Unsupervised feature extraction based on uncorrelated approach

engineering


Introduction
Based on the job performed and the kind of experience given to the learning agent, several classes of machine learning techniques are utilized.The data points are depicted in a d-dimensional space, which establishes a perspective of the dataset, in conventional machine learning techniques.Selecting a proper interpretation for the data has required a huge amount of work.By extracting a set of dimensions or by transferring the characteristics onto another space, a representation may be created.A representation's appropriateness is evaluated in light of specific criteria on the characteristics of the raw data, including information gain and variance, or in light of the intended job.These procedures are frequently known as pre-processing techniques.
The methodologies for feature extraction in the current literature generally fall into two categories: supervised learning and unsupervised learning.Since labels are frequently unavailable, the focus is on unsupervised feature extraction.The techniques that are frequently used, unsupervised feature extraction techniques are frequently used: Principal Component Analysis (PCA) [7] [24], Independent Component Analysis (ICA) [20], MultiDimensional Scaling (MDS) [28], ISOMAP [4], Locally Linear Embedding (LLE) [38], Laplacian Eigenmaps (LE) [5], Locality Preserving Projection (LPP) [14], Neighbourhood Preserving Embedding (NPE) [13], and Isometric projection (IsoP) [17].These strategies have served as the foundation for several improved algorithms.A common framework may be used to reformulate each of the aforementioned algorithms.
The motivation behind developing the Uncorrelated Neighborhood Preserving Embedding (UNPE) algorithm for feature extraction is to address the limitations of traditional dimensionality reduction techniques and enhance their effectiveness in capturing the underlying structure of highdimensional data.The key contributions of this work are: • Preserving Local Structure: UNPE aims to preserve the local structure of the data, meaning that neighboring data points in the high-dimensional space should remain close to each other in the reduced-dimensional space.
• Uncorrelated Feature Extraction: The main contribution of the UNPE algorithm is its ability to extract uncorrelated features.This is achieved by incorporating the constraint of uncorrelatedness during the feature extraction process.
• Theoretical Analysis: It provides theoretical insights and analysis of the uncorrelated NPE algorithm.This includes discussion on the mathematical foundations, convergence properties, optimization techniques of the algorithm.
• Experimental Validation: Experimental results and evaluations to demonstrate the effectiveness and superiority of the Uncorrelated NPE algorithm compared to LLP, LLE, PCA, LE dimensionality reduction techniques.
A different category of strategies concentrates on utilising several viewpoints at once rather than picking the The number of clusters   ∈ ℝ The  ℎ data of dataset  Actual dataset matrix  ()  The  ℎ view actual dataset matrix  Data embedding matrix best one.To address this issue, several solutions have been proposed.Fisherface(FLD) [23] and Maximum Margin Criterion (MMC) [33], [18] are two common approaches.Fisherface initially uses PCA [7][1] to reduce the dimensions and create a nonsingular within-class scatter matrix.an improved version from different angle, MMC [45] may be seen as an effective and reliable feature extraction criteria in place of Fisher's criterion.Unsupervised feature techniques or dimensional reduction technique as proposed in UNPE play a crucial role in applications such as diabetic retinopathy classification [37], real-time plant disease detection in agricultural settings [42], and the development of hybrid models for wind speed forecasting [2] and crude oil time series prediction [25].These methods facilitate the extraction of relevant features from data, enabling their application in various machine learning and predictive tasks The remainder of this work is structured as follows: Related work is presented in Section II.Background Research is discussed in section III.Framework for dimensionality reduction is developed in Section IV.Theoretical analysis is presented in section V. Experiments and the outcomes are analysed in section VI. Conclusions are presented in section VII.

