To fully utilize the advances in omics technologies and achieve a more comprehensive understanding of human diseases, novel computational methods are required for integrative analysis of multiple types of omics data. Here, we present a novel multi-omics integrative method named Multi-Omics Graph cOnvolutional NETworks (MOGONET) for biomedical classification. MOGONET jointly explores omics-specific learning and cross-omics correlation learning for effective multi-omics data classification. We demonstrate that MOGONET outperforms other state-of-the-art supervised multi-omics integrative analysis approaches from different biomedical classification applications using mRNA expression data, DNA methylation data, and microRNA expression data. Furthermore, MOGONET can identify important biomarkers from different omics data types related to the investigated biomedical problems. Our understanding of human disease can be improved by integrating the abundance of high throughput biomedical data. Here, the authors use deep learning methods successfully used on images to integrate various types of omics data to improve patient classification and identify disease biomarkers. Therefore, there is a need for novel integrative analysis methods to effectively take advantage of the interactions and complementary information in multi-omics data.The rapid advancement in high-throughput biomedical technologies has enabled the collection of various types of “omics” data with unprecedented details. Genome-wide data for different molecular processes, such as mRNA expression, DNA methylation, and microRNA (miRNA) expression, can be acquired for the same set of samples, resulting in multiple omics (multi-omics) data for various studies of diseases. While each omics technology can only capture part of the biological complexity, integrating multiple types of omics data can provide a more holistic view of the underlying biological processes. Specifically, for human diseases, existing studies have demonstrated that integrating data from multiple omics technologies can improve the accuracy of patient clinical outcome prediction comparing with only using a single type of omics data10. With the rapid development of personalized medicine, curated datasets with detailed annotations that characterize the phenotypes or traits of the samples are becoming more widely available. Therefore, there is an increasing interest in supervised multi-omics integration methods that can identify disease-related biomarkers and perform predictions on new samples. Early attempts of supervised data integration methods for biomedical classification tasks include feature concatenation-based strategies and ensemble-based strategies. On the one hand, concatenation-based methods integrated different omics data types by directly concatenating the input data features to learn the classification model5. On the other hand, ensemble-based methods integrated the predictions from different classifiers, each trained on one type of omics data individually1. However, these methods failed to consider the correlations among different omics data types and could be biased toward certain omics data types. Recently, more supervised multi-omics data integration methods focused on exploiting the interactions across different omics data types have been proposed. For example, van de Wiel et al.6 introduced an adaptive group-regularized ridge regression method that incorporated methylation microarray data and curated annotations of methylation probes for cervical cancer diagnostic classification. Singh et al.4 proposed Data Integration Analysis for Biomarker discovery using Latent cOmponents (DIABLO) by extending the sparse generalized canonical correlation analysis to a supervised setting, which could seek common information across multiple omics types while discriminating between different phenotypic groups.A great number of methods have been proposed over the years to perform multi-omics data integration for various problems. However, most existing efforts focus on unsupervised multi-omics data integration without the additional information of sample labels13. For example, Huang et al.2 integrated the features of mRNA expression and miRNA expression data with additional clinical information at hidden layers for better prognosis prediction in breast cancer. However, these existing methods are based on fully connected networks, which did not exploit the correlations between samples effectively through similarity networks. Moreover, while current deep learning-based methods often integrate different omics data at the input space13 or the learned feature space12, different types of omics data can also present unique characteristics at the high-level label space. Therefore, it is crucial to utilize the correlations across different classes and different omics data types to further boost the learning performance.With the continuous advancement of deep learning in various tasks, more and more multi-omics integration methods begin to take advantage of the high learning capability and flexibility of deep neural networks (NN)To this end, we introduce MOGONET, a multi-omics data analysis framework for classification tasks in biomedical