SynOmics: integrating multi-omics data through feature interaction networks
Muhtasim Noor Alif, Wei Zhang
TL;DR
SynOmics is a new framework that improves multi-omics data integration using graph networks, leading to better biomedical predictions.
Contribution
Introduces SynOmics, a graph convolutional network framework that models both within- and cross-omics feature interactions for improved integration.
Findings
SynOmics outperforms existing methods in multi-omics integration across biomedical classification tasks.
The framework captures both intra-omics and inter-omics dependencies through parallel learning.
It shows potential for biomarker discovery and clinical applications.
Abstract
The integration of multi-omics data is essential for achieving a comprehensive understanding of molecular systems and enhancing the performance of predictive models in biomedical research. However, many existing models have limited capacity to capture cross-omics feature interactions, which hinders the depth of integration. In this study, we introduce SynOmics, a graph convolutional network framework designed to improve multi-omics integration by constructing omics networks in the feature space and modeling both within- and cross-omics dependencies. By incorporating both omics-specific networks and cross-omics bipartite networks, SynOmics enables simultaneous learning of intra-omics and inter-omics relationships. Unlike traditional approaches that rely on early or late integration strategies, SynOmics adopts a parallel learning strategy to process feature-level interactions at each…
Genes, proteins, chemicals, diseases, species, mutations and cell lines named across the full text — each resolved to its canonical identifier and authoritative record.
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Figure 5| Name | Definition |
|---|---|
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| Number of samples |
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| Number of features of omics1 |
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| Number of features of omics2 |
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| Number of classes |
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| Feature matrix of omics1 |
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| Feature matrix of omics2 |
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| Adjacency matrix for omics1 features |
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| Adjacency matrix for omics2 features |
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| Bipartite incidence matrix from omics1 to omics2 |
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| Bipartite incidence matrix from omics2 to omics1 |
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| Weight matrix of |
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| Weight matrix of |
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| Weight matrix of |
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| Weight matrix of |
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| Intra-omics output feature matrix of the |
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| Intra-omics output feature matrix of the |
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| Inter-omics output feature matrix of the |
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| Inter-omics output feature matrix of the |
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| Aggregated output feature matrix of the |
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| Aggregated output feature matrix of the |
| Dataset | Target variables and number of samples |
|---|---|
| B | ER+{333}, ER-{106} |
| R | HER2+{70}, HER2-{364} |
| C | PR+{303}, PR-{134} |
| A | TN{67}, non-TN{334} |
| Stage-1{208}, Stage-2{711}, Stage-3{207}, Stage-4{22} | |
| L | Short survival duration (<13 months){54} |
| U | Long survival duration (>56 months){53} |
| A | Short disease-free duration (<12 months){25} |
| D | Long disease-free duration (>28 months){87} |
| Stage-1{237}, Stage-2{107}, Stage-3{69}, Stage-4{21} | |
| O | Short survival duration (<16 months){31} |
| Long survival duration (>50 months){77} | |
| V | Short disease-free duration (<27 months){74} |
| Long disease-free duration (>43 months){25} |
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|---|---|---|---|---|---|---|---|---|---|
| ER | AUC | 0.9684 | 0.9612 | 0.9243 | 0.9543 |
| 0.9405 | 0.9422 | 0.9308 |
| MCC |
| 0.8420 | 0.7967 | 0.8236 | 0.8455 | 0.8168 | 0.7920 | 0.7452 | |
| HER2 | AUC | 0.8532 |
| 0.8033 | 0.7859 | 0.8398 | 0.8143 | 0.8265 | 0.7217 |
| MCC |
| 0.4733 | 0.4744 | 0.4204 | 0.4286 | 0.4505 | 0.4591 | 0.2710 | |
| PR | AUC | 0.9210 |
| 0.8732 | 0.9195 | 0.8457 | 0.8813 | 0.8783 | 0.8494 |
| MCC | 0.7275 |
| 0.6811 | 0.7265 | 0.4197 | 0.6813 | 0.6500 | 0.6105 | |
| TN | AUC |
| 0.9577 | 0.9109 | 0.9627 | 0.9611 | 0.9312 | 0.9472 | 0.9219 |
| MCC | 0.7588 | 0.7636 | 0.6986 |
| 0.7725 | 0.7122 | 0.7001 | 0.6639 |
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|---|---|---|---|---|---|---|---|---|---|---|
| L | SD | AUC |
| 0.6567 | 0.5444 | 0.6297 | 0.5806 | 0.6528 | 0.6330 | 0.6313 |
| U | MCC |
| 0.3328 | 0.2072 | 0.2088 | 0.0018 | 0.2921 | 0.1941 | 0.2044 | |
| A | DFD | AUC | 0.6043 | 0.5814 | 0.5838 | 0.5596 |
| 0.6035 | 0.5627 | 0.5932 |
| D | MCC |
| 0.2647 | 0.2741 | 0.0878 | 0.1605 | 0.2868 | 0.0573 | 0.0842 | |
| O | SD | AUC |
| 0.6592 | 0.6161 | 0.6593 | 0.4755 | 0.6355 | 0.6423 | 0.4739 |
| MCC | 0.3455 |
| 0.2614 | 0.1891 | −0.0010 | 0.2730 | 0.0878 | −0.0364 | ||
| V | DFD | AUC |
| 0.4401 | 0.4472 | 0.3991 | 0.5522 | 0.4394 | 0.4473 | 0.4117 |
| MCC |
| 0.0338 | 0.0421 | −0.1626 | 0.0370 | 0.0176 | −0.0252 | −0.0491 |
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| BRCA Stage | AUC | 0.5430 |
| 0.5456 | 0.5034 | 0.4984 | 0.5156 | 0.5269 | 0.5163 |
| MCC | 0.0595 |
| 0.0388 | 0.0423 | 0.0015 | 0.0366 | 0.0007 | 0.0109 | |
| LUAD Stage | AUC | 0.5331 | 0.5452 | 0.5091 |
| 0.5328 | 0.4976 | 0.5337 | 0.5482 |
| MCC | 0.0451 | 0.0622 | −0.0080 | 0.0067 | 0.0047 | −0.0045 | 0.0435 |
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| AUC | MCC | ||||||
|---|---|---|---|---|---|---|---|
| Both | Intra | Inter | Both | Intra | Inter | ||
| BRCA | ER |
| 0.9612 | 0.9582 |
| 0.8388 | 0.7791 |
| HER2 |
| 0.8440 | 0.5993 |
| 0.4542 | 0.1955 | |
| PR |
| 0.9156 | 0.6995 |
| 0.7246 | 0.3815 | |
| TN |
| 0.9594 | 0.8377 | 0.7588 |
| 0.4912 | |
| LUAD | SD |
| 0.6686 | 0.5734 |
| 0.3379 | 0.2378 |
| DFD |
| 0.5332 | 0.5732 |
| 0.2272 | 0.2737 | |
| OV | SD |
| 0.6247 | 0.5411 |
| 0.2834 | 0.2498 |
| DFD |
| 0.4451 | 0.4485 |
| 0.0313 | 0.0295 |
- —National Science Foundation10.13039/100000001
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Taxonomy
TopicsBioinformatics and Genomic Networks · Advanced Graph Neural Networks · Graph Theory and Algorithms
Introduction
Biological systems and diseases are influenced by complex interactions among various omics types. To gain a deeper understanding of complex diseases, it is crucial to comprehend how these omics types interact both within and across their respective layers [1, 2]. Multi-omics integration is essential for effectively capturing these interactions [3, 4]. With the advancement of high-throughput sequencing technologies, researchers can now generate high-quality, precise multi-omics data [5]. While each omics technology provides only a partial view of biological complexity, integrating multiple omics types offers a more comprehensive understanding of the underlying biological processes. However, this integration remains a challenging task due to the complex relationships among the different omics types.