Related work
In this section, we first discuss various notations, then provide a quick summary of popular feature extraction methods, and last discuss the framework of graph embedding and its extensions, which enable the reformulation of these methods, [8].Notations are as shown in table 1 and summary of different feature extraction algorithm are as shown in Table 2 and 3.
Feature selection (FS) and Feature Extraction (FE) in the field of dimensionality reduction (DR), two primary methods are commonly employed.FS is recognized as a crucial technique, particularly given the exponential growth of data generation.It effectively addresses dimensionality problems by reducing redundancy, eliminating irrelevant data, and enhancing the result interpretability.On the other hand, FE focuses on identifying the most distinctive, informative, and compact set of features to improve data processing and storage efficiency.
Zebari et al. [49] provides a comprehensive exploration of both FS and FE techniques.It delves into the intricate details of various papers, including the algorithms and approaches utilized, the datasets employed, the classifiers applied, and the achieved results.
Within the context of graph embedding and extensions, Techniques, including PCA, ISOMAP, LLE, LE, LPP [14], NPE, and IsoP, are all recast in [43].The algorithms primarily consist of two phases such as creating a matrix of graph similarity and using that matrix to solve an optimization issue.Table 2 and Table 3 provide an overview of current developments in the feature extraction and dimensionality reduction fields.
Biagetti et al. [6] employ four popular machine learning algorithms, namely k-nearest neighbor (kNN), decision tree (DT), support vector machine (SVM), and naive Bayes, to facilitate the classification process.The Encephalographic (EEG) signal, a noninvasive measurement of brain activity, holds promise in early detection of neurodegenerative disorders like Alzheimer's Disease (AD) in older adults.To this end, this study explores the classification of AD using EEG signals with the aid of the Robust-Principal Component Analysis (R-PCA) feature extraction algorithm.
Huang et al. [19], have introduced a unique and robust formulation of Principal Component Analysis (PCA) called Double L2, p-norm based PCA (DLPCA) is introduced as a feature extraction technique.Unlike traditional approaches, DLPCA integrates both the minimization of reconstruction error and the maximization of variance into a unified framework.The focus is on learning a latent subspace that effectively connects the transformed features with the original features, ensuring the robustness of the method by incorporating the L2, p-norm distance metric for both reconstruction error and data variance.It demonstrates improved applicability and suitability for feature extraction tasks.
Researchers have proposed numerous strategies to combat spammers, who continually adapt to evade detection.To enhance detection effectiveness, algorithms and feature extraction techniques have been developed.For instance, Murugan et al. [35] use a hybrid approach that combines logistic regression with Principal Component Analysis (PCA) to extract crucial features and maximize detection rates.This approach addresses the evolving tactics employed by spammers.
Yang et al. [44] present a diagnostic system based on Support Vector Machines (SVM) operates in three stages.It begins with Principal Component Analysis (PCA) to reduce data dimensionality.Next, it optimizes SVM parameters with the differential evolution algorithm.Finally, the system classifies tumors, evaluating performance with multiple indices: accuracy, sensitivity, specificity, and ROC-AUC.
Feature Extraction Algorithms (FEAs) play a crucial role in addressing the challenges associated with highdimensional data, enabling machine learning algorithms to effectively operate.Anowar et al. [3] aim to comprehensively explore and evaluate representative FEAs, both conceptually and empirically, the efficacy of the FEAs is evaluated based on classification accuracy and computational speed.
Zhang et al. [51] proposed a Semi-Supervised local Multi-Manifold Isomap (SSMM-Isomap) is a semi-supervised learning framework that combines linear embedding with joint learning of local nonlinear manifold features using both labeled and unlabeled data.Its goal is to minimize distances within the same manifold while maximizing distances between different manifolds.It preserves local topology structures by considering neighborhood reconstruction error.To efficiently handle new data, it introduces a feature approximation error, allowing direct extraction of local manifold features using a learned linear extractor.
Locally linear embedding (LLE) is a well-established nonlinear dimensionality reduction technique that finds extensive application in machinery fault diagnosis.LLE focuses on reducing the dimensions of a given dataset by leveraging its underlying geometric structure, which plays a pivotal role in determining the quality of the embedding result.However, traditional LLE algorithms rely on the Ordinary Least Squares (OLS) method to calculate the geometry structure, making the embedding result susceptible to noise interference.
To address this issue, a robust variant of LLE, known as robust LLE (RLLE), has been developed by Zhang et al. [50].RLLE employs advanced techniques such as Least Angle Regression and Elastic Net (LARS-EN) to compute the local structure, thereby enhancing the algorithm's robustness against noise.Additionally, a novel fault diagnosis methodology is proposed, combining RLLE with Support Vector Machine (SVM) classifiers, to tackle machinery fault diagnosis challenges.
The complex nature of industrial process data, characterized by its large volume, high dimensionality, and nonlinearity, presents significant challenges in achieving timely and accurate fault diagnosis in industrial processes.In response to these challenges, that presents a novel and effective fault diagnosis approach utilizing an enhanced global and local Dimensionality Reduction (DR) method called Discrimination Locality Preserving Projections integrated with Sparse Autoencoder (SAEDLPP).
He et al. [15] SAEDLPP begins with a Sparse Autoencoder (SAE) to extract global data characteristics.Then, Discrimination Locality Preserving Projections (DLPP) focuses on local information.This fusion of SAE and DLPP ensures both global and local feature preservation.SAEDLPP enhances fault diagnosis for industrial data, capturing broad context and fine details, improving accuracy.
The Locality Preserving Projections (LPP) [14] algorithm is a popular linear dimensionality reduction technique widely employed in various applications, including face recognition.However, a notable limitation of LPP is that its projection matrix is not orthogonal, posing challenges for reconstruction and other related tasks.To address this issue, Orthogonal LPP (OLPP), aims to obtain an orthogonal projection matrix through a step-by-step procedure.However, the computational complexity increases significantly.
Wang et al. [39], present a novel and efficient approach called Fast Orthogonal LPP (FOLPP) that overcomes the limitations of the original LPP algorithm and reduces the computational burden compared to OLPP.FOLPP simultaneously minimizes the locality of data points while maximizing the globality, all under the constraint of orthogonality.By incorporating this orthogonal constraint during the optimization process, FOLPP achieves a balance between preserving local information and capturing global patterns in the data.
Xie et al. [41], proposed Low-Rank Sparse Preserving Projections (LSPP), a dimensionality reduction approach, combines manifold learning and low-rank sparse representation.It preserves geometric structure and handles data corruptions, offering robust features.This approach addresses challenges in real-world applications, contributing to dimensionality reduction.
Jiang et al. [21], developed a hybrid hierarchical fault diagnosis method is introduced for efficient bearing fault detection.It combines the Scale-Variable Dispersion Entropy (SVDE) with the Parametric t-distributed Stochastic Neighbor Embedding (Pt-SNE) algorithm, merging statistical analysis and machine learning principles.The methodology involves multiple stages: Initially, SVDE and statistical analysis evaluate the bearing's health state.When a fault is detected, a mixed-scale model, integrating CEEMDAN and SVDE, is constructed to further assess the fault condition.This comprehensive approach offers an effective solution for bearing fault diagnosis.
Terahertz-based material identification poses challenges in dimensionality reduction due to the sensitivity of traditional methods such as Principal Component Analysis (PCA), Local Preserving Projection (LPP), and Local Linear Embedding (LLE) to the number of nearest neighbor samples and their limited consideration of class differences.These limitations can hinder subsequent clustering models and lead to inaccurate clustering results.
To address these challenges, Yi et al. [46], proposed an improved approach for Terahertz spectral recognition by combining the t-distributed Stochastic Neighbor Embedding (t-SNE) and an enhanced Fuzzy C-means (FCM) algorithm.The t-SNE method transforms the high-dimensional sample distribution into a Gaussian distribution, while representing the low-dimensional coordinates as a t-distribution.This approach elongates the distance between clusters, relieving congestion and enhancing the discrimination between classes.
The selection of text features plays a fundamental and crucial role in text mining and information retrieval.Traditional approaches to feature extraction often rely on manual crafting, which can be a time-consuming process.However, with the advent of deep learning, it has become possible to automatically learn effective feature representations directly from training data.This has opened up new avenues in feature extraction for various applications, particularly in text mining.The algorithm combines the tdistributed Stochastic Neighbor Embedding (t-SNE) method for dimensional reduction and the FCM clustering to effectively recognize different substances through terahertz spectrum.
Investigating the influence of t-SNE hyperparameters on clustering performance could optimize the feature extraction process.