applications. MOGONET unifies omics-specific learning with multi-omics integrative classification at the label space. Specifically, MOGONET utilizes graph convolutional networks (GCN) for omics-specific learning. Comparing with the fully connected NN, GCN takes advantage of both the omics features and the correlations among samples described by the similarity networks for better classification performance. Besides directly concatenating the label distribution from each omics data type, MOGONET also utilizes View Correlation Discovery Network (VCDN) to explore the cross-omics correlations at the label space for effective multi-omics integration. To the best of our knowledge, MOGONET is the first supervised multi-omics integrative method that utilizes GCNs for omics data learning to perform effective class prediction on new samples. We demonstrated the capabilities and versatility of MOGONET through a wide range of biomedical classification applications, including Alzheimer’s disease patient classification, tumor grade classification in low-grade glioma (LGG), kidney cancer type classification, and breast invasive carcinoma subtype classification. We also showed the necessity of integrating multiple omics data types as well as the importance of combining both GCN and VCDN for multi-omics data classification through comprehensive ablation studies. Moreover, we demonstrated that MOGONET can identify important omics signatures and biomarkers related to the investigated biomedical problems.We introduce MOGONET, a supervised multi-omics integration framework for biomedical classification tasks Fig. . After pTo demonstrate the effectiveness of MOGONET, we applied the proposed method on four different biomedical classification tasks using four different datasets: ROSMAP for Alzheimer’s Disease (AD) patients vs. normal control (NC) classification, LGG for grade classification in low-grade glioma (LGG), KIPAN for kidney cancer type classification, and BRCA for breast invasive carcinoma (BRCA) PAM50 subtype classification. Three types of omics data , DNA methylation data (meth), and miRNA expression data (miRNA)) were used for classification to provide comprehensive and complementary information on the diseases. Only samples with matched mRNA expression, DNA methylation, and miRNA expression data were included in our study. The details of the datasets are listed in Table 15. The PAM50 breast cancer subtype information of TCGA BRCA patients was acquired through TCGAbiolinks16. Different omics data in the ROSMAP dataset were acquired from AMP-AD Knowledge Portal17. ROSMAP is composed of ROS and MAP, which both are longitudinal clinical-pathologic cohort studies of AD from Rush University19.Omics data of LGG, KIPAN, and BRCA, as well as the grade information of LGG patients, were acquired from The Cancer Genome Atlas Program (TCGA) through Broad GDAC Firehose. PAM50 is a 50-gene signature that classifies breast cancer into five molecular subtypes: normal-like, basal-like, human epidermal growth factor receptor 2 (HER2)-enriched, Luminal A, and Luminal B13, analysis of AD with multiple omics data types is an emerging field. For example, Jiang et al.20 analyzed mRNA and miRNA expression data to identify active transcription factors and miRNA regulatory pathways in AD to better understand the pathology of AD. Humphries et al.21 combined RNA sequencing and DNA methylation data to identify gene networks specific to late-onset AD. However, methods that directly address the accurate identification of AD patients from normal age-matched people with machine learning algorithms are still limited. Here, to demonstrate the generalization ability of MOGONET to different diseases and medical applications, we also applied MOGONET on AD patient classification using the ROSMAP dataset, where AD patients and NC subjects were selected for the classification task in our experiment.For kidney cancer type classification, the differences among chromophobe renal cell carcinoma (KICH), clear renal cell carcinoma (KIRC), and papillary renal cell carcinoma (KIRP) can be clearly observed in the omics data. Therefore, kidney cancer type classification is the simplest task among these classification tasks and serves more as a proof-of-concept experiment for multi-class applications. On the other hand, while multi-omics integration methods have been well studied for cancersWe compared the classification performance of MOGONET with existing supervised multi-omics integration algorithms. We also performed comprehensive ablation studies to demonstrate the necessity of different components in MOGONET. To compare the effectiveness of different multi-omics integration methods, we randomly selected 30% of the samples in a dataset as the test set and the remaining 70% of the samples as the training set. The test set was constructed by preserving the class distribution in the original dataset. To evaluate the performance of the compared methods, we used accuracy (ACC), F1 score (F1), and area under the receiver operating characteristic curve (AUC) for binary classification tasks, and we used accuracy (ACC), average F1 score weighted by support (F1_weighted), and macro-averaged F1 score (F1_macro) for multi-class