modernization of the country’s development in the field of public health in the future. Mammography plays a central part in the early detection of breast cancers because it can show changes in the breast years before a patient or physician can feel them. Research has shown that annual mammograms lead to early detection of breast cancers when they are most curable, and breast-conservation therapies are available. Women, beginning at age 40, should screen for mammography every year. In this research, we use the mammographic mass dataset from UCI is available on the UCI website. (https://archive.ics.uci.edu/ml/datasets/Mammographic+Mass?fbclid=IwAR1TL44iSKmqXX6PMiSVjqGVZRD-suQTPEVsejq01SUylZwildNu7UWEPZQ). This dataset contains a BI-RADS assessment, the patient’s age, and three BI-RADS attributes: shape of mass, margin of mass, and density of mass together with the ground truth (the severity field) for 516 benign and 445 malignant masses that have been identified on full-field digital mammogram collected at the Institute of Radiology of the University Erlangen-Nuremberg between 2003 and 2006. These datasets can indicate how well a several computer-aided diagnosis (CAD) system performs compared to the radiologists [28]. After 167 missing attribute values from 7 BI-RADS, 5 ages, 31 shapes, 48 margins, and 76 densities were removed before the training process. The following Table 1 shows the overview of all attributes.
In 2021, Parvez et al. [29] showed many machine learning predictive models to classify breast cancer using this mammographic mass dataset. The following Table 2 shows the comparison of our algorithm 1 consider in two constrain closed convex sets and with machine learning predictive models by Parvez et al. [29] after feature engineering.
From Table 2, the results show that our algorithm 1 when constrain closed convex set by was used gives the highest accuracy 85.03% after removing rows with missing values and outliers. We next explain how our algorithm 1 optimizes weight parameter in training data for machine learning. We focus on extreme learning machine (ELM) by using 5-fold cross-validation [30]. The ELM method is defined as follows: assume that is a set of training data with distinct samples such that is an input training data and is a target. Finding optimal output weight using the output function is the objective of the ELM method. The following output function is for single-hidden layer feed-forward neural networks (SLFNs) with hidden nodes:
where is an activation function and and are parameters of weight and finally the bias, respectively. The optimal output weight at the -th hidden node is found by setting the hidden layer output matrix as follows:
We also assume an optimal output weight such that , where is the training target data. For solving linear system , we use the least square problem when the Moore-Penrose generalized inverse of is not easy to find. To ovoid overfitting in the machine learning, we consider constrain least square problem in two different closed convex subsets of as follows:
where , such that is regularization parameters. Setting , , , , and for our algorithm 1 to solve the problem (3.1).
We use four evaluation metrics: accuracy, precision, recall, and F1-score [31] as explained below for comparing the performance of the classification algorithms.
where is the true negative, is the false positive, is the false negative, and is the true positive.
For avoiding model overfitting, we consider accuracy and loss plots. This research, we use the following binary cross-entropy loss function:
where is the -th scalar value in the model output, is the corresponding target value, and is the number of scalar values in the model output.
For comparison with other existing methods from the literature, the necessary parameters of each algorithm are chosen in Table 3. The extrapolation parameter of algorithm 1 () and algorithm (10) is in the following from:
where is a number of iterations that we want to stop and for algorithm (6).
Sigmoid is set as an activation function with hidden nodes , and four evaluation metrics of each algorithm are shown in Table 4.
Table 4 shows that our algorithm 1 with constrain closed convex set is the highest F1-score, precision, recall, and accuracy efficiency. Additionally, our algorithm 1 with constrain closed convex set has the lowest number of iterations. The optimal-fitting of our algorithm 1 is shown by considering the training and validation loss with the accuracy.
From Figures 1 and 2, we observe that both of algorithm 1 and have optimal-fitting models. This means that the algorithm suitably learns the training dataset and generalizes well to predict the severity of mammographic mass based on BI-RADS assessment, the patient’s age, shape, margin, and density of mass.
Remark 2. Since a matrix in ELM was generated by a finite dataset that contains real numbers, thus we can see from Table 4 that our Algorithm 1:. which requires a norm estimation of the bounded linear operator of , gives more efficiency than the algorithm (10) of Wang and Yu [24].
4. Conclusion and Discussion
Nowadays, there are many studies interested in the accuracy of artificial intelligence (AI) for the detection of breast cancer in mammography screening programme. Some believe that artificial intelligence (AI) has helped improve radiologists’ performance and provides results equivalent or superior to those of radiologists’ alone such as reduce the volume in screen-reading without affecting cancer detection substantially [32]. Although there are some issues that should be more explore including possible factors on recall and interval cancers [33], Freeman et al. [34] performed the systematic review of test accuracy and concluded that there is inadequate evidence in judgement of accuracy of artificial intelligence (AI) in detecting breast cancer on screening mammography. There is still small researches, which could not be representing the real effect of artificial intelligence (AI) in clinical practice or where on the clinical pathway AI might be of most benefit.
This paper presented an applying inertial modified relaxed Mann algorithms for split feasibility problems for extreme learning machine based on BI-RADS assessment, the patient’s age, and three BI-RADS attributes for predicting the severity of mammographic mass lesion to assist the physician regarding making decision about whether to go for biopsy or not. The comparison with other machine learning models and existing algorithms for split feasibility problems shows that our algorithm provides the highest performance value of 85.03% accuracy, 82.56% precision, 87.65% recall, and 85.03% F1-score. Moreover, considering training and validation loss, and the accuracy plots show that our algorithm has good fit model.
Data Availability
The mammographic mass dataset from UCI is available on the UCI website (http://archive.ics.uci.edu/ml/datasets/mammographic+mass).
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
The writing, which includes review and editing, was done by P.N.. Review and editing were in charge of W.K.. Writing, which includes the original draft and software, was the responsible of W.C.. All authors have read and agreed to the published version of the manuscript.
Acknowledgments
This research was supported by the National Research Council of Thailand and University of Phayao (N42A650334) and Thailand Science Research and Innovation, University of Phayao (FF66-UoE).