Zhang et al. [51] 2018
Semi-supervised local multimanifold Isomap by linear embedding for feature extraction The method aims to extract discriminative and robust features from data by combining Isomap, a popular nonlinear dimensionality reduction algorithm, with a linear embedding approach.
Performing sensitivity analysis and parameter tuning experiments could lead to better insights into the stability and robustness of the method.Analyze and identify the ideal parameter settings for the various forms of data.

Xie et al. [41] 2018
Low-rank sparse preserving projections for dimensionality reduction The algorithm aims to address the challenge of dimensionality reduction in high-dimensional data by simultaneously preserving the intrinsic geometric structure and learning a robust representation that can effectively handle data with corruptions and outliers.
Evaluating its performance under varying levels of noise could help assess its ability to handle real-world data with varying degrees of contamination.Deep learning, as a novel feature extraction method, has demonstrated remarkable achievements in the field of text mining.Its key differentiating factor from conventional methods lies in its ability to automatically learn features from large-scale datasets, without the need for handcrafted features that heavily rely on prior knowledge of designers.This enables deep learning to harness the power of big data, as it autonomously learns feature representations by leveraging the vast amount of information contained within the data.Unlike traditional approaches, which often struggle to effectively utilize big data due to the limitations of handcrafted features, deep learning automatically adapts and learns millions of parameters to extract meaningful features.
The capability of deep learning to automatically learn feature representations from big data has revolutionized the field of text mining.By effectively leveraging the abundance of data and capturing intricate patterns and relationships, deep learning has shown tremendous potential in improving the accuracy and effectiveness of various text mining tasks.Its ability to extract complex and meaningful features from raw text data has paved the way for advancements in natural language processing, sentiment analysis, document classification, and many other areas of research and industry.
Liang et al. [30], outlines the common methods used in text feature extraction first, and then expands frequently used deep learning methods in text feature extraction and its applications.
In recent decades, the field of image processing has played a crucial role in addressing and resolving challenges in medical imaging.Medical imaging techniques are essential for visualizing the internal and external structures of the human body.These techniques aid in medical diagnosis, disease analysis, and the development of datasets comprising both normal and abnormal medical images.Medical imaging can be categorized into two types: invisible-light medical imaging and visible-light medical imaging.While visiblelight medical imaging can be easily understood by a layperson, invisible-light medical imaging requires interpretation by a trained radiologist.The analysis of medical images, regardless of the type, often necessitates the processes of segmentation and feature extraction.
Various techniques exist in the domain of medical imaging, Chowdhary et al. [9], focus specifically on tumor detection through mammograms or magnetic resonance imaging (MRI).A range of segmentation and feature extraction methods employed in the preprocessing of medical images has been discussed.Segmentation is a vital step in isolating specific regions of interest, such as tumors, from the surrounding background, allowing for more accurate analysis.Feature extraction techniques aim to capture relevant information and characteristics from the segmented regions, enabling effective classification and diagnosis.