classification tasks. We evaluated all the methods on five different randomly generated training and testing splits, and the mean and standard deviation of the evaluation metrics across these five experiments were reported.6. The implementation in the GRridge R package was used. (8) Block PLSDA. Block PLSDA is a multi-omics integration method that projects data to latent structures with discriminant analysis. Block PLSDA integrates multiple types of omics data measured on the same set of samples to classify a discrete outcome. Block PLSDA is one of the supervised analysis methods included in DIABLO4. (9) Block sPLSDA. Block sPLSDA is block PLSDA with additional sparse regularization, which can select relevant features from the dataset. It is also a supervised analysis method within DIABLO. Implementations in the mixOmics R package22 were used for block PLSDA and block sPLSDA. Block PLSDA and block sPLSDA represent the state-of-the-art approaches for supervised multi-omics integration and classification. Among the tested methods, KNN, SVM, Lasso, RF, XGBoost, and NN were trained with the direct concatenation of the preprocessed multi-omics data as input. All methods were trained with the same preprocessed data. The classification results for ROSMAP, LGG, BRCA, and KIPAN are shown in Tables We compared the classification performance of MOGONET with the following nine existing classification algorithms for omics data: 1) K-nearest neighbor classifier (KNN). Label predictions were made by voting of KNN in the training data. 2) Support vector machine classifier (SVM). (3) Linear regression trained with L1 regularization (Lasso). In Lasso, an individual model was trained to predict the probability of each class, and the class predicted with the highest probability was selected as the final prediction of the class label for the entire model. 4) Random forest classifier (RF). (5) Gradient tree boosting-based classifier implemented in the XGBoost package (XGBoost). (6) Fully connected NN classifier. Deep fully connected NN were trained with cross-entropy loss. (7) Adaptive group-regularized ridge regression (GRridge) Support  Random f K-nearesFrom Tables 15. Note that for the same method, results in Table Different subtypes of BRCA using PAM50 classification may indicate different tumorigenesis mechanisms, and some subtypes may be closer in terms of molecular features than the others. Therefore, we further evaluated the performance of different methods on BRCA PAM50 subtype classification with the following two additional definitions of labels while trained with the labels of five subtypes. One was the binary classification of normal-like vs. non-normal-like subtypes, where the non-normal-like category included the rest four different subtypes  NN_NN: fully connected NN with the same number of layers and the same dimensions of hidden layers as the GCN part in MOGONET were used for omics-specific classification. A fully connected NN with the same number of layers as VCDN was used for multi-omics integration. However, instead of constructing the cross-omics discovery tensor, label distribution from each omics data type was directly concatenated to a vector as the input of the multi-omics integration network. (2) NN_VCDN: the omics-specific classification component was the same as NN_NN without utilizing GCNs. The multi-omics integration component utilized VCDN, which was the same as MOGONET. (3) MOGONET_NN: the omics-specific classification component utilized GCN, which was the same as MOGONET. The multi-omics integration part was the same as NN_NN without utilizing VCDN. Note that MOGONET_NN itself is also a novel approach. To the best of our knowledge, there is no existing method that applies GCNs to supervised multi-omics data classification problems.3 = 8 in MOGONET. While VCDN can effectively utilize the cross-view correlation in the label space, such advantage could be limited when the number of distinct labels was small. Moreover, in the application of VCDN to human action recognition, its advantage compared with NN became more obvious when dealing with complex datasets with over ten classes23. Nevertheless, exploring cross-view correlations was still essential to multi-omics classification as we observed that MOGONET produced better results than MOGONET_NN in all classification tasks under different evaluation metrics. Another interesting observation was that while MOGONET consistently outperformed MOGONET_NN, NN_VCDN failed to consistently outperform NN_NN in all the classification tasks. One possible explanation of this is related to the construction of the cross-omics discovery tensor. Since the input of VCDN was constructed by multiplying the class probabilities predicted by each omics-specific classifier, the prediction noise or error might be amplified if the omics-specific classifiers were not effective. Therefore, GCNs were needed for effective omics-specific learning to fully exploit the advantages of VCDN, and these two components could be trained jointly to achieve superior results for multi-omics classification tasks.As shown in Tables While we used three omics data types in our classification tasks, MOGONET can also be extended to accommodate