Triad-LMF: a hierarchical low-rank multimodal fusion framework for robust cancer subtype classification using multi-omics data.

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Learn more: PMC Disclaimer | PMC Copyright Notice BMC Bioinformatics . 2026 Mar 26;27:118. doi: 10.1186/s12859-026-06378-3 Triad-LMF: a hierarchical low-rank multimodal fusion framework for robust cancer subtype classification using multi-omics data Xingyue Tan 1, # , Xiran Chen 1, # , Renjie Tian 2 , Qinyu Cai 3, 4 , Miaoyuan Jiang 1 , Dongqiu Yang 1 , Lei Zhang 1, ✉ Author information Article notes Copyright and License information 1 School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400070 China 2 Institute of Computer Vision and Traffic Image Understanding, School of Information Science and Engineering, Chongqing Jiaotong University, Chongqing, 400070 China 3 School of Life Sciences, Westlake University, Hangzhou, 310030 China 4 Department of Bioinformatics, School of Basic Medical Sciences, Chongqing Medical University, Chongqing, 400016 China ✉ Corresponding author. # Contributed equally. Received 2025 Aug 26; Accepted 2026 Jan 13; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13231787  PMID: 41888653 Abstract Cancer heterogeneity poses a major challenge for accurate molecular subtype classification. Conventional methods often fail to exploit complementary information across multiple omics modalities, leading to overfitting on high-dimensional data and limited representation of subtype heterogeneity. To address this, we propose Triad-LMF, a multi-omics integration framework based on low-rank multimodal fusion to improve classification accuracy. Triad-LMF harmonizes heterogeneous omics data and integrates information through a two-stage hierarchical fusion strategy. Local Pairwise Fusion and Global Triadic Fusion are combined via the Two-Feature and Three-Way LMF modules, enabling a gradual transition from local modality interactions to global feature integration. Experimental results show that Triad-LMF consistently outperforms existing methods. UMAP visualization confirms enhanced subtype separability, and SHAP-based analysis highlights biologically meaningful features. Across independent datasets, Triad-LMF demonstrates strong generalization, offering a robust and interpretable framework for multi-omics-driven cancer subtype classification. Keywords: Cancer subtyping, Deep learning, Supervised learning, Low-rank multi-modal fusion, Hierarchy network Introduction The high heterogeneity of cancer makes its molecular subtype classification a key cornerstone of precision medicine [ 1 ]. In recent years, the rapid development of multi-omics data has provided unprecedented opportunities to uncover molecular mechanisms of cancer, identify subtype-specific biomarkers, and guide personalized therapy [ 2 ]. By integrating multimodal data such as gene expression, epigenetics, proteomics, and pathology images, researchers can more comprehensively depict the biological characteristics of cancer. However, the high dimensionality of multi-omics data, the complex interactions between modalities,and data heterogeneity pose severe challenges to traditional computational methods. Therefore, developing novel integration frameworks that efficiently fuse these data is urgently needed to advance cancer subtype classification and precision medicine. Supervised learning methods hold a significant position in cancer subtype classification models due to their ability to substantially improve classification Accuracy by leveraging labeled data. These methods mine subtype-specific patterns by integrating gene expression, mutation data, and other omics features, demonstrating excellent performance across cancer types such as breast cancer and colorectal cancer [ 3 – 5 ]. By optimizing model architectures and feature selection strategies, they enhance robustness against sparse or high-dimensional data, providing strong support for precise diagnosis and treatment. However, the limitations of existing methods cannot be overlooked: many adopt a one-time fusion strategy, making it challenging to capture dynamic interactions among modalities [ 4 , 6 ]; handling high-dimensional data often leads to overfitting, especially pronounced in deep learning models [ 7 , 8 ]. These shortcomings constrain the further application of multi-omics data in precision medicine. Based on the challenges above, we believe the fundamental issue of current models lies in the lack of hierarchical fusion mechanisms and explicit modeling of modality interactions, which