NPE
In contrast to PCA, Neighbourhood Preserving Embedding (NPE) preserves the local neighbourhood structure on the data manifold [29].The two key processes in NPE [13] are calculating edge weights and linear projections.The definition of the objective function is and   are data points of node i and node j in the adjacency graph.  is weights of the edge from node i to j and it is 0 when there is no edge.
where   is where m is the total number of set of points.Further computing the linear projection by solving eigenvector problem.
where  is the transformation vector The reduced dimension reduction objective function results in where Y is d dimensional vector and column vectors A which is  = ( 0 ,  1 , ⋯ ,  −1 ) and A is  x  matrix which is solution of Equation 3 organised in order of eigenvalues.

UDLPP
Uncorrelated discriminant locality preserving projections (UDLPP) [48], maximizes the distance of projected samples from various clusters while preserving the geometric structure of samples that belong to the same cluster in low-dimensional space [10] [22].The objective function is as follows: where X is the matrix of order  × ,  = [ 1 ,  2 …   ] L is the Laplacian matrix [26], W is the similarity matrix, G is uncorrelated constraint,  1 ,  2 ,   are column vectors [40] ordered according to their corresponding eigenvalues

ULPP
Uncorrelated Locality Preserving Projections (ULPP) [26], is to identify the best transformation while preserving the data set's inherent geometry.The objective function is as follows: where  1 ,  2 , …   be the projected low-dimension space.V is a transformation vector.L is the Laplacian matrix.X is the original matrix.B=G-W where W is the similarity matrix.

Proposed Framework
In the context of UNPE, our objective is to maintain both the data set's local structure and its between-locality information.UNPE is the name for this method of linear dimension reduction.In general, UNPE aims for a perfect transformation that keeps the data set's natural geometry.Weight matrix   having the weight of the edge from node i to node j, and 0 if there is no such edge.Let the anticipated low-dimension space be represented by the  1 ,  2 , ...  .The following is UNPE objective function, assuming that V is a transformation vector with the formula  =   , by simple algebra formulation, here adjacency weight matrix   is calculated with considering  and  spatial coordinates, the minuend of objective function can be reduced to: where X is set of data points and B is difference of sum of different values.
From the algorithm NPE we obtain, Simplifying the Equation 11, where fixed weights   , while optimizing the coordinates   .where  = ( −  ) ( −  ) clearly, the matrix   is symmetric and semi-positive definite [16].
Substituting Equations 12 and 10 in Equation 9 results in, The limitation is then taken into account.Additionally applying the condition      =  in accordance with [48].
The solution, UNPE is defined as maximizing constraint problem as follows: The generalised eigenvalues problem's maximum eigenvalues solutions provide the vectors   that maximise the objective function, Once V is obtained, the linear features are extracted by  =   .

Theoretical analysis
In this section, the UNPE algorithm is examined from three perspectives.Initially, the algorithm 1's convergence patterns are presented, followed by a discussion of the challenges related to parameter selection and computation time.

Convergence Analysis
First, we show that the algorithm is convergent with respect to UNPE.The objective function of Equation 13decreases as the number of iterations rises, which causes the Algorithm 1 to converge.The Equation 10 is solved at a local maximum using the Algorithm 1.
Theorem 1 provides the UNPE algorithm's convergence.).Therefore the objective function decreases with direct proportion to the number of iteration.According to the constraint of convergence analysis        = 1 ( = 1, 2, 3, 4…, ), we obtain For all i and j, the following inequality holds Combining Equation 29and the constraint The following inequality exists:

Complexity Analysis
Three different optimization problems may be derived from UNPE.The eigen-decomposition method is used to address the issue, and its computational complexity is ( 3 ) and ( 3 ), where  is the size of the data and  is the dataset's dimensionality.( 2 ) is the computational complexity of the n separate issues as a whole.In conclusion, UNPE's computational complexity is ( •( 3 ,  3 )), where T is the number of iterations [34].
Table 4 provides an overview of the temporal and spatial complexity approachs.(i) Assume that the newly decreased dimension d is a tiny constant; the future iterations are ignored.Data with three perspectives have cubic temporal complexity according to the UNPE.When dealing with data that spans more than three views, these procedures takes longer time to complete.(ii) Compared to PCA and LE, UNPE has more spatial complexity.Similar to LPP, UNPE needs more room to maintain the tensor.

Parameter determination
The UNPE's sensitivity to parameters is investigated on this enormous data set.The annotation accuracy of UNPE is studied, then the parameters k, , and  are modified.This analysis shows that there is a little variation in UNPE performance.It works well with a smaller k and the synthetic data set.Without sacrificing generality, set k = 3 for all of the data sets used in the experiment.The cross-validation result is consistent with the UNPE's high annotation accuracy even when k=3 and k=2.The convergence of UNPE is also confirmed using this set of data.The UNPE objective function values are shown against iterations using [32].The values of the optimisation function that correspond to the first few projections all converge after a number of rounds.However, compared to smaller data sets, this enormous data set's convergence performance is slower.
NPE is used to get the highest annotation accuracy on this subset.UNPE outperforms PCA statistically by a

Experiment
The suggested approaches will be tested in this section using a common unsupervised method task, namely clustering, due to their nature of unsupervised algorithms.In the first experiment, results on convergence behaviour are presented.The second experiment uses the k-means clustering findings on several datasets utilising different quantities of extracted characteristics to assess UNPE's projection abilities.The third category includes k-means clustering results on multi-view data sets, which are used to assess UNPE's projection capability [31] [54].These experiments are conducted using the Python programming language on a computer system with 16GB of RAM, and the corresponding code can be accessed through the GitHub link4 .