different numbers of omics data types. To demonstrate the effectiveness of MOGONET with different choices of data modalities, we compared its performance with other methods on the BRCA dataset using only two types of omics data: mRNA expression data and DNA methylation data , MOGONET with two types of omics data , and the omics-specific GCNs trained with single-omics data type before integration . The results are shown in Fig. k, which determines the threshold of affinity values adaptively when constructing the weighted sample similarity networks for omics-specific GCNs . From Fig. k did influence the classification performance of MOGONET as the performance fluctuated with the change of k. However, MOGONET was still robust to the change of k as it consistently outperformed existing methods under different k values. The only exception was when k = 7 in the ROSMAP dataset. In this case, while GRridge yielded higher AUC than MOGONET, MOGONET still produced higher ACC and F1 than GRridge.One important hyper-parameter in MOGONET is CNs (Eq. ). In our24 in the mixOmics R package22 to identify important features in the ROSMAP, BRCA, and LGG datasets  terms that are significantly enriched in a set of genes. To account for multiple testings and control the false discovery rate (FDR), the Benjamini–Hochberg procedure was applied, and the adjusted p values were reported.For genes of the top-ranked mRNA expression features and genes inferred from the top-ranked DNA methylation features, we applied the ToppGene SuiteAPLN and KIF5A were significantly enriched, including apelin receptor binding  and central region of growth cone . Moreover, the apelin domain was also significantly enriched by the genes identified from mRNA expression (p = 1.15E−2). Apelin has been proposed as a promising target for AD27. Apelin are expressed in various parts of the central nervous system in humans28 and plays an important role in the pathogenesis of AD27. For example, it has been shown that apelin may involve in the regulation of Tau phosphorylation and amyloid-β accumulation, thus affects the pathophysiology of AD29. Moreover, as a key isoform of kinesin-1, KIF5A is critical in facilitating anterograde mitochondrial transport in neurons30. Wang et al.31 also reported a potential role of KIF5A deficiency in AD-relevant axonal mitochondrial traffic abnormalities and suggested therapeutic value in AD treatment through restoring KIF5A function. For genes related to the identified DNA methylation features by MOGONET, several GO terms related to the inflammatory process were significantly enriched, including myeloid leukocyte activation , positive regulation of cytokine secretion , and positive regulation of inflammatory response . Several studies have suggested the involvement of inflammation in AD pathogenesis by showing the increased levels of inflammatory cytokines in AD33. It has also been shown that the secretion of cytokines and chemokines could regulate the activity of microglia and astrocytes in AD, which plays a critical role in inflammation and neurodegeneration34. On the other hand, for biomarkers identified by sPLSDA, some GO terms related to the solute carrier family were significantly enriched, such as amino acid sodium symporter activity . Moreover, APLN and another kinesin-1-related gene KIF5B were also identified.For AD patient classification, eight mRNA features, five DNA methylation features, and 17 miRNA features were identified by MOGONET as the top 30 important biomarkers Table . For gen35 found that the expression level of hsa-miR-423 was significantly altered in the hippocampus and medial frontal gyrus for early and late-stage AD patients comparing to control samples, where both hippocampus and medial frontal gyrus were regions primarily affected by AD pathology. In addition, Nagaraj et al.36 reported that hsa-miR-33a was deferentially expressed in blood plasma between AD patients and age-matched controls. For identified mRNA expression and DNA methylation biomarkers, Hohman et al.37 identified the lower expression of TMC4 was associated with the amyloid deposition-related decline in executive function. Besides, the overexpression of CDK18 could regulate the phosphorylation of Tau protein in human brain while hyper-phosphorylated Tau is known to be associated with the pathology of AD38.Moreover, highly-ranked genes and miRNAs identified by MOGONET have also been shown to be associated with AD. Cogswell et al.p = 3.51E−2) and response to progesterone . For example, the progesterone receptor is often used as a positive prognostic marker in estrogen receptor-α (ERα)+ breast cancers39. Mohammed et al.40 further demonstrated that an activated progesterone receptor could function as a proliferative brake in ERα+ breast tumors via modulation of ERα chromatin binding and transcriptional activity. Several GO terms related to the solute carrier family were also significantly enriched, such as neurotransmitter transmembrane transporter activity  and symporter activity . Among the identified solute carrier family genes, SLC6A14 has been shown to be one of the glucose metabolism-related genes that were downregulated by metformin in triple-negative breast cancer (TNBC)41. Additionally, prosaposin receptor activity  was significantly enriched for genes related to the identified DNA methylation features by MOGONET. Wu et al.42 demonstrated that prosaposin could upregulate estrogen receptor alpha expression through the mitogen-activated protein kinase (MAPK)-signaling pathway and suggested that prosaposin may be involved in breast cancer development and progression. On the other hand, for genes identified from mRNA expression data by sPLSDA, no significantly enriched GO term was found. For DNA methylation features identified by sPLSDA, several biological process terms were significantly enriched, including positive regulation of MAPK cascade  and regulation of glucose metabolic process . Studies have shown that hormone-bound steroid receptors activate different complex MAPK-related pathways in breast cancer cells44. It has also been shown that glucose and factors related to glucose metabolism could contribute to breast cancer development45.For BRCA PAM50 subtype classification, 15 mRNA features, nine DNA methylation features, and six miRNA features were identified by MOGONET as the top 30 important biomarkers Table . For gen46 demonstrated that SOX11 was critical for regulating the expression of many genes that define the basal-like subtype. They also demonstrated that SOX11 was related to the invasion and migration of basal-like breast tumors. FABP7 has also been shown to be related to different breast cancer subtypes. Cordero et al.47 uncovered the critical role of FABP7 in metabolic reprogramming of HER2+ breast cancer cells as well as HER2+ breast cancer brain metastasis. Zhang et al.48 identified a novel subgroup within the basal-like breast tumors with higher expression of FABP7 that showed significantly better clinical outcomes. For identified miRNA biomarkers, there have been several studies investigating the association between miRNA-205 and breast cancer. Specifically, miRNA-205 is generally downregulated and exhibits a tumor-suppressive function in breast cancer49. While miRNA-205 expression is decreased in breast cancer, the relative levels of downregulation vary across different subtypes. For example, miRNA-205 is upregulated in estrogen/progesterone+ breast cancer compared with HER2+ breast cancer50, while TNBCs usually express the least miRNA-205 among different subtypes52. Moreover, different studies also showed that metastatic breast cancers had lower expression levels of miRNA-205 than non-metastatic breast cancers53. In addition, miRNA-187 was identified as an independent prognostic factor in breast cancer, where its overexpression was associated with a more aggressive phenotype54.Moreover, highly-ranked genes and miRNAs identified by MOGONET have also been shown to be associated with breast cancer. For example, Shepherd et al.The rapid advancement of omics technologies has enabled personalized medicine using molecular-level data with unprecedented details. Previously, labeled biomedical data have been scarce due to the high expense for collecting and annotating data and the lack of knowledge about subtypes of diseases. Consequently, most existing multi-omics integration methods focus on unsupervised methods without additional phenotypic information and try to extract biological insights from the identified clusters of samples. However, thanks to the rapid development of omics technologies and personalized medicine, as well as large consortium studies such as TCGA and ROSMAP, labeled omics datasets with detailed annotations are becoming available at an unprecedented volume and speed. Therefore, it has become more and more important to take advantage of these labeled omics data to better predict essential phenotypes or traits  on new samples. To this end, we propose MOGONET, a supervised multi-omics integration method for biomedical classification tasks based on deep multi-view learning, where we consider each omics data type as a view of the samples. We utilized GCNs for omics-specific learning and VCDN for multi-omics integration at the high-level label space. MOGONET also effectively identified meaningful potential biomarkers in each omics data type that showed a strong association with the diseases. To summarize, MOGONET is an innovative deep learning-based multi-omics classification algorithm with both superior performance and good interpretability.Comparing to fully connected networks, GCNs can utilize both the features and the geometrical structures of the data. While commonly used fully connected networks can only be trained on structured data, GCNs can also generalize NN for arbitrarily structured graphs. This suggests that our GCN-based method is flexible and can be potentially generalized to include more data types to boost the classification performance in the future. We also demonstrated that VCDN can effectively classify multi-omics data by integrating the omics-specific classification produced by GCNs at the label space. As each element in the input of VCDN is constructed by multiplying class probabilities from different classifiers, VCDN might be more sensitive to the noise or error produced in omics-specific learning. Therefore, effective omics-specific classification through GCNs is