limits the depth of biological pattern mining, while overfitting issues undermine model generalization. Therefore, this study proposes the Triad-LMF model, which introduces the following innovative designs to achieve breakthroughs in multi-omics data integration and subtype classification: Triad-LMF first explicitly models modality-specific interactions through the Two-Feature LMF module, followed by high-level integration via the Three-way LMF module. This hierarchical design overcomes the limitations of traditional one-step fusion. Compared with existing methods, this strategy captures dynamic interactions between modalities through stepwise fusion, significantly improving classification performance. Low-rank decomposition techniques are introduced at all fusion stages. By enforcing low-rank constraints, the nonlinear modeling capability is enhanced while the number of parameters is significantly reduced, effectively alleviating overfitting issues caused by high-dimensional data and providing a feasible solution for large-scale multi-omics data analysis. Triad-LMF prioritizes learning joint representations of specific modality pairs through the Two-Feature LMF module, generating interpretable interaction features that improve model robustness. It also offers biological insights into the contribution of modality pairs to subtype classification, deepening the exploration compared to existing methods and providing new perspectives for identifying subtype-specific biomarkers. Related work Unsupervised learning Unsupervised learning methods mine potential subtypes through clustering or dimensionality reduction. For example, [ 9 ] proposed a kernel fusion approach that integrates gene expression, miRNA expression, and isoform data to construct similarity kernel matrices, thereby efficiently clustering breast cancer and lung cancer subtypes. This method uses kernel functions to capture nonlinear relationships between modalities and demonstrates high subtype discrimination capability on the TCGA dataset. However, its simple fusion strategy makes it difficult to model hierarchical interactions between modalities, limiting the analysis of complex biological patterns. Liu et al. [ 10 ] developed a multiview spectral clustering method (MRF-MSC) that fuses multi-omics data through multi-smooth representations and combines spectral clustering to uncover subtype structures, showing excellent performance across multiple cancer types in the TCGA dataset. Liu et al. [ 11 ] proposed SMMSN (Self-supervised Multi-fusion Strategy Network), which integrates multi-omics data through graph convolutional networks and stacked autoencoders using dual self-supervision for label-free cancer subtype discovery, demonstrating significant clinical relevance in survival analysis and case studies across eight cancer datasets. Supervised learning To achieve higher classification Accuracy, supervised learning methods leverage labeled data to improve cancer subtype classification performance significantly. BCDForest by [ 3 ] optimizes subtype classification of small-scale gene expression data through multi-granularity scanning and enhances model robustness to sparse data via random forests and multi-scale feature selection strategies; however, its performance is limited when generalized to large-scale multi-omics datasets. DeepCSD by [ 5 ] employs a minimalist feedforward neural network combined with gene expression and mutation data to identify colorectal cancer subtypes efficiently. It benefits from low computational complexity but has limited capacity to capture complex biological features. Li et al. [ 4 , 12 ] utilize graph convolutional networks (GCNs) to integrate multi-omics data by constructing inter-modality relation graphs, demonstrating excellent performance in breast cancer and pan-cancer classification. These works exploit graph neural networks to mine topological relationships between modalities but suffer from high model complexity, easily leading to overfitting. [ 6 , 13 , 14 ] optimize classification and biomarker identification through GCNs and graph attention networks (GATs), combining attention mechanisms to prioritize key features, further enhancing cancer subtype classification Accuracy and interpretability. Sun et al. [ 15 ] perform multi-class classification using co-expression modules, mining gene co-expression patterns to strengthen the model’s biological relevance, but remains insufficient in modality fusion. Yin et al. [ 8 ] propose a robust GCN framework that stabilizes high-dimensional data through