Dataset and evaluation criteria
There are four different kinds of matrix data sets [16] to assess the performance of the UNPE.They consist of imaging data such as Coil20, Pie, chemical data such as Tox, and biology data such as Prostate-GE.They are between 171 and 2414 in size.The dimensionality varies from 674 and 15154.There are between 2 and 68 courses.Additionally, with some preparation, Table 5 lists the specific statistical characteristics of these data sets [53].
The Columbia University Image Library (Coil20) dataset is a widely used benchmark dataset in the field of computer vision and machine learning for evaluating image classification and object recognition algorithms.It was developed by the Center for Research on Intelligent Systems at Columbia University.Coil20 consists of 20 object classes, each captured in various poses under varying lighting conditions.The dataset contains a total of 1,440 images, with 72 images per class.The images are captured in a controlled environment, where objects are placed on a turntable and photographed from different angles.
Key features of the Coil20 dataset are as follows, (i) Object Variation: The dataset exhibits variations in pose, rotation, and illumination, making it challenging for image classification algorithms to correctly identify objects across different viewpoints.
(ii) Controlled Environment: The images are captured in a controlled setting, ensuring consistent backgrounds and lighting conditions for each object.
(iii) Object Classes: The 20 object classes include items such as cups, bottles, toys, and other everyday objects.
(iv) Turntable Captures: Objects are placed on a turntable, which rotates to capture images from different angles.Each object is photographed at increments of 5 degrees, resulting in 72 images per class.
(v) Gray-Scale Images: The images are typically provided in grayscale format, which simplifies preprocessing and analysis but also reduces the color information available to algorithms.
The Pose, Illumination, and Expression (Pie) Database is a widely used dataset in the field of computer vision and pattern recognition.It was developed at Carnegie Mellon University (CMU) and is designed to study the effects of pose, illumination, and facial expressions on face recognition algorithms.
Key features of the CMU Pie Database are as follows, (i) Variations in Pose, Illumination, and Expression: The dataset captures variations in pose (different angles), illumination conditions (varying lighting conditions), and facial expressions (different emotions).These variations present challenges that real-world face recognition systems may encounter.
(ii) Subjects and Images: The CMU PIE Database includes images of 68 subjects, each captured under different conditions.For each subject, there are approximately 13 images under varying pose, illumination, and expression conditions.
(iii) Pose Variation: The dataset covers a wide range of pose variations, capturing faces at different angles and orientations.
(iv) Illumination Variation: Images are captured under different lighting conditions, including varying levels of brightness and direction of light sources.
(v) Expression Variation: The dataset includes images depicting various facial expressions, such as smiling, neutral, and other emotional states.
(vi) Benchmarking Face Recognition Algorithms: The CMU PIE Database serves as a benchmark for evaluating the robustness and performance of face recognition algorithms under challenging conditions involving pose, illumination, and expression variations.
The TOX datset share insights into early drug safety testing, with a particular emphasis on the evaluation of in vitro cardiotoxicity.It is a significant resource in the field of computational toxicology and cheminformatics.It is designed to facilitate the prediction of potential toxicity and biological activity of chemical compounds using machine learning and predictive modeling techniques.The dataset consists of thousands of chemical compounds, each characterized by its chemical structure, molecular features, and descriptors.It contain 4 different class of dataset with size of 171.
The Prostate-GE dataset is a valuable resource provided by the Clinical Proteomics Program Databank (CPTAC), which is a program initiated by the National Cancer Institute (NCI) with the goal of advancing the understanding of cancer biology through proteomics research.The Prostate-GE dataset specifically focuses on proteomic data related to prostate cancer.
Key features of the Prostate-GE dataset from the Clinical Proteomics Program Databank are as follows, (i) Proteomics Data: The dataset contains proteomic data, which includes information about the proteins present in samples from prostate cancer tissues.Proteomics is the study of the complete set of proteins in a biological sample, providing insights into the molecular mechanisms and characteristics of diseases like prostate cancer.
(ii) Clinical Relevance: The Prostate-GE dataset is designed to be clinically relevant, providing insights into the molecular basis of prostate cancer development, progression, and potential therapeutic targets.Understanding the proteomic profiles of prostate cancer tissues can lead to improved diagnostic and treatment strategies.
(iii) Large-Scale Analysis: The dataset likely includes a significant number of samples from prostate cancer patients, allowing researchers to perform large-scale analyses to identify protein biomarkers, molecular pathways, and protein expression patterns associated with prostate cancer.
(iv) Potential Applications: Researchers can use the Prostate-GE dataset for various applications, including biomarker discovery, identifying potential drug targets, understanding the heterogeneity of prostate cancer, and exploring the relationship between proteomic alterations and clinical outcomes.
(v) Data Integration: The dataset might be accompanied by clinical and genomic data, enabling integrative analyses that combine proteomics information with other types of molecular and clinical data for a comprehensive view of prostate cancer.
(vi) Research Advancements: The availability of the Prostate-GE dataset empowers researchers and the scientific community to advance our understanding of prostate cancer biology and contribute to the development of personalized and targeted approaches for diagnosis and treatment.
The assessment of UNPE techniques for clustering is carried out based on experimental findings using three distinct measures [12], containing the F-score, Normalised Mutual Information (NMI), and clustering accuracy (ACC) are used to assess UNPE performs [47][36] [27].