needed to fully utilize the superiority of VCDN. Through ablation studies, we demonstrated that both GCNs and VCDN were essential to effective multi-omics data classification, while GCNs might play a more essential role in the biomedical classification tasks in this paper.c classes and m different data types, an individual GCN can be trained for each data type. For multi-omics classification, the label distribution generated by each omic-specific GCN can be integrated by either direct concatenation as in MOGONET_NN or constructing a cm-dimensional cross-omics discovery vector with the similar fashion as in MOGONET. Therefore, MOGONET is a supervised multi-omics classification framework that can be generalized to accommodate many different omics data types.While we only utilized mRNA expression, DNA methylation, and miRNA expression data for the multi-omics classification tasks in this paper, both the omics-specific GCNs and the multi-omics integration component can be extended to accommodate different or more types of data. Specifically, for a classification task with MOGONET is a framework for classification tasks with multi-omics data. The workflow of MOGONET can be summarized into three components: (1) preprocessing. Preprocessing and feature preselection were performed on each omics data type individually to remove noise, artifacts, and redundant features that may deteriorate the performance of the classification tasks. (2) Omics-specific learning via GCNs. For each omics data type, a weighted sample similarity network was constructed from the omics features. Then, a GCN was trained using both the omics features and the corresponding similarity network for omics-specific learning. (3) Multi-omics integration via VCDN. A cross-omics discovery tensor was calculated using the initial class probability predictions from all the omics-specific networks. A VCDN was then trained with the cross-omics discovery tensor to produce the final predictions. VCDN can effectively learn the intra-omics and cross-omics label correlations in the higher-level label space for better classification with multi-omics data. MOGONET is an end-to-end model, where both omics-specific GCNs and VCDN are trained jointly. We describe each component in detail in the following sections.To remove noise and experimental artifacts in the data and better interpret the results, proper preprocessing for omics data is essential. First, for DNA methylation data, only probes corresponding to the probes in the Illumina Infinium HumanMethylation27 BeadChip were retained for better interpretability of the results. The number of features for each dataset and each omics data type is listed in Table F-value was calculated sequentially using the training data to evaluate whether a feature was significantly different across different classes, where FDR controlling procedures were applied for multiple-testing compensation. However, selecting too few features might also result in only selecting highly correlated features, which could potentially restrain the models from taking advantage of complementary information from diverse features. To avoid this situation, we determined the number of preselected features for each omics data type with an additional rule, i.e., the first principal component of the data after feature preselection should explain <50% of the variance. We also demonstrated that MOGONET could produce consistent results under a wide range of different numbers of preselected features ( is the input of the lth layer and Wl)( is the weight matrix of the lth layer. σ(⋅) denotes a non-linear activation function. For effective training of GCNs, Kipf and Welling55 further modified the adjacency matrix A as:I is the identity matrix.By viewing each sample as a node in the sample similarity network, the goal of each GCN in MOGONET is to learn a function of features on a graph A is constructed by calculating the cosine similarity between pairs of nodes and edges with cosine similarity larger than a threshold ϵ are retained. Specifically, Aij, which is the adjacency between node i and node j in the graph, is calculated as:xi and xj are the feature vectors of node i and node j, respectively. i and j. The threshold ϵ is determined given a parameter k, which represents the average number of edges per node that are retained including self-connections:I(⋅) is the indicator function and n is the number of nodes. The parameter k for generating the adjacency matrix in Eq.  can be trained with Xtr and ith row in ith training sample and c denotes the number of classes in the classification task. Therefore, both the features and the geometrical structure of the training data are utilized in learning the classification task.Although GCNs have been widely utilized in unsupervisedxte and the training samples Xtr. Therefore, given Xtrte, GCN(⋅), we have xte.For a new test sample entclass1pt{minimaith omics data type, a omic-specific GCN, GCNi(⋅), is trained with training data jth row in jth training sample from the ith omics data type. Therefore, the loss function for GCNi(⋅) can be written as:LCE(⋅) represents the cross-entropy loss function. jth training sample and kth element in the vector In MOGONET, to perform