regularization optimization, achieving excellent performance in pan-cancer subtype classification. Bu et al. [ 16 ] model molecular mechanisms via gene interaction networks, significantly improving subtype classification Accuracy. Although these supervised methods excel in Accuracy, their one-time fusion strategies struggle to model dynamic interactions between modalities, and handling high-dimensional data often leads to overfitting issues. Semi-supervised learning In scenarios with limited annotated data, semi-supervised learning offers a compromise solution. DeepType by [ 1 ] integrates supervised classification and unsupervised clustering, effectively leveraging the limited labeled information in the TCGA dataset through joint optimization of labeled and unlabeled data, demonstrating excellent breast cancer subtype classification performance. Gao et al. [ 2 ] mined nonlinear features using a semi-supervised model that combines autoencoders and regularization techniques, enhancing the model’s generalization ability on small sample datasets. Lupat et al. [ 17 ] utilized multi-omics autoencoders to predict breast cancer subtypes, uncovering latent features through nonlinear dimensionality reduction and improving model robustness. Jiang et al. [ 18 ] optimized classification performance by incorporating consistency regularization, enhancing classification Accuracy by constraining model consistency on unlabeled data. Jose et al. [ 19 ] modeled gene mutation data via mutation networks combined with graph neural networks to optimize classification, showing potential in pan-cancer subtype classification. However, these fusion methods still exhibit insufficient modeling of modality interactions, making it difficult to capture the complex relationships in multi-omics data comprehensively. Methods This paper proposes a deep learning framework called Triad-LMF based on Low-rank Multimodal Fusion (LMF). Inspired by [ 20 ], this method efficiently fuses heterogeneous omics data through modality-specific low-rank factors, overcoming the limitations of traditional tensor fusion methods and generating a unified low-dimensional representation for downstream cancer subtype identification. Figure 1 shows the framework of Triad-LMF. We describe each block in more detail in the following. Fig. 1. Open in a new tab Framework of Triad-LMF, illustrating how multiple omics modalities are processed by modality-specific encoders and integrated through local pairwise fusion and global triadic fusion to predict cancer subtypes Modility-specific feature extractor encoder Multi-omics data typically exhibit high dimensionality, heterogeneity, and noise interference, which pose significant challenges for feature extraction. For example, the RNA-seq data in the BRCA dataset contains 16,383 features, where RPPA includes only 223 features. This discrepancy in dimensionality and data noise may obscure underlying biological patterns. To address this, we designed a Modality-Specific Embedding Subnetwork. Given a set of cancer multi-omics data X = { X 1 , X 2 ,..., X N }, , where N is the number of datasets, X i is the input data of modality i , with as the input dimension of modality. To extract the hidden representations of nonlinear features for each modality. Each omics is assigned with Modality-specific Encoder . For each omics modality, we employ an encoder network composed of three fully connected layers to extract high-level latent representations. The input features are first normalized using Batch Normalization to stabilize training and mitigate internal covariate shift. A dropout layer is then applied to reduce the risk of overfitting by randomly deactivating neurons during training. Each encoder layer performs a linear transformation followed by a ReLU activation to introduce non-linearity and enhance feature expressiveness. Specifically, given an input feature vector, the encoding process can be expressed as: where denotes the weight matrix of the -th linear layer, represents Batch Normalization, and denotes the Dropout operation. Low-rank multimodal fusion Integrating representations from multiple modalities to capture cross-modal interaction information in multi-omics data analysis is crucial for achieving precise cancer subtype classification. However, traditional multimodal fusion methods often result in exponential growth in computational complexity when dealing with high-dimensional data, which is impractical for real-world applications. We designed Low-rank Multimodal Fusion modules that significantly reduce