Convergence Result
Convergence graphs provide a way to compare the convergence behavior of different algorithms or parameter settings.A steeper decrease in the objective function might indicate faster convergence or better performance.A graph of the objective function versus iterations is a visual tool that helps assess the convergence behavior of the algorithms.It provides insights into whether the algorithm is progressing towards an optimal solution, how quickly it is converging, and whether any adjustments are needed for better performance.
Convergence graphs can also help determine when to stop the algorithm.If the graph shows that the objective function value has stabilized and is not changing significantly, it might be an indication that further iterations are unnecessary.Specifically detailing the changes in objective function value over iterations.The convergence graph on Coil-20 dataset of UNPE algorithm is shown in Figure 1.
The objective function values are shown on the y-axis, while the iterations are shown on the x-axis, it is observed that UNPE converges on Coil-20 datasets after a number of iterations (within six) and consistent with the value of the objective function as iterations progress, it indicates that the algorithm is converging towards an optimal solution.These experimental results prove the effectiveness of the approach.[11].

Experiment on Coil-20 dataset
The results have undergone testing using the Coil-20 dataset, demonstrating their stability by three parameters.They F-score results are displayed in Figure 2(a), NMI in Figure 3(a) and Accuracy in Figure 4(a).Through iterative optimization, UNPE automatically learns the appropriate view weights.
Our suggested techniques UNPE for clustering are evaluated for the experimental findings using three distinct metrics: F-score, Normalised Mutual Information (NMI) and clustering ACCuracy (ACC).Output of different feature extraction algorithm on different dataset is shown as shown in Figure 4.
As seen from Fig. 4, we have the following conclusions.(i) UNPE leads the competing techniques in the majority of extracted dimensions when measuring the effectiveness of feature extraction by the clustering accuracy.(ii) Three metrics, including F-score, Accuracy and NMI, each and every points to the benefits of our algorithm.(iii) UNPE performs the best on these data sets among the several characteristics that may be derived.(iv) The effectiveness of algorithms for feature extraction do not improve at the same rate as does the quantity of features extracted.The notion that extra characteristics may make the representations redundant is supported by the evidence.

Experiment on Pie dataset
The results of F-score are shown in Figure 2(c), NMI in Figure 3(c) and Accuracy as in Figure 4(c).Using its various characteristics, including clustering accuracy, NMI, and F-score for the sample dataset, UNPE's performance is assessed.For dimension reduction, many unsupervised methods including LPP, LLE, PCA [7], and LE are utilised.This suggests that the outcomes of UNPE are consistent.

Experiment on Tox dataset
The samples of 100 were obtained for the Pie dataset, which also includes 3 perspectives.The results of F-score are shown in Figure 2(b), NMI in Figure 3(b) and Accuracy in Figure 4(b).According to the comparative results, UNPE regularly outperforms the other techniques and is quite stable.This finding shows that our technique may effectively project raw characteristics.The UNPE approach can generate more discriminating low-dimensional representations from unsupervised multi-view data.

Experiment on Prostate-GE dataset
The observations of 400 samples were included in a generated synthetic dataset with 3 viewpoints [2, 2, 2] are the feature sizes.To determine the effect of UNPE, the experiments are conducted using Prostate-GE dataset in a way similar to that of earlier datasets.The F-score findings are shown in Figure 2

Performance Metrics
In this work, three metrics-cluster F-score, NMI, and cluster accuracy-are used.
(i) The accuracy of an assessment is assessed using the F-score or F-measure.It is calculated from the test's recall and precision, where recall represents the proportion of true positive results to all positive results, including those that were incorrectly labelled as positive, and precision represents the proportion of true positive findings to all samples that should have tested positive.While recall is typically referred to as sensitivity, precision is sometimes referred to as a positive predictive value.
(ii) The Normalised Mutual Information (NMI), a commonly used performance metric, measures the amount of information exchanged between two variables-the true labels and the projected labels.It calculates the entropynormalized similarity between the actual labels of the data points and their predicted labels.Consideration is given to both the diversity of the clusters and the grouping's accuracy.
(iii) The cluster accuracy statistic counts the number of data points in each cluster that were successfully categorized.It is determined by classifying each cluster according to the majority of its corresponding data points, and then comparing the anticipated class labels to the actual class labels.