Over time, a wide range of methods have been developed for integrating multi-omics data, enabling various downstream applications such as disease and cancer research, biomarker discovery, and drug development [3, 4, 6–8]. These approaches leverage complementary information from multiple omics types—such as genomics, transcriptomics, proteomics, and metabolomics—to provide deeper biological insights and improve the accuracy of predictions in biomedical research. Since molecular structures and interactions often exhibit graph-like patterns [9–11], the rise of graph-based deep learning techniques has led to the widespread use of graph neural networks (GNNs) [12] for multi-omics data integration [13–15].
Wang et al. [16] introduced the Sample Similarity Network Fusion (SNF) approach to create a unified graph representation across multiple omics types. Although SNF does not involve deep learning, it has been widely adopted to generate joint networks that can serve as inputs to GNN models. Building on this, Li et al. [17] proposed MoGCN , which extends SNF by incorporating a graph convolutional network (GCN) [18] for predictive modeling. MoGCN employs an early integration strategy using autoencoders to directly combine input omics layers and learn a joint representation. However, this approach does not separately model omics-specific relationships, potentially overlooking important intra-omics information. In contrast, late integration methods aim to capture omics-specific interactions before aggregating information across omics types. For example, MOGONET [19] uses omics-specific GCNs to learn intra-omics representations and then integrates these embeddings through a correlation network. Similarly, SUPREME [20] trains omics-specific GCNs independently before merging the resulting embeddings for final prediction. While these approaches typically rely on fixed sample similarity networks, MOGLAM [21] employs a dynamic graph convolutional approach that updates the similarity networks as the model learns omics-specific interactions. Several recent approaches leverage attention mechanisms to fuse heterogeneous omics. For example, OmicsFormer [22] tokenizes modality-specific features and employs multi-head self-attention to model long-range, cross-omics dependencies, yielding a joint sample representation without prespecifying a similarity graph. MOGAT [23], in contrast, builds per-omics sample networks and applies graph attention to learn neighborhood weights within each modality, followed by attention-based (or gated) fusion across modalities, producing a final embedding tailored to the prediction task.
Most existing models construct sample similarity networks to facilitate learning by sharing information across samples [17, 19, 20, 23–25]. While this approach can be effective, it has several notable limitations. One major drawback is its inability to capture the inherent interactions among omics features and molecular entities—interactions that exist independently of individual samples and are fundamental to biological systems. Although sample similarity networks may reflect broad population-level patterns, they often overlook critical feature-level mechanisms such as gene regulation and pathway dependencies, resulting in the loss of important biological insights. Furthermore, omics data are typically characterized by high dimensionality and a limited number of samples. This imbalance, where the number of features greatly exceeds the number of samples, poses a significant challenge for effective integration. In such cases, sample similarity networks often fail to capture the complex dynamics and relationships that underlie real-world biological systems.
Another limitation of traditional approaches is their tendency to model only intra-omics relationships using graph-based learning, while treating inter-omics dependencies separately through domain-agnostic architectures such as multilayer perceptrons (MLPs) or standard autoencoders [17, 19, 20, 25]. This decoupled design restricts the model’s ability to capture coordinated signals across omics types, ultimately limiting the quality of the learned representations. In biological systems, omics layers are tightly interconnected—genomic alterations can drive transcriptomic changes, which may in turn influence epigenetic modifications or proteomic responses. Ignoring these interdependencies or modeling them in a disjointed manner can result in fragmented or biologically shallow representations. GNNs offer a promising alternative by enabling structured information flow across omics layers. However, their application in cross-omics integration remains limited, primarily due to the challenge of defining biologically meaningful interaction networks between heterogeneous omics types—e.g. linking mRNA targets to their regulatory microRNAs (miRNAs). While intra-omics networks often leverage well-established resources such as functional interaction databases or co-expression networks, cross-omics relationships are typically less well-defined. Constructing these inter-omics connections often requires integrating prior biological knowledge, curated interaction databases (e.g. miRNA-mRNA targeting), or statistical inference from multi-omics datasets. In the absence of robust strategies for defining these cross-layer links, the full potential of GNNs for cross-omics integration remains underutilized.
In this study, we introduce SynOmics, a GCN-based framework designed to enhance multi-omics integration through feature-level learning. Our method integrates both intra-omics and inter-omics information by leveraging omics-specific networks and cross-omics bipartite networks. By operating in the feature space rather than the sample space, SynOmics provides a more nuanced representation of biological interactions. The bipartite networks offer a structured approach for capturing inter-omics information flow [26–29]. Unlike conventional early or late integration strategies, SynOmics employs a parallel learning approach to jointly model within- and cross-omics patterns. Experimental results demonstrate that SynOmics performs robustly across a variety of tasks, highlighting its potential for heterogeneous data integration, clinical biomarker discovery, and improved clinical diagnostics.
Materials and methods
In this section, we first describe the standard formulations of graph convolutional and bipartite graph convolutional layers. We then present SynOmics, our proposed framework for robust multi-omics integration, which leverages feature-level graph convolution both within and across omics types. Finally, we outline the evaluation methods used to assess the performance of SynOmics.
Graph convolution
Omics data often exhibit graph-like structures due to complex regulatory mechanisms, making GCNs [18] well-suited for modeling these interactions through efficient node embedding and message propagation. Accordingly, SynOmics employs graph convolution to capture intra-omics interactions within each omics type.
In traditional GCNs, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathbf{X})\end{document} is an attributed graph, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{V} = {v_{1}, v_{2},..., v_{n}}\end{document} is the set of nodes, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{E}\end{document} is the set of edges between the nodes, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{X} = {x_{1}, x_{2},..., x_{n}} \in \mathbb{R}^{n \times d}\end{document} is the feature matrix, where each row \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} x_{i}\end{document} represents the feature vector of node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} v_{i}\end{document} . The adjacency matrix for an unweighted graph is defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{A} \in {0,1}^{n \times n}\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} A_{ij}=1\end{document} if there is an edge between node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} v_{i}\end{document} and node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} v_{j}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} A_{ij}=0\end{document} otherwise. Then the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} l\mathrm{th}\end{document} layer operations are specified by:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \mathbf{Z}^{(l+1)}= \sigma(\hat{\mathbf{A}}\mathbf{Z}^{(l)}\mathbf{W}^{(l)})\end{align*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}^{(l+1)}\end{document} is the output feature matrix of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} {l\mathrm{th}}\end{document} layer, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \sigma (.)\end{document} is the non-linear activation function, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \hat{\mathbf{A}}\end{document} is the normalized version of the adjacency matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{A}\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{W}^{(l)}\end{document} is the associated weight matrix of layer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} l\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}^{(l)}\end{document} is the input feature matrix and notably, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}^{(0)}=\mathbf{X}\end{document} . The normalized adjacency matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \hat{\mathbf{A}}\end{document} is defined by:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \hat{\mathbf{A}}=\mathbf{D}^{-1/2}\tilde{\mathbf{A}}\mathbf{D}^{-1/2}\end{align*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \tilde{\mathbf{A}}=\mathbf{A}+\mathbf{I}\end{document} denotes the adjacency matrix with self connections, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{D}\end{document} is the degree matrix of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{A}\end{document} .