computational complexity by introducing modality-specific low-rank factors to address this issue. We propose two variants for this purpose: Local Pairwise Fusion for pairwise modality fusion Global Triadic Fusion is used to integrate all modalities. Local pairwise fusion Given the representation of two modalities and , respectively, the fused output representation generated by the two feature LMF is , where r is the rank parameter. To introduce a bias term and enhance the flexibility of the representation, we first extend the hidden representations, as defined in Eqs. ( 1 ) and ( 2 ): 1 2 where 1 is an all-one vector. The fusion process uses modality specific low-rank factors and , where is the output dimension. The intermediate fusion tensor is calculated as following Eqs. ( 3 ) and ( 4 ): 3 4 where and are modality-specific intermediate representations. Then, cross-modal interaction characteristics are captured through element-wise multiplication, as formulated in Eq. ( 5 ): 5 where is the joint tensor. The final output is obtained by extracting the first dimension and transposing, as formulated in Eq. ( 6 ): 6 Therefore, the output of Local Pairwise Fusion generates a compact feature vector of dimension r for each sample, where r is the low-rank parameter reflecting the degree of compression of the fused representation. This representation not only efficiently preserves cross-modal correlations but also provides high-quality intermediate features for subsequent Global Triadic Fusion multimodal integration. Global triadic fusion Given three representations , , produced by the Modility-Specific Feature Extractor Encoder, The final output representation of the Global Triadic Fusion is , with as the output dimension, which we set to the number of specific cancer subtype categories. It is worth noting that the Global Triadic Fusion does not directly fuse the original hidden representations produced by the Modality-Specific Feature Extractor Encoder. Instead, it is built upon the results of three bimodal low-rank fusions described in Sect. 3.2 . This hierarchical design is first used to extract local interactions between modalities through pairwise fusion, and then applied to integrate global features via trimodal fusion. This enhances the model’s robustness and expressive power in scenarios with modality imbalance. The implementation of Global Triadic Fusion relies on a preceding pairwise fusion step. The specific process is as follows: Before Global Triadic Fusion, three Local Pairwise Fusions respectively fuse modality pairs to generate intermediate representations as shown in Eqs. ( 7 )–( 9 ): 7 8 9 where denotes the Local Pairwise Fusion, which outputs , r is the rank parameter, and each output is a compact feature vector of dimension r corresponding to each sample. Take the three pairs of fused intermediate representations as the input of the Global Triadic Fusion. Similarly, extend its hidden representation as shown in Eqs. ( 10 )–( 12 ): 10 11 12 where is an all-one vector, and the extended representations are denoted as . Define low-rank factors for each intermediate representation as modality-specific factors , , , where is the rank parameter of the three-modality fusion, and is the final output dimension. Then, we use low-rank factors to transform the expanded intermediate representation, as shown in Eqs. ( 13 )–( 15 ): 13 14 15 where represents pairs of intermediate tensors. Next, integrate the three intermediate tensors through element-wise multiplication, as shown in Eq.  ( 16 ) : 16 where is the joint tensor that captures the global interaction information among all modalities. The joint tensor generates the final output through a weighted sum and bias calculation, as shown in Eq.  ( 17 ): 17 where is the fusion weight matrix, is the bias vector, and the reshaped output is the feature vector corresponding to each sample with dimension . Compared to Local Pairwise Fusion, Global Triadic Fusion performs hierarchical fusion, which not only preserves local interaction characteristics but also achieves efficient integration of global information. Hierarchical fusion integrator In the Triad-LMF framework, the fusion of multi-omics data is realized through a hierarchical structure to efficiently integrate heterogeneous information and generate a unified representation suitable for cancer subtype classification. The Hierarchical Fusion Integrator, a core component of this layered design, integrates Local Pairwise Fusion and Global Triadic Fusion step-by-step via the hierarchy, transitioning from local