F-score
The Coil-20, Pie, TOX, and Prostate-GE datasets' Fscore performance results for the UNPE method are shown in Figure .2(a).The F-score values are computed with the 10 to 55 different extracted attributes.It is compared with the most advanced algorithms, including the LPP [14], LE [5], LLE [38], PCA [7] methods.On the coil-20 dataset, the Fscore value for the UNPE method is 0.1412%, 1.26% greater than that of the current LPP [14]and PCA [7] techniques.In a similar vein, it outperforms LE and LLE algorithms.Furthermore, when the number of features varies from 10 to 35, UNPE has performed between 1 and 2 percent better than LE, LLE, and PCA [7] techniques.The UNPE method performs better than PCA algorithm at feature numbers 10, 15, 20, 25, 30, 35, 40, 45, and 50 by 1.60%, 2.00%, 1.63%, 1.37%, 1.42%, 1.51%, 1.59%, 1.04%, and 1.12%, respectively.The F-score continuously rises as the extracted feature count rises from 5 to 50.The pair-wise correlation between the variables makes the UNPE method more relevant.It is observed throughout the experiment that there are variances in the k-means clustering algorithm's k value and feature number.The k values, which in this instance is 4, directly affect the F-score numbers.
Similar to this, Pie and TOX datset feature extraction performance is compared using the k-means method.As with the Pie dataset, four distinct methods are utilised for comparison.The F-score performance of UNPE on the Pie dataset for 5 features is shown in 2(c), and it is 0.531, which is 1.14% better than the PCA and LPP methods and significantly better than the LE and LLE algorithms, respectively.The UNPE also outperforms the PCA method by 0.54%, 1.27%, 0.58%, 0.18%, 1.52%, 0.51%, 1.74% 1.71% respectively, when the number of features increases from 10, 15, 20, 25, 30, 35, 45, and 50.It performs better than other algorithms, such as LE, LLE, and LPP, according to [14].In the experiment, it was found that the number of features increased steadily along with the F-score values.This is an account of local neighborhood structures are preserved by the UNPE algorithm in each view, directly affecting the Fscore measurement.
The F-score outcomes for the TOX dataset are shown in Fig. 2(b).UNPE performs 5 to 9 percentage points better than the PCA [7] method.The UNPE F-score is better than PCA at features 5,10,15,20,25,30,35,40,45, and 50 by 0.42%, 1.75%, 0.73%, 1.02% 1.35%, 1.17%, 0.00%, 1.74%, 1.63%, 0.53%.Similar it performs better than LPP by 0.5% to 1.75%.The number of features grows in direct proportion to algorithm performance.The local neighbourhood structures in each view of UNPE that are consistent across views and are in charge of delivering better performance as the number of features increases.The F-score values for the Prostate-GE dataset are shown in Fig. 2(d).UNPE beats PCA method by 4 to 7 percentage points in the experiment, where the number of extracted features spans from 5 to 50.The UNPE Fscore is superior than PCA at features 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 in terms of 0.32%, 1.65%, 0.72%, 1.02% 1.35%, 1.17%, 0.03%, 1.74%, 1.63%, 0.53%.Similar it performs better than LPP by 0.5% to 1.75 %.The number of features increases in direct proportion to the efficiency of the algorithm.The UNPE algorithm in each view maintains the local neighborhood structures, which has an immediate impact on the F-score calculation.

NMI
For the Coil-20 dataset, the UNPE algorithm is compared with other algorithms such as LPP [14], LE [5], LLE [38], PCA [7].Evaluation of results with the NMI parameter is shown in Figure 3(a).In Figure 3(a), UNPE substitutes PCA algorithm on Coil-20 dataset by 0.41% to 1.56 % for extracted feature counts between 10 and 55.In a similar vein, UNPE outperforms PCA algorithm by 0.48% to 1.83 % .In addition, UNPE offers much superior outcomes than LE and LLE algorithms.It functions consistently over a variety of extracted characteristics.The NMI values rise along with the amount of features, which is a certain indication of pairwise correlation and high-order correlation alignment in the UNPE method.

Accuracy
Regarding clustering accuracy, Figure .4(a) compare the UNPE algorithm's performance to that of the other four methods, namely LPP [14], LE [5], LLE [38], PCA [7].There are between 5 and 50 features.The UNPE performs better than the PCA (0.33%to 1.65%) and the LPP(0.28 % to 1.70%).It performs much better than the LE, LLE, and LPP algorithms.With the addition of more features, it consistently improves performance.It describes uncorrelated concepts that are used to eliminate extraneous features from the UNPE method following pairwise and high-order feature correlation.
The performance of the UNPE algorithm on the Pie dataset is shown in Figure 4(c), which is similar to the prior dataset.On the Pie dataset, compares the performance of the UNPE method with that of the other four algorithms, namely LPP [14], LE [5], LLE [38], PCA [7].The UNPE algorithm performs substantially better than LE, LLE, and LPP in terms of clustering accuracy, outperforming them by a factor of 1% to 1.5% compared to PCA and LPP.Similarly, UNPE outperforms MCCA method by 0.58%, 0.63%, 0.72%, 1.44%, 0.35%, 0.47%, 0.36%, 1.26%, 1.15%, 0.97%, respectively, for extended feature numbers 10, 15, 20, 25, 30, 35, 40, 45, and 50.UNPE performs well in directly proportion to the increasing number of features.When the two correlations are combined, numerous perspectives may be seen with more flexibility, and cluster accuracy performance is improved.
When comparing the UNPE accuracy metric to the TOX dataset, the LPP method performs better with a difference between 0.18% and 1.32% and the PCA algorithm between 0.17 % and 1.94 %.In comparison to LPP, LLE, and LE algorithms, UNPE yields better useful results.UNPE outperforms LE, LLE, and LPP with a larger percentage of difference.The performance of UNPE improves with feature count, and it is clearly evident that the algorithm considers the local neighbourhood structures of each view into account.