Bipartite graph convolution
To account for the influence of one omics type on another, we extend GCNs to bipartite networks for modeling cross-omics interactions. This approach, known as bipartite graph convolution (BGCN), has been adopted in prior studies [30–33] and enables structured information flow between two distinct datasets.
Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{U}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{V}\end{document} be two sets of nodes, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{U}= {u_{1}, u_{2},..., u_{n}}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{V}= {v_{1}, v_{2},..., v_{m}}\end{document} . Then, the bipartite graph is defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{B}=(\mathcal{U}, \mathcal{V}, \mathcal{E_{B}})\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{E_{B}}\in \mathcal{U}\times \mathcal{V}\end{document} is the set of edges between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{U}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{V}\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{B}{u} \in {0,1}^{n\times m}\end{document} the incidence matrix from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{U}\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{V}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{B}{v} \in {0,1}^{m\times n}\end{document} the incidence matrix from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{V}\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{U}\end{document} , which is notably the transpose of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{B}{u}\end{document} for an undirected graph. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} {\mathbf{B}{u}}{(ij)}=1\end{document} if there is an edge between nodes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} u{i}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} v_{j}\end{document} , and 0 otherwise. We define the adjacency matrix of the bipartite graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{B} \in {0,1}^{(n+m)\times (n+m)}\end{document} as:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \mathbf{B} = \begin{pmatrix} \mathbf{0}_{u,u} & \mathbf{B}_{u} \\ \mathbf{B}_{v} & \mathbf{0}_{v,v} \end{pmatrix}\end{align*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{0}{u,u} \in {0}^{n\times n}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{0}{v,v} \in {0}^{m\times m}\end{document} are zero matrices. Given the incidence matrices, we define the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} l\mathrm{th}\end{document} layer of BGCN as following:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathbf{Z}_{v \to u}^{(l+1)}=\sigma(\hat{\mathbf{B}}_{u} \mathbf{Z}_{v}^{(l)} \mathbf{W}_{u}^{(l)}) \end{align*}\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathbf{Z}_{u \to v}^{(l+1)}=\sigma(\hat{\mathbf{B}}_{v} \mathbf{Z}_{u}^{(l)} \mathbf{W}_{v}^{(l)}) \end{align*}\end{document}where, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}{u}^{(l)} \in \mathbb{R}^{n \times d}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}{v}^{(l)} \in \mathbb{R}^{m \times d}\end{document} are the feature matrices of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{U}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{V}\end{document} , respectively. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}{v \to u}^{(l+1)} \in \mathbb{R}^{n \times d^{\prime}}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}{u \to v}^{(l+1)} \in \mathbb{R}^{m \times d^{\prime}}\end{document} are the hidden representations of nodes in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{U}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{V}\end{document} , aggregated from the opposite domain. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \sigma (.)\end{document} denotes the non-linear activation function. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \hat{\mathbf{B}}{u}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \hat{\mathbf{B}}{v}\end{document} are the normalized incidence matrices of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{B}{u}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{B}{v}\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{W}{u}^{(l)} \in \mathbb{R}^{d \times d^{\prime}}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{W}{v}^{(l)}\in \mathbb{R}^{d \times d^{\prime}}\end{document} are the associated weight matrices. The adjacency matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{B}\end{document} in equation (3) is normalized in the same way as equation (2), and then the incidence matrices are extracted to get the normalized incidence matrices \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \hat{\mathbf{B}}{u}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \hat{\mathbf{B}}{v}\end{document} .
SynOmics
SynOmics is a supervised framework for multi-omics integration in biomedical classification tasks (Fig. 1). It employs graph convolution for intra-omics learning and BGCN for modeling inter-omics regulatory interactions. Unlike approaches that rely on sample-level similarities, SynOmics focuses on biologically meaningful regulatory links between features, such as miRNA regulation of mRNA expression, to capture more relevant signals. The framework operates on feature-level networks, where nodes represent molecular features and edges represent their biological relationships. To support this, we adapt the standard GCN and bipartite GCN formulations accordingly. The mathematical notations used in SynOmics are summarized in Table 1.
SynOmics integrates two or more omics using omics-specific feature networks and inter-omics bipartite graphs; per layer it applies two intra-omics and two inter-omics GCNs to produce modality-wise intra/inter embeddings that are (a) combined in the weighted message passing layer or (b) aligned in the dual alignment layer, and (c) the overall architecture concatenates the final-layer outputs from either (a) or (b) and feeds them to a fully connected predictor (schematics use reduced-size matrices).
With the updated mathematical notations, we define the intra-omics graph convolution as:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathbf{Z}_{u}^{(l+1)} = \sigma(\mathbf{W}_{u}^{(l)}\hat{\mathbf{A}}_{u}\mathbf{H}_{u}^{(l)}) \end{align*}\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathbf{Z}_{v}^{(l+1)} = \sigma(\mathbf{W}_{v}^{(l)}\hat{\mathbf{A}}_{v}\mathbf{H}_{v}^{(l)}) \end{align*}\end{document}It is important to note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{H}{u}^{(0)}=\mathbf{X}{u}^{T} \in \mathbb{R}^{p \times n}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{H}{v}^{(0)}=\mathbf{X}{v}^{T} \in \mathbb{R}^{q \times n}\end{document} are the inputs for the first layer. Equations (6) and (7) enable information propagation by aggregating feature information from each node in one omics layer to its connected neighbors, including itself through self-connections.
The inter-omics BGCN is defined as:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathbf{Z}_{v \to u}^{(l+1)}=\sigma(\mathbf{W}_{v \to u}^{(l)} \hat{\mathbf{B}}_{u} \mathbf{H}_{v}^{(l)}) \end{align*}\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathbf{Z}_{u \to v}^{(l+1)}=\sigma(\mathbf{W}_{u \to v}^{(l)} \hat{\mathbf{B}}_{v} \mathbf{H}_{u}^{(l)} ) \end{align*}\end{document}Equations (8) and (9) enable cross-omics information propagation from nodes in one omics layer to their connected counterparts in the other omics layer, as defined by the bipartite network. Specifically, in equation (8), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}^{(l+1)}{v \to u}\end{document} aggregates information for omics1 by collecting signals from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{H}{v}^{(l)}\end{document} , the hidden representation of omics2 from the previous layer, using the normalized bipartite incidence matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \hat{\mathbf{B}}_{u}\end{document} . We apply the Leaky ReLU activation function [34] to obtain hidden representations. SynOmics executes the operations defined in equations (6) through (9) within a single layer, enabling parallel learning of intra- and inter-omics interactions as an alternative to staged integration strategies.
Training
To investigate how training methodology influences downstream performance, we explore two alternative strategies for training SynOmics. The first approach, “weighted message passing, balances the contributions of the intra- and inter-omics modules, providing a straightforward end-to-end training mechanism. This strategy is computationally efficient, faster to train, and easily scalable, making it ideal for rapid experimentation. The second approach, “dual alignment,involves pretraining the GCN layers to enable the model to learn more detailed and comprehensive feature representations. By fine-tuning omics features before applying them to downstream tasks, this method enhances generalization and robustness. The details of these two training strategies are described in the following subsections.
Weighted message passing
For effective integration, it is important to recognize that the two omics signals being considered may not contribute equally to the final representation. These differences can arise from variations in biological relevance to the task, data quality (e.g. noise levels or sparsity), or the strength and consistency of the signal. For example, one omics layer may contain highly informative patterns that are strongly correlated with the target phenotype, while the other may be more heterogeneous or affected by experimental noise, potentially diluting the predictive signal.