modality interactions to global information integration. Given the imput multimodal omics data and , its corresponding hidden representations are denoted as , and , which are generated by the Modality-Specific Embedding Subnetwork. Here, B represents the batch size, and denotes the hidden dimension of each modality. The Hierarchical Fusion Integrator employs a two-stage strategy: Two-Feature LMF module: Three Local Pairwise Fusion modules are applied to modality pairs, producing intermediate representations that capture local inter-modal interactions. Three-way LMF module: A Global Triadic Fusion module integrates the above intermediate representations to generate the final output, which can directly support cancer subtype classification tasks. The Hierarchical Fusion Integrator optimizes the contributions of sparse modalities and balances the representation weights across modalities. Its performance will be validated in subsequent experiments. Pseudocode Algorithm 1 summarizes the Triad-LMF framework for cancer subtype classification. Each modality is encoded, pairwise fused, then combined via a global triadic fusion to predict cancer subtypes. Algorithm 1 Triad-LMF for Cancer Subtype Classification Open in a new tab Experiments and results Dataset We applied the proposed model to classification tasks on three different datasets. The first dataset is breast invasive carcinoma (BRCA), which includes CNV, mRNA, RPPA, and PAM50 subtype labels (including Her2, LumA, and LumB subtypes), sourced from the literature [ 4 ]. The second dataset is used for molecular subtype classification of STAD (including CIN, MSI, GS, and EBV subtypes), provided by the literature [ 16 ]. Both datasets contain a sufficient number of samples and a reasonable number of subtypes. Additionally, for BRCA cancer subtype classification, we introduced a third dataset: the PAM50 subtype classification of breast invasive carcinoma (BRCA_2) from TCGA. Compared to the BRCA dataset, which covers broader molecular levels, BRCA_2 focuses more on RNA regulatory mechanisms. All datasets are summarized in Table 1 . Table 1. Summary of the BRCA, STAD, and BRCA_2 datasets, including molecular subtype categories and the number of features for each data modality (e.g., RPPA, CNV, RNA-seq, methylation, and miRNA) Dataset Categories Number of features BRCA Basal Her2 LumA LumB RPPA CNV RNA-seq 112 53 248 98 223 16383 16383 STAD CIN MSI GS EBV methylation miRNA mRNA 173 59 50 19 3287 702 4089 BRCA_2 Basal Her2 LumA LumB SNV miRNA RNA-seq 118 40 356 136 6 6510 2616 Open in a new tab It is worth noting that in the BRCA_2 dataset, the number of features in the SNV modality is only 6, showing a significant imbalance compared to miRNA (6510 features) and RNA-seq (2616 features). Such extreme differences in feature dimensions across modalities are common in multi-omics data. This significant imbalance among modalities may weaken the model’s ability to capture sparse modalities when using traditional multimodal fusion methods (such as feature concatenation or tensor outer product), as high-dimensional ones can overshadow the contributions of low-dimensional modalities. The Triad-LMF model effectively addresses this challenge through its unique design, and its efficiency and robustness in imbalanced modality scenarios will be discussed in detail in Section 3. We preprocessed the data before applying the Triad-LMF model to ensure quality and consistency. First, patient samples with more than 10% missing information in any modality were excluded from the dataset. This threshold was chosen based on standard practices reported in the literature [ 8 ] while retaining sufficient samples to support statistical analysis. Second, features with more than 10% missing values across all patient samples were removed to avoid impairing the model’s feature extraction capabilities due to incomplete information. These preprocessing steps transformed the multimodal data into a high-quality set, laying a solid foundation for modality-specific embedding and multimodal fusion. Evaluation metrics To comprehensively quantify Triad-LMF’s performance in cancer subtype multiclass classification tasks, we selected the following standard evaluation metrics. where TP refers to samples that actually belong to a certain class and are correctly predicted as that class, while TN refers to samples that the model correctly identifies as not belonging to the target class. For multi-class tasks, any class other than the positive class is defined as negative. FP refers to samples the model