Other performance Measures
The cluster evaluation results in terms of the Rand Index and Silhouette Index for different dimensionality reduction algorithms for two datasets coil-20 and Pie dataset, as shown in Figure 6.The Rand Index measures the similarity between the true clustering and the predicted clustering.A higher Rand Index indicates better clustering quality.In the Figure 6(a) and 6(b), the UNPE algorithm appears to achieve higher Rand Index values compared to the other algorithms, suggesting that it produces more accurate and consistent clustering.The Silhouette Index assesses the quality of clusters based on their separation and cohesion.A higher Silhouette Index indicates well-separated and dense clusters.The Figure 6(c) and 6(d) demonstrates that the UNPE algorithm yields higher Silhouette Index values, indicating that it generates clusters with better internal cohesion and separation between clusters.
The effectiveness of the UNPE method was evaluated using a statistical t-test against PCA (Principal Component Analysis) and LPP (Locality Preserving Projections) on two different datasets, Coil-20 and Pie.Three performance indicators have been evaluated using the t-test statical method, cluster accuracy, NMI (Normalised Mutual Information), and F-score.The t-test findings shows that the p-values are between 0.035 and 0.046 indicating that UNPE results

Conclusion
Uncorrelated Neighbourhood Preserving Embedding (UNPE), is a novel manifold learning technique proposed in this study and offers two outstanding properties.(i) The local neighbourhood structure on the data manifold is preserved.(ii) The simple uncorrelated constraint is then implemented to integrate statistically uncorrelated features.The results of the experiments show that UNPE has a promising performance.The UNPE method generates projections and minimises redundant feature extraction by using the uncorrelated condition.The UNPE algorithm is compared with the LPP [14], LE [5], LEE [38], PCA [7] algorithms.In the experiment, the convergence is assessed using four different datasets.The experiment uses the k-means method to get the performance measures F-score, NMI, and Clustering Accuracy for all four datasets; nevertheless, the UNPE algorithm outperforms LPP and PCA algorithm by 1% to 2%.
Quantum computing-based UNPE is one of the most recent developments in quantum algorithms for related dimensionality reduction approaches.By using the benefits of quantum computing, such as exponential speedup in certain operations, these algorithms perform better than conventional algorithms.

Figure 2 :
Figure 2: k-means F -score on four datasets with various numbers of extracted features.The extracted feature is on the x-axis.

Figure 5 (
Figure 5(a) and Figure 5(e) showcase the clustering visualizations without applying any feature extraction techniques to the TOX and ProstateGE datasets, respectively.Subsequently, Figures 5(b), 5(c), and 5(d) demonstrate the cluster visualizations after applying feature extraction methods, specifically UNPE, PCA (Principal Component Analysis), and LPP (Locality Preserving Projections), to the TOX dataset.Meanwhile, Figures 5(f), 5(g), and 5(h) display the cluster visualizations after applying the same feature extraction techniques to the ProstateGE dataset.The visual results are compelling as they illustrate that the UNPE algorithm consistently produces more effective clustering outcomes compared to PCA and LPP for both datasets.These visualizations highlight the superior performance of UNPE in achieving well-defined and distinct clusters, making it a favorable choice for feature extraction in clustering tasks involving the TOX and ProstateGE datasets.

Figure 3 :
Figure 3: k-means NMI on four datasets with various numbers of extracted features.The extracted feature is on the x-axis.

Figure 4 :
Figure 4: k-means accuracy, on four datasets with various numbers of extracted features.The extracted feature is on the x-axis.
(b)  .UNPE outperforms PCA and LPP in terms of NMI performance by 0.5% to 1.5%.Similarly it outperforms LLE and LE.It unmistakably shows that the retrieved features in the UNPE technique have high order variance correlation.

Figure 5 :
Figure 5: Clustering visualization with different feature extraction algorithms with TOX and ProstateGE dataset.The extracted feature is on the x-axis and y -axis.

Figure 6 :
Figure 6: Rand index and silhouette index graph of coil-20 and pie datasets

Table 1 Notations
Size of the data collection Dimensionality of the vth view's data set  Data set size  Dimensionality of the processed data collection

Table 2
Literature survey on dimensionality reduction methods

Table 4
Complexity analysis of the Dimension Reduction Methods

Table 5
Characters of Different Data Sets