To account for these disparities, we introduce a mechanism that balances the contributions of intra-omics and inter-omics information when forming the final node representation. Specifically, we use a weighted integration approach to combine the outputs of the intra-omics and inter-omics GCN modules, as shown in Fig. 1a. This weighted combination allows the model to regulate the influence of each module on the final representation, effectively controlling the flow of information between within-omics and cross-omics signals.
To implement this, we take the hidden features learned from the intra-omics GCNs ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}{u}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}{v}\end{document} ) and the inter-omics GCNs ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}{v \to u}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{Z}{u \to v}\end{document} ), and integrate them using a weighted sum, defined as:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathbf{H}_{u}^{(l+1)} = k \times \mathbf{Z}_{u}^{(l+1)} + (1-k) \times \mathbf{Z}_{v \to u}^{(l+1)} \end{align*}\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathbf{H}_{v}^{(l+1)} = k \times \mathbf{Z}_{v}^{(l+1)} + (1-k) \times \mathbf{Z}_{u \to v}^{(l+1)} \end{align*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} k \in (0, 1)\end{document} is a tunable hyperparameter that controls the relative contributions of intra-omics and inter-omics hidden features. The resulting combined feature representations, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{H}{u}^{(l+1)}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{H}{v}^{(l+1)}\end{document} , are then used as input for the next layer of SynOmics.
At the final layer, we concatenate the hidden representations and apply a fully connected layer for prediction, as illustrated in Fig. 1c. This operation is defined as:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \hat{\mathbf{Y}}=\sigma((\mathbf{H}_{u}^{(L)}||\mathbf{H}_{v}^{(L)})^{T}\mathbf{W})\end{align*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \hat{\mathbf{Y}} \in \mathbb{R}^{n \times nc}\end{document} denotes the predictions, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \sigma (.)\end{document} is the non-linear activation function, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} L\end{document} indicates the last layer, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{W} \in \mathbb{R}^{(p+q) \times nc}\end{document} is the associated weight matrix, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} nc\end{document} represents the number of classes. We use the sigmoid activation function for binary classification and the softmax activation function for multi-class classification. The model is trained using binary cross-entropy ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} BCE\end{document} ) loss for binary tasks and categorical cross-entropy ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} CCE\end{document} ) loss for multi-class classification:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathcal{L} = BCE(Y_{i}, \hat{Y_{i}}) \text{; binary classification} \end{align*}\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathcal{L} = CCE(Y_{i}, \hat{Y_{i}}) \text{; multi-class classification} \end{align*}\end{document}Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} Y_{i}\end{document} denotes the ground truth label, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \hat{Y_{i}}\end{document} represents the predicted output for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} i\mathrm{th}\end{document} sample.
This approach enables end-to-end training, where the entire model—including the GCN layers and the fully connected layer—is optimized jointly to generate the final predictions.
Dual alignment
The hidden representations generated by the intra-omics and inter-omics modules correspond to the same omics type but are learned through distinct pathways. The intra-omics GCN captures dependencies within a single omics layer, while the inter-omics GCN gathers complementary information from a different omics layer via the bipartite network. Although both modules aim to represent the same set of omics features, they encode different aspects of the data. To ensure consistency between these perspectives, we introduce an alignment loss that minimizes the discrepancy between their outputs, encouraging convergence in a shared feature space. Moreover, to preserve the integrity of the original omics data during transformation, we incorporate a reconstruction loss, which penalizes deviations between the learned intra-omics hidden features and the original inputs. This helps maintain biological interpretability and prevents excessive distortion during feature learning. Together, these loss functions promote coherent integration while preserving meaningful structure in the omics data (Fig. 1b). To implement this, we initially train the intra-omics and inter-omics GCN modules independently in an unsupervised manner, without using ground truths from the downstream prediction task. This decoupling allows each module to focus solely on learning structural and relational patterns in the data. By separating representation learning from task-specific supervision, we ensure that the learned features retain generalizable biological signals. The total loss function for this pretraining phase is defined as:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathcal{L}= \alpha \times \mathcal{L}_{A} +(1-\alpha) \times \mathcal{L}_{R} \end{align*}\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathcal{L}_{A} = MSE(\mathbf{Z}_{u}^{(l)}, \mathbf{Z}_{v \to u}^{(l)}) + MSE(\mathbf{Z}_{v}^{(l)}, \mathbf{Z}_{u \to v}^{(l)}) \end{align*}\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*} & \mathcal{L}_{R} = MSE(\mathbf{Z}_{u}^{(l)},\mathbf{X}_{u}^{T}) + MSE(\mathbf{Z}_{v}^{(l)}, \mathbf{X}_{v}^{T}) \end{align*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{L}{A}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathcal{L}{R}\end{document} denote alignment loss and reconstruction loss respectively. The hyperparameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \alpha \in (0,1)\end{document} controls the trade-off between the two losses, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} MSE\end{document} refers the mean squared error. After pretraining the GCN modules, we proceed to train SynOmics in an end-to-end manner, where the predictive layer is same as equation (12). The model is then optimized using the loss functions defined in equations (13) and (14) for binary and multi-class classification tasks, respectively.
Extending to \documentclass[12pt]{minimal}
\usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \end{document} omics types
SynOmics is designed to be flexible, supporting the integration of any number of omics data types rather than being limited to just two. To achieve this, we construct an intra-omics network for each of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} M\end{document} omics types, along with bipartite networks for every pairwise combination of omics types. For each omics type, the corresponding feature matrix and intra-omics network are processed by an omics-specific GCN to learn intra-omics representations. In parallel, for every omics pair, the two associated feature matrices and their bipartite network are passed through a bipartite GCN to capture cross-omics interactions. After one layer of processing, each omics type yields one hidden representation capturing intra-omics dependencies and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} M-1\end{document} hidden representations that capture its relationships with the other \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} M-1\end{document} omics types. These \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} M\end{document} representations are then aggregated using assigned weights and passed to the next layer. In the final layer, the hidden representations from all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} M\end{document} omics types are combined and passed through a fully connected layer to generate the final prediction.
Network construction
We compute pairwise affinities between feature vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{x}{i}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{x}{j}\end{document} using one of the following three options; the user selects the option and any hyperparameters.
(i) Cosine similarity
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& c(\mathbf{x}_{i},\mathbf{x}_{j}) = \frac{\mathbf{x}_{i}^{\top}\mathbf{x}_{j}}{\lVert \mathbf{x}_{i}\rVert_{2}\,\lVert \mathbf{x}_{j}\rVert_{2}}\,.\end{align*}\end{document}(ii) Pearson correlation (PCC)
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& r(\mathbf{x}_{i},\mathbf{x}_{j}) = \frac{\sum_{k=1}^{n}(x_{ki}-\bar x_{i})(x_{kj}-\bar x_{j})} {\sqrt{\sum_{k=1}^{n}(x_{ki}-\bar x_{i})^{2}}\, \sqrt{\sum_{k=1}^{n}(x_{kj}-\bar x_{j})^{2}}} \,.\end{align*}\end{document}(iii) Gaussian (radial basis function) AQAU: Please provide the appropriate expansion for “RBF directly in the text. kernel
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& s(\mathbf{x}_{i},\mathbf{x}_{j}) = \exp\!\left(-\frac{\lVert \mathbf{x}_{i}-\mathbf{x}_{j}\rVert_{2}^{2}}{2\sigma^{2}}\right), \qquad \sigma>0 \,.\end{align*}\end{document}For each omics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} m\in {u,v,\ldots }\end{document} , we build an undirected intra-omics adjacency \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{A}^{(m)}\end{document} by thresholding the chosen affinity:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& A^{(m)}_{ij} = \begin{cases} 1, & a(\mathbf{x}_{i},\mathbf{x}_{j}) \ge \varepsilon_{m} \\ 0, & \text{otherwise} \end{cases}\end{align*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} a\in {c,r,s}\end{document} denotes cosine, PCC, or Gaussian, respectively, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{x}{i},\mathbf{x}{j}\end{document} are the feature vectors of nodes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} v_{i}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} v_{j}\end{document} , respectively. All intra-omics adjacencies are normalized according to equation (2).