incorrectly predicts as belonging to the target class, and FN refers to samples that the model incorrectly predicts as not belonging to the target class. Accuracy , Precision , Recall , and are the most commonly used classification performance metrics based on the aforementioned TP , TN , FP , and FN . Considering the potential class imbalance in the dataset, we primarily report and Accuracy to evaluate model performance comprehensively. To further ensure the reliability of the results, all performance comparisons are validated for significance using the Wilcoxon rank-sum test. Experimental setup The implementation of Triad-LMF is based on the PyTorch framework, and training and testing were conducted on an NVIDIA GeForce RTX 3060 GPU. To ensure the model’s stability and generalization capability, we performed a systematic hyperparameter optimization on the validation set through grid search, focusing on key parameters such as the learning rate, local fusion rank, global fusion rank, and dropout rate. Table 2 lists the candidate value ranges for the hyperparameters. Table 2. Hyperparameter optimization settings for Triad-LMF, showing the candidate ranges evaluated for learning rate, batch size, network architecture, regularization, and dropout Hyperparameter Candidate value range Learning rate 1e − 3, 5e − 4, 1e − 4, 5e − 5 Batch size 32, 64, 128 Hidden layer 64, 128, 256 2, 4, 8 4, 8, 16 Weight decay 1e − 4, 1e − 5, 1e − 6 Dropout 0.1, 0.3, 0.5 Open in a new tab Triad-LMF uses the cross-entropy loss function and minimizes the loss through the Adam optimizer. To prevent overfitting, we introduce an early stopping mechanism that terminates training when the validation loss does not decrease for 10 consecutive epochs. Additionally, it should be noted that Dropout is not mentioned in the table; we adjust the corresponding Dropout rates according to the characteristics of each modality to balance regularization effects. All experiments are repeated five times, reporting the metrics’ mean and standard deviation to ensure the results’ robustness and reproducibility. As shown in Table 3 , the final selected hyperparameter combination achieves excellent classification performance on the validation set. Table 3. Experimental setup and training hyperparameters used for the Triad-LMF model, including optimization settings, network dimensions, and early stopping criteria Hyperparameter Value Hyperparameter Value Framework PyTorch Weight decay 1e-5 Optimizer Adam 128 Learning rate 1e-4 r 4 Batch size 64 8 Training epochs 100 Early stopping 10 Weight decay 1e-5 Early stopping patience 10 Hardware platform NVIDIA GeForce RTX 3060 Open in a new tab During training, we apply batch normalization to the preprocessed features to reduce input feature covariate shift. Moreover, we need to avoid overfitting in high-dimensional feature space since our datasets have a relatively small sample size. We applied Dropout regularization during training Triad-LMF outperforms existing methods for comparative cancer subtype diagnosis To rigorously evaluate the performance of Triad-LMF, we compared it against a series of baseline methods and related advanced multi-omics fusion approaches on the BRCA and STAD datasets. The baseline methods were constructed as follows: Naive multimodal fusion: directly concatenating feature vectors from all modalities, then inputting the concatenated high-dimensional features into traditional machine learning classifiers for training. Leading multi-omics fusion methods: Triad-LMF was further compared with MOGONET [ 6 ], DeepMo [ 21 ], and MoGCN [ 4 ]. The same training–testing data splits were used across all comparative methods to ensure fairness in all experiments. Performance evaluation uniformly used the metrics defined in Section 3.1. The experimental results are shown in Table 4 and Fig. 2 . On both the BRCA and STAD datasets, Triad-LMF consistently outperformed all baseline and comparative methods on key evaluation metrics. On the BRCA dataset, Triad-LMF achieved an F1 score of 0.8613, while the best-performing baseline method, RF, scored 0.8435. Similarly, as shown in Table 5 for the STAD dataset, Triad-LMF reached an Accuracy of 0.8761, surpassing MOGONET’s 0.8197 and DeepMo’s 0.8462. These results demonstrate that our proposed hierarchical fusion strategy can effectively integrate information from different omics data types, thereby significantly improving the Accuracy of cancer subtype classification. Table 4. Performance comparison of Triad-LMF and baseline methods for cancer subtype classification