omic-specific classification, we construct a multi-layer GCN for each omics data type. Specifically, for the 65. However, it is always challenging to align various views properly without causing negative influence. On the other hand, VCDN23 can exploit the higher-level cross-omics correlations in the label space, as different types of omics data can provide unique class-level distinctiveness. VCDN is designed to learn the higher-level intra-view and cross-view correlations in the label space and has shown significant improvements in human action recognition tasks. In MOGONET, we utilize VCDN to integrate different omics data types for classification. Moreover, while the original form of VCDN was designed for samples with two views23, we further generalize it to accommodate an arbitrary number of data types and demonstrated with three types of omics data: mRNA expression, DNA methylation, and miRNA expression.Existing methods utilizing multi-view data for biomedical classification tasks either directly concatenate features from different views or learn to fuse data by learning the weight of each view or fusing features from different views in a low-level feature spacejth sample from different omics data types Cj is calculated as:ath entry of Since mRNA expression data, DNA methylation data, and miRNA expression data are used in our experiments, for simplicity, we first demonstrate how to extend VCDN to accommodate three views. For the predicted label distribution of the Cj is reshaped to a c3 dimensional vector cj and is forwarded to VCDN(⋅) for the final prediction. VCDN(⋅) is designed as a fully connected network with the output dimension of c. The loss function of VCDN(⋅) can be written as:Then, the obtained tensor VCDN(⋅) could reveal the latent cross-view label correlations and help to improve the learning performance. By utilizing VCDN(⋅) to integrate initial predictions from different types of omics data, the final prediction made by MOGONET is based on both the omics-specific predictions and the learned cross-omics label correlation knowledge.To this end, m omics data types, each element in Cj can be calculated as:Extension of MOGONET to a different number of views can be performed in a similar fashion. For data with Cj is reshaped to a cm dimensional vector, and VCDN(⋅) can be trained in the same way as Eq. . The loss function for NN(⋅) can be written as:NN(⋅) shares the same number of layers as VCDN(⋅), while the dimensionality of the input data is different.For MOGONET_NN in the ablation study, the label distribution from each omics data type is directly concatenated to a longer vector as the input of the multi-omics integration network γ is a trade-off parameter between the omics-specific classification loss and the final classification loss from VCDN(⋅). We set γ = 1 in all our experiments. MOGONET is an end-to-end model, and all networks are trained jointly. For training MOGONET, we first pretrain each of the omics-specific GCN individually to get a good initialization of the GCN. Then, during one epoch of the training process, we first fix VCDN(⋅) and update GCNi(⋅), i = 1, 2, 3 for each omics data type to minimize the loss function L. Then we fix the omics-specific GCNs and update VCDN(⋅) to minimize L. Omics-specific GCNs and VCDN are updated alternatively until convergence.In summary, in our experiments where three omics data types are used, the total loss function of MOGONET can be written as:ablation approach has been widely adopted for feature importance ranking and feature selection in NN67. Using this approach, we analyzed the contribution of each feature in different types of omics data by assigning the feature to zero and calculating the classification performance decrease on the test set comparing to using all the features. Features with the largest performance drop were considered to be the most important ones. We used the F1 score to measure the performance drop for binary classification tasks and F1_macro for multi-class classification tasks. To account for the randomness in the training process, we performed five repeated experiments in a dataset and summarized the results by summing up the performance decrease in the repeated experiments. Since each omics data type carried the same importance in MOGONET during the construction of the cross-omics discovery tensor, while different numbers of features were preselected for training in different omics data types, we further scaled the feature importance of each omics data type by the number of preselected features. Finally, we ranked the importance of each feature.Identifying biomarkers is essential for interpreting the results and understanding the underlying biology in biomedical applications. Over the years, there have been extensive studies on determining the importance of features for NN. Since the input of MOGONET is scaled to  during preprocessing, we can remove the signal from a feature by setting it to zero. Therefore, the importance of a feature to the classification task can be measured by the performance decrease after the feature is removed. Such Further information on research design is available in the Supplementary InformationPeer Review FileReporting Summary