We incorporate curated inter-omics interaction networks from established biological databases as prior knowledge to guide training of inter-omics modules, providing known molecular interactions, regulatory relationships, and functional associations. When such priors are unavailable for a pair of omics, we construct the bipartite (cross-omics) adjacency by applying the same affinity options \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} a\in {c,r,s}\end{document} between features from the two omics and thresholding at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \varepsilon \end{document} ; the resulting matrix is assembled as in equation (3) and normalized per equation (2).
Evaluation methods
Disease classification
To evaluate the binary classification performance of SynOmics, we use two key metrics: AUROC (Area Under the Receiver Operating Characteristic Curve) and MCC (Matthews Correlation Coefficient).
AUROC provides a comprehensive measure of the model’s ability to distinguish between positive and negative classes across all possible classification thresholds. It evaluates performance based on predicted probabilities, making it particularly informative prior to applying any threshold. This is especially valuable for imbalanced datasets, as it reflects the model’s overall capacity to separate classes probabilistically.
In contrast, MCC evaluates performance after a classification threshold has been applied to convert probabilities into discrete class labels. It considers all components of the confusion matrix, offering a balanced and interpretable summary of prediction quality. This makes MCC particularly effective for assessing final classification outcomes in imbalanced settings.
Since our datasets consist of real-world samples that often exhibit imbalanced class distributions, we selected AUROC and MCC as the most suitable evaluation metrics. To determine the optimal classification threshold, we use Youden’s Index [35], derived from the ROC curve. This method maximizes the sum of sensitivity and specificity, providing a balanced approach to classification performance. Once the optimal threshold is identified, we apply it to the model’s probability predictions to obtain class labels, which are then used to compute the MCC.
Survival analysis
We train a Cox proportional hazards model with an Elastic Net penalty [36] to assess the association between patients’ overall survival and their omics profiles. The Elastic Net penalty combines the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} L_{1}\end{document} -norm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} L_{2}\end{document} -norm penalties in a weighted manner by maximizing the following log-likelihood function:
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& logL(\boldsymbol{\beta})-\alpha\left (r \Sigma_{i=1}^{m}|\beta_{i}| + \frac{1-r}{2}\Sigma_{i=1}^{m}{\beta_{i}}^{2}\right)\end{align*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} L({\boldsymbol{\beta }})\end{document} is the partial likelihood of the Cox model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \alpha \geq 0\end{document} is a regularization parameter controlling overall shrinkage, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} r \in [0,1]\end{document} determines the relative contributions of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} L_{1}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} L_{2}\end{document} penalties. The coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \beta {i}\end{document} corresponds to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} i\end{document} -th genomic feature among the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} m\end{document} features in the omics data. To evaluate model performance, we define high-risk and low-risk groups based on the prognostic index ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} PI\end{document} ) computed from the independent test set. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} PI\end{document} represents the linear component of the Cox model, calculated as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} PI={\boldsymbol{\beta }}^{T} \mathbf{X}{test}\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \mathbf{X}_{test}\end{document} is the test set omics profile, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \boldsymbol{\beta }\end{document} is the risk coefficient vector estimated from the model fitted on the training set. Survival outcomes are visualized using Kaplan-Meier survival plots [37]. To construct the high-risk and low-risk groups for these plots, we divide the ordered \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} PI\end{document} values from the test set such that each group contains an equal number of samples. We then use the log-rank test to compare the survival distributions of the two groups and assess whether the observed difference in overall survival is statistically significant.
Experiments
This section introduces the datasets and networks used in our study, followed by a comparative evaluation of SynOmics against existing integration models. We then assess its performance in survival prediction and biomarker identification. Finally, we analyze the impact of a key hyperparameter and conduct an ablation study to evaluate the individual contributions of intra- and inter-omics learning.
Datasets and networks
We applied SynOmics to three TCGA datasets: breast invasive carcinoma (BRCA) [38], lung adenocarcinoma (LUAD) [39], and ovarian serous cystadenocarcinoma (OV) [40]. RNA-seq mRNA expression and miRNA expression data were obtained from the UCSC Xena Hub [41]. DNA methylation data and copy number variation (CNV) data were also collected from the same source to enable experiments involving more than two omics types. For mRNA expression, we used \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} log_{2}(x + 1)\end{document} transformed RSEM normalized count data. For miRNA expression, we used \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} log_{2}(x + 1)\end{document} transformed RPM values. DNA methylation profiles, generated using the Illumina Infinium HumanMethylation450 platform, consisted of beta values representing the ratio of methylated to total probe intensity at each locus. For CNV, we used TCGA segmented copy-number profiles (hg19) generated from Affymetrix SNP 6.0 arrays and processed with Circular Binary Segmentation, with common germline CNV probes removed via Broad GDAC. We used the segment mean ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} log_{2}(\text{copy-ratio})\end{document} ) values, mapping segments to genes for downstream analysis. Clinical data for all three cancer types were retrieved from cBioPortal [42]. We obtained the mRNA-miRNA interaction network from TargetScanHuman [43], which provides context++ scores to quantify regulatory relationships between miRNAs and their gene targets.
In the BRCA dataset, patients were classified based on receptor status (Table 2), including estrogen receptor (ER+ vs. ER−), human epidermal growth factor receptor 2 (HER2+ vs. HER2−), progesterone receptor (PR+ vs. PR−), and triple-negative status (TN vs. non-TN). Triple-negative breast cancer patients test negative for all three standard receptors: ER, PR, and HER2. For the LUAD and OV datasets, patients were categorized into short- and long-survival groups based on overall survival time. Patients who survived less than a predefined threshold were placed in the short-survival group, while those who survived longer were classified into the long-survival group. A similar thresholding method was applied to stratify patients into short- and long-duration groups based on disease-free survival. Patients who experienced recurrence before the threshold were assigned to the short-duration group, while those with recurrence after the threshold or no recurrence at all were classified into the long-duration group. Thresholds were selected to ensure that each group contained at least 20 samples. Table 2 summarizes the thresholds used for the LUAD and OV datasets.
In addition to binary classification tasks, we evaluated our model on multi-class classification using cancer stage data from the BRCA and LUAD cohorts. Each dataset includes four stages, representing progressively advanced levels of disease. Table 2 also reports the number of samples associated with each target variable.