on the BRCA dataset, evaluated using accuracy, precision, recall, and F1-score (mean ± standard deviation), bold denotes the optimal results of Triad-LMF Method Accuracy Precision Recall F1 DT 0.8350 ± 0.021 0.7500 ± 0.030 0.7730 ± 0.028 0.7614 ± 0.027 KNN 0.6117 ± 0.035 0.3960 ± 0.045 0.3382 ± 0.040 0.3608 ± 0.042 GNB 0.5728 ± 0.038 0.3233 ± 0.048 0.3441 ± 0.045 0.3336 ± 0.046 RF 0.8830 ± 0.015 0.8710 ± 0.018 0.8174 ± 0.020 0.8435 ± 0.019 SVM 0.8447 ± 0.018 0.8379 ± 0.020 0.7773 ± 0.022 0.8068 ± 0.021 MoGCN 0.8002 ± 0.016 0.7919 ± 0.034 0.8015 ± 0.010 0.7967 ± 0.018 DeepMo 0.8626 ± 0.025 0.8079 ± 0.017 0.7871 ± 0.010 0.7871 ± 0.010 MOGONET 0.8738 ± 0.016 0.8837 ± 0.017 0.7644 ± 0.019 0.8196 ± 0.018 Triad-LMF 0.8947 ± 0.012 0.8950 ± 0.013 0.8299 ± 0.014 0.8613 ± 0.013 Open in a new tab Fig. 2. Open in a new tab Performance comparison of Triad-LMF and baseline methods on the BRCA and STAD datasets under standard supervised learning settings, evaluated using accuracy, precision, recall, and F1-score, where Triad-LMF consistently achieves superior performance across all metrics Table 5. Performance comparison of Triad-LMF and baseline methods for cancer subtype classification on the STAD dataset, evaluated using accuracy, precision, recall, and F1-score (mean ± standard deviation), bold denotes the optimal results of Triad-LMF Method Accuracy Precision Recall F1 DT 0.7377 ± 0.025 0.7480 ± 0.030 0.6614 ± 0.032 0.7022 ± 0.030 KNN 0.6885 ± 0.028 0.7456 ± 0.035 0.6038 ± 0.038 0.6675 ± 0.036 GNB 0.7049 ± 0.027 0.7438 ± 0.034 0.6057 ± 0.037 0.6679 ± 0.035 RF 0.8852 ± 0.014 0.8355 ± 0.017 0.8636 ± 0.016 0.8493 ± 0.015 SVM 0.8525 ± 0.016 0.8143 ± 0.019 0.8163 ± 0.018 0.8153 ± 0.017 MoGCN 0.8033 ± 0.033 0.8023 ± 0.008 0.8033 ± 0.012 0.8028 ± 0.007 DeepMo 0.8462 ± 0.014 0.8660 ± 0.034 0.8608 ± 0.008 0.8634 ± 0.017 MOGONET 0.8197 ± 0.018 0.8618 ± 0.020 0.7727 ± 0.021 0.8149 ± 0.020 Triad-LMF 0.8761 ± 0.013 0.8445 ± 0.015 0.8217 ± 0.016 0.8330 ± 0.014 Open in a new tab Revealing the latent space structure of Triad-LMF through UMAP To qualitatively analyze the Triad-LMF model’s ability to learn discriminative data representations during the hierarchical fusion process, we employed UMAP technology to reveal the model’s progressive refinement of cancer subtype discriminatory information through two-dimensional visualization. UMAP is a nonlinear dimensionality reduction method that effectively preserves both local and global topological structures of high-dimensional data, making it suitable for evaluating the discriminability of latent space representations [ 22 ]. Specifically, we extracted embedding vectors from four key stages of Triad-LMF’s hierarchical fusion architecture: three Local Pairwise Fusion stages and one Global Triadic Fusion stage. Based on test samples from the BRCA and STAD datasets, these vectors reflect the model’s representational capability for cancer subtype classification at different fusion stages. We applied UMAP to project the high-dimensional embedding vectors into a two-dimensional space to assess samples’ clustering characteristics and inter-cluster separability in the latent space. As shown in Fig. 3 , during the Local Pairwise Fusion stages, the model already demonstrated preliminary subtype differentiation ability, with samples of the same subtype forming loose clusters and cluster boundaries exhibiting some level of separability. However, the final latent space representation after Global Triadic Fusion significantly outperformed the earlier stages, evidenced by tighter subtype clusters and more precise boundaries between clusters. These visualizations reveal the mechanism by which Triad-LMF progressively enhances representational discriminability through modal synergy, validating the effectiveness of its hierarchical fusion architecture in integrating multi-modal information. Fig. 3. Open in a new tab UMAP visualizations of learned representations for the STAD ( A ) and BRCA ( B ) datasets, comparing local pairwise fusion results (left three columns) with global triadic fusion (rightmost column), where global triadic fusion yields clearer class separation Ablation study To quantitatively evaluate the contributions of each key component in the Triad-LMF framework–notably the Local Pairwise Fusion and Global Triadic Fusion–to the overall model performance, we designed and conducted a series of ablation experiments using the BRCA dataset. These experiments investigated the impact on performance metrics by sequentially removing certain components of Triad-LMF or replacing specific