The miRNA expression and DNA methylation datasets contained missing values, which we addressed using K-nearest neighbor (KNN) imputation with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} K=5\end{document} . Omics datasets typically include thousands of features, many of which exhibit low variability and offer limited analytical value. To reduce noise and improve model performance, we retained only the most variable features, as these are more likely to capture biologically meaningful patterns. Specifically, we selected the top 1000 most variable features for mRNA expression, the top 200 for miRNA expression, and the top 1000 for DNA methylation. To further ensure biological relevance, we retained only features with known interactions based on the provided biological networks. We then split the dataset by allocating 80% for training and 20% for testing. The training set was further divided, reserving 20% for validation and using the remaining 80% for model training. After splitting, we standardized the training, validation, and test sets independently to avoid data leakage, scaling each set based on its own Z-scores. This preprocessing results in the input feature matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} X_{m} \in \mathbb{R}^{n \times d}\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} n\end{document} is the number of samples, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} d\end{document} is the number of features, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} m \in {1, 2,...,M}\end{document} denotes the omics type. To ensure robust evaluation, we repeated the data split 100 times and reported the average performance metrics across these runs. For intra-omics network construction, we use cosine similarity, which yielded the best performance among the tested methods (see Supplementary Section S3). This procedure helps ensure that our results reflect overall model performance rather than being influenced by random variation in a single split.
Cancer outcome prediction
We compared the classification performance of SynOmics with six existing state-of-the-art multi-omics integration deep learning models: MOGONET [19], MoGCN [17], SUPREME [20], MOGLAM [21], OmicsFormer [22] and MOGAT [23]. These methods focus on integrating multi-omics signals for downstream tasks, using either graph-based models (e.g. GNNs) or attention/Transformer-style architectures. The classification results for the BRCA, and LUAD and OV datasets are shown in Tables 3 and 4, respectively.
As shown in Table 3, SynOmics variants generally outperform other models on BRCA. For ER status, SUPREME attains a slightly higher AUC (+0.27%), plausibly because it leverages patient-similarity networks that emphasize global ranking. When discriminative structure is concentrated at the patient/sample level, methods operating on patient-similarity graphs (e.g. SUPREME, MoGCN) can gain a modest AUC advantage. However, SynOmics achieves a higher MCC (+0.58%), reflecting stronger binary decision accuracy. This likely stems from its ability to capture fine-grained cross-omics cues, aided by thresholding (e.g. Youden’s Index). For TN status, MoGCN slightly surpasses SynOmics in MCC (+3.31%). This is consistent with MoGCN’s sample-level perspective, which captures broad between-group separations when class cues are well expressed on the patient graph. Architecturally, SUPREME and MoGCN learns over sample-similarity networks. By contrast, SynOmics is intentionally feature-centric, yielding advantages when the predictive signal is distributed over feature-level mechanisms and when cross-omics relations are informative—situations common in heterogeneous, high-dimensional multi-omics where sample-only graphs can miss regulatory dependencies. In such settings, SynOmics has shown strong, consistent performance.
In the LUAD and OV datasets (Table 4), SynOmics demonstrates strong overall performance. SUPREME slightly leads in AUC for LUAD disease-free duration, but both SynOmics variants outperform all other models in the remaining settings.
We also evaluate SynOmics on a multi-class classification task using BRCA and LUAD cancer stage information, as shown in Table 5. Multi-class classification is generally more challenging than binary classification, and thus we observe a performance drop across all models compared to their binary classification results. However, SynOmics with dual alignment outperforms all other methods in BRCA stage prediction, achieving higher AUC and MCC scores. Although not the best, SynOmics remains competitive on LUAD stage prediction.
Survival prediction
We conducted an overall survival prediction analysis using the BRCA, LUAD, and OV datasets to assess the quality of the hidden representations generated by SynOmics. For this analysis, we extracted the hidden representations produced by SynOmics before the predictive layer and input them into the Cox proportional hazards model, as outlined in Section Survival analysis. We set the relative weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} r\end{document} in equation (22) to 0.5 to combine the subset selection property of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} L_{1}\end{document} -norm with the regularization strength of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} L_{2}\end{document} -norm. The Kaplan–Meier survival plots and log-rank test \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} p\end{document} -values in Fig. 2 demonstrate that the hidden representations generated by SynOmics possess strong prognostic ability in distinguishing between high-risk and low-risk groups. For example, at approximately 5 years (60 months), the high-risk group in the BRCA dataset exhibited a survival probability of 69.84% (95% CI: 47.78%–83.99%), whereas the low-risk group showed a higher survival probability of 90.91% (95% CI: 50.81%–98.67%). In the LUAD dataset, the high-risk group demonstrated a markedly lower survival probability of 9.09% (95% CI: 0.54%–33.29%) compared to 81.82% (95% CI: 44.74%–95.12%) in the low-risk group. Similarly, for the OV dataset, the high-risk group had a survival probability of 27.27% (95% CI: 6.52%–53.89%), while the low-risk group maintained a survival probability of 80.81% (95% CI: 42.35%–94.85%). All of these separations were statistically significant ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} P\end{document} -value <.05). As indicated by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} P\end{document} -values, none of the datasets could significantly distinguish between high-risk and low-risk groups in their original form (see Supplementary Section S1). In contrast, the hidden representations generated by SynOmics demonstrate improved prognostic capability. They enable a significant separation between the two risk groups, highlighting their effectiveness for downstream survival analysis tasks.
Survival analysis based on the learned hidden representations for (a) BRCA (ER status), (b) LUAD (survival duration), and (c) OV (survival duration); numbers in parentheses give low-/high-risk group sizes, and p-values are from log-rank tests comparing overall survival between the two groups.
Biomarker identification
SynOmics begins with individual omics types and combines intra- and inter-layer signals to learn a unified hidden representation that captures complex relationships between omics features. This representation not only supports disease classification but also facilitates the identification of discriminative biomarkers. To evaluate its effectiveness, we conducted a feature-level analysis to examine how well the hidden representation distinguishes between different lung cancer (LUAD) stages.
Our input data consisted of 1200 features—1000 from mRNA expression and 200 from miRNA expression. Throughout the SynOmics layers, we maintained the same feature-space dimension to maximize the benefits of information sharing through the feature networks. As a result, the output hidden representation retained the same number of features, allowing for a direct comparison between the input and output features of SynOmics. To identify the most predictive features, we conducted ANOVA tests on both the 1200-dimensional input and output feature sets to determine which features, in both the original and encoded spaces, are significantly associated with cancer stage. Specifically, we grouped the input and output feature expressions based on their corresponding stages and conducted ANOVA tests to determine which features exhibited the most statistically significant ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} P\end{document} -value <.05) differences in expression. This analysis revealed 8 significant features in the original dataset and 33 significant features in the hidden representation, demonstrating that SynOmics enhances omics data representation by uncovering additional features strongly associated with cancer progression. For further analysis, we categorized patients into high- and low-expression groups based on each significant feature identified by SynOmics and conducted log-rank tests to evaluate their ability to distinguish survival differences between these groups.
Figure 3a provides a violin plot illustrating the expression pattern of one representative significant feature (among the 33 identified by SynOmics) across advancing cancer stages, while Fig. 3b shows patient survival probabilities for high- and low-expression groups based on the same significant feature. This feature exhibited a statistically significant variation in expression levels across cancer stages, as determined by the ANOVA test ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} P\end{document} -value =.0183). The mean expression values showed a progressive increase across stages, with stage 1 having a mean of 0.2701 (95% CI: 0.1368–0.4035), stage 2 with 0.6138 (95% CI: 0.1857–1.0419), stage 3 with 0.6400 (95% CI: 0.0868–1.1931), and stage 4 reaching 1.2625 (95% CI: 0.0903–2.4347). This pattern suggests a potential up-regulation of the feature with disease progression. Survival analysis further demonstrated the prognostic utility of the representative feature. At \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \sim \end{document} 5 years (60 months), the high-risk group showed a survival probability of 32.94% (95% CI: 16.72%–50.18%), whereas the low-risk group showed a survival probability of 59.46% (95% CI: 20.67%–84.25%). The log-rank test confirmed that this separation was statistically significant ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} P\end{document} -value =.0037).
Representative biomarker identification results: (a) violin plot of a SynOmics-selected feature across cancer stages (p-value from ANOVA); (b) Kaplan–Meier survival comparing high vs. low expression groups with sample sizes in parentheses (p-value from log-rank test).
To enhance biological clarity, we systematically linked the 33 stage-associated latent features to genes and pathways. For each latent feature, we computed Spearman correlations between the latent score and all 1000 genes and applied Benjamini–Hochberg False Discovery Rate (FDR) [44] within-latent to obtain signed gene lists; the sign of each latent’s association with ordinal stage (I–IV) was used to label genes and pathways as up-with-stage or down-with-stage. For pathway interpretation, we performed pre-ranked GSEA [45] on all genes ranked by signed correlation for each latent using Reactome 2022 [46]. GSEA returns a normalized enrichment score (NES), FDR, and the leading-edge genes driving each enrichment; we report pathways at FDR \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \le 0.05\end{document} . Across the stage-linked features we frequently observe enrichment of extracellular-matrix/adhesion processes and cellular stress/inflammatory responses among the up-with-stage signals, together with reductions in epithelial/lineage programs among the down-with-stage signals (e.g. a representative factor shows Extracellular Matrix Organization, NES \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \approx \end{document} 1.87, FDR \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \approx \end{document} 0.036, with leading-edge genes COL11A1, ADAM12, MFAP5 [47–49]; others highlight stress/stimulus responses with EGLN3, SLC7A11, IGFBP1 [50–52]). Complete per-latent summaries, top gene lists (up/down with stage), and significant pathway annotations are provided in Supplementary Data S1–S3. Together, these results indicate that the representative feature may serve as both a stage-associated and prognostically relevant biomarker, with robust evidence from both expression variability and survival stratification.
Hyperparameter tuning
In the “weighted message passing module of SynOmics, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} k\end{document} determines the relative contribution of intra-omics and inter-omics features when combining the two sources of information, as described in equations (10) and (11). If the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} k\end{document} is too large, the hidden representation tends to reflect mostly the intra-omics dynamics, with minimal influence from the other omics types. Conversely, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} k\end{document} is too small, the hidden representation predominantly captures information from other omics sources, neglecting the intra-omics dynamics. To effectively capture both types of dynamics, a balance between the two sources of information is essential. Figure 4 shows the model’s performance (AUC and MCC scores) across different values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} k\end{document} for the three datasets. It is evident that lower values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} k\end{document} lead to poorer performance, while performance improves as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} k\end{document} increases. The best results are typically observed when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} k\end{document} is in the range of 0.4 to 0.5. However, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} k\end{document} becomes too large, performance declines again, particularly in the LUAD and OV datasets. This suggests that neither intra-omics nor inter-omics dynamics alone provides optimal performance; rather, combining both sources leads to the best results.
SynOmics performance vs. the intra-/inter-omics weight k: AUC (top) and MCC (bottom) for BRCA (ER status), LUAD (survival duration), and OV (survival duration) across k values controlling the relative contribution of intra- and inter-omics representations.
The analysis of the hyperparameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \alpha \end{document} in the “dual alignment module, along with other hyperparameter settings, is available in Supplementary Section S6.
Ablation study
To assess the relative importance of the intra-omics and inter-omics modules in SynOmics, we conduct an ablation study to better understand the contributions of each component. In this study, we train SynOmics using only the intra-omics or inter-omics modules individually.
For intra-omics learning, we focus exclusively on constructing omics-specific networks and applying the corresponding GCNs, without incorporating any cross-omics interactions through the bipartite network. In this configuration, only the GCNs defined in equations (6) and (7) are trained, while the bipartite GCNs in equations (8) and (9) are excluded. As a result, the model relies entirely on intra-omics representations, with no involvement of weighted message passing or the dual alignment module.
In contrast, for inter-omics learning, we rely solely on the bipartite interaction networks and their associated GCNs to model cross-omics relationships. In this case, omics-specific networks are not used, and only the bipartite GCNs in equations (8) and (9) are trained. As with the intra-omics setup, the weighted message passing and dual alignment modules are omitted, since integration across omics types is not performed. The outcomes of this ablation study are summarized in Table 6. The table clearly shows that the model trained with both modules outperforms the versions trained with just one module. When trained to capture only intra-omics relationships, the model experiences a noticeable performance drop. However, the drop is even more significant when the model is trained solely on inter-omics relationships. This suggests that focusing only on inter-omics dynamics, without considering intra-omics interactions, is insufficient for effective multi-omics integration. While considering only intra-omics relationships may yield decent results, the most effective multi-omics integration requires accounting for both intra- and inter-omics dynamics.
Scaling SynOmics to three and four omics types
To demonstrate the extensibility of SynOmics, we increased the inputs from two to three and four omics types. Beyond mRNA and miRNA expression, we incorporated DNA methylation as a third modality and copy-number variation (CNV) as a fourth. Adding modalities reduces the usable cohort for supervised learning because complete measurements across all sources are required. Under this constraint, BRCA would suffer a substantial loss of cases, whereas LUAD retained aligned DNA-methylation and CNV profiles after prior quality control. We therefore report the three-omics and four-omics analyses on LUAD to enable a fair, like-for-like comparison with the two-omics baseline. For inter-omics pairs involving DNA methylation and CNV, we construct the networks using cosine similarity, which achieved the best performance among the tested methods (see Supplementary Section S3).
As can be observed in Fig. 5, performance improves as complementary modalities are added, based on boxplots aggregated over 100 independent train/validation/test splits. Relative to two omics, median AUC increases from 0.669 to 0.678 with three omics (+1.35%) and to 0.711 with four omics (+6.28%); the step from three to four omics is +4.87%. Median MCC rises from 0.325 to 0.365 with three omics (+12.31%) and to 0.378 with four omics (+16.31%), with a +3.56% gain from three to four. The largest jump occurs when adding CNV (fourth omics), indicating additional, non-redundant signal beyond expression and methylation. Detailed three-omics results, including benchmarks against baseline methods, are provided in the Supplementary Section S5.
SynOmics performance on LUAD vs. number of omics: AUC (left) and MCC (right) boxplots over 100 splits (medians annotated with red bars; boxes indicate interquartile ranges) show monotonic gains from two (mRNA+miRNA) to three (+DNA methylation) to four (+CNV) omics, with the largest jump from three to four.
Discussion
Multi-omics integration is essential for uncovering coordinated biological mechanisms across layers such as genomics, transcriptomics, proteomics, and metabolomics. Traditional methods that rely on sample correlations often fail to capture the true feature-level relationships that drive biological processes. Sample correlation, while useful in some contexts, does not necessarily reflect the true dynamics of feature-level interactions within biological systems. Sample networks are typically built on the assumption that there exists homogeneity within samples and consistent relationships across varying biological states. This assumption can be erroneous, especially in low-sample, high-dimensional contexts typical of omics data, where sample variability may overshadow true biological signals.
Rather than depending on correlations between samples, SynOmics emphasizes feature-level interactions, improving its resilience to changes in sample size and variability. It achieves cross-omics integration through bipartite networks, which represent relationships between features across omics types and can be built using known interactions or computed similarities. This approach is better equipped to incorporate contextual information—such as the regulatory role of miRNA in mRNA expression—compared to context-agnostic methods that use fully connected layers or autoencoders. Experimental results demonstrate the superiority and effectiveness of SynOmics. Notably, its survival prediction capabilities indicate its potential for patient survival analysis. Effective multi-omics integration should consider both intra-omics and inter-omics interactions, as evidenced by the ablation studies conducted with SynOmics. The model adeptly captures how expression levels correlate with different disease stages.
In cohorts where subtype signal is mostly organized at the patient/sample level, methods that learn over patient-similarity networks (e.g. MoGCN via SNF, SUPREME) can show modest advantages on ranking metrics. SynOmics is intentionally feature-centric, modeling intra-omics relations and inter-omics bipartite links to capture cross-omics regulatory cues, and therefore performs best when feature-level mechanisms drive separability. These complementary inductive biases explain occasional gaps (e.g. TN in BRCA) alongside SynOmics’ strong performance in heterogeneous multi-omics settings. Practically, an adaptive or hybrid strategy may be beneficial: exploit patient-graph structure when it dominates, and rely on SynOmics’ cross-omics modeling when regulatory signals are key.
The design of SynOmics might appear to make the framework overly dependent on external inter-omics priors. In light of this, we explicitly evaluated this via a perturbation analysis of the inter-omics network (see Supplementary Section S2). Randomly rewiring 10%–50% of inter-layer edges led to only minor fluctuations in AUC/MCC, indicating robustness to imperfect or noisy prior knowledge; SynOmics benefits from reasonable priors but does not critically rely on exact inter-omics connectivity. We further implemented a hybrid inter-omics network that blends curated prior interactions with a cohort-specific, data-driven graph (see Supplementary Section S2). Empirically, the purely data-driven variant attains performance close to the prior-only counterpart, indicating that reliable edges can be inferred from the data itself. Nevertheless, when high-quality priors are available, assigning them non-trivial weight in the hybrid graph consistently provides a small but meaningful gain, and thus is recommended in practice.
SynOmics operates at the feature level, so its computational complexity is largely determined by the feature size of the omics data used. Specifically, the complexity is mathematically expressed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} O(lp^{2}n)\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} l\end{document} represents the number of layers, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} p\end{document} represents the feature size of the omics type with the larger dimension, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} n\end{document} represents the sample size. The quadratic nature of this complexity with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} p\end{document} can be significant, especially when dealing with large feature sets. However, applying dimensionality reduction techniques can help mitigate this issue by reducing the number of features. In many omics datasets, a large portion of features have low variance and contribute minimally to key tasks such as disease or survival prediction. We experimented with three feature-selection strategies based on this—(i) variance-based thresholding, (ii) selecting high-variance features that are correlated with the labels, and (iii) minimum-redundancy–maximum-relevance—and found that variance-based thresholding performed best (see Supplementary Section S4). In this study, we therefore focused on using features with high variance to train the model. We also incorporated early stopping in our model training based on validation loss. If the validation loss does not improve over a specified number of epochs (e.g. 20 epochs), we consider the loss to be saturated and terminate the training. This approach also helps reduce training time. As a result, our model achieved the shortest training time compared to the others. The average training times for SynOmics, MOGONET, MoGCN, SUPREME, MOGLAM and MOGAT were 5.35, 131.10, 15.30, 174.32, 199.37, and 54.9 s respectively, measured on a machine with an AMD Ryzen 9 5950X CPU, 128 GB RAM, and a single NVIDIA RTX A4500 (20 GB) GPU. We also evaluated the feature–runtime trade-off (see Supplementary Section S4) and found that selecting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \sim \end{document} 1000–2000 features offers an optimal balance between performance and computation.
Parallel integration of information across all omics layers—rather than at the beginning or end of the process—enables SynOmics to create a cohesive, integrated representation. To further improve the performance of SynOmics, it may be advantageous to explore advanced network creation techniques [53–55] beyond traditional methods such as cosine similarity or correlation, and to implement more sophisticated imputation strategies [56–60] instead of relying solely on KNN. Establishing an effective bipartite network between omics types presents challenges, but when accomplished, it can greatly improve model performance by accurately reflecting the multilayered interactions among diverse biological features. In future work, we plan to extend SynOmics for application in single-cell and spatial multi-omics, where spatial coordinates and cell-level similarities can be incorporated through appropriate network modeling.
In conclusion, our findings highlight that SynOmics is a flexible and effective model for multi-omics integration, enabling a more in-depth understanding of complex diseases. By leveraging the dynamic relationships within omics data, the model enhances the accuracy of disease prediction and uncovers deeper insights into how different molecular factors contribute to disease progression. SynOmics can identify key biomarkers, which are essential for advancing medical research. These biomarkers have the potential to guide the development of targeted therapies, leading to more effective drugs and treatments. Overall, SynOmics provides an effective and scalable approach to multi-omics integration, offering a path toward more accurate biological interpretation and clinically actionable outcomes.
Key Points
- SynOmics integrates multi-omics data at the feature level using both intra-omics networks and inter-omics bipartite networks.
- SynOmics incorporates two alternative integration approaches and adopts a parallel learning strategy for joint representation learning.
- SynOmics consistently outperforms state-of-the-art methods in classification tasks and demonstrates robust performance in survival prediction.
- SynOmics identifies biologically relevant features associated with disease progression and differential expression across stages.
Supplementary Material
Data_S1_latent_stage_bbaf595
Data_S2_latent_gene_correlations_bbaf595
Data_S3_latent_gsea_results_bbaf595
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The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1Karczewski KJ, Snyder MP. Integrative omics for health and disease. Nat Rev Genet 2018;19:299–310. 10.1038/nrg.2018.429479082 PMC 5990367 · doi ↗ · pubmed ↗
- 2Yan J, Risacher SL, Shen L. et al. Network approaches to systems biology analysis of complex disease: Integrative methods for multi-omics data. Brief Bioinform 2018;19:1370–81.28679163 10.1093/bib/bbx 066PMC 6454489 · doi ↗ · pubmed ↗
- 3Hasin Y, Seldin M, Lusis A. Multi-omics approaches to disease. Genome Biol 2017;18:1–15. 10.1186/s 13059-017-1215-128476144 PMC 5418815 · doi ↗ · pubmed ↗
- 4Subramanian I, Verma S, Kumar S. et al. Multi-omics data integration, interpretation, and its application. Bioinf Biol Insights 2020;14:117793221989905. 10.1177/1177932219899051 PMC 700317332076369 · doi ↗ · pubmed ↗
- 5Goodwin S, Mc Pherson JD, Mc Combie WR. Coming of age: ten years of next-generation sequencing technologies. Nat Rev Genet 2016;17:333–51. 10.1038/nrg.2016.4927184599 PMC 10373632 · doi ↗ · pubmed ↗
- 6Athieniti E, Spyrou GM. A guide to multi-omics data collection and integration for translational medicine. Comput Struct Biotechnol J 2023;21:134–49. 10.1016/j.csbj.2022.11.05036544480 PMC 9747357 · doi ↗ · pubmed ↗
- 7Nicora G, Vitali F, Dagliati A. et al. Integrated multi-omics analyses in oncology: a review of machine learning methods and tools. Front Oncol 2020;10:1030. 10.3389/fonc.2020.0103032695678 PMC 7338582 · doi ↗ · pubmed ↗
- 8Sun YV, Yi-Juan H. Integrative analysis of multi-omics data for discovery and functional studies of complex human diseases. Adv Genet 2016;93:147–90. 10.1016/bs.adgen.2015.11.00426915271 PMC 5742494 · doi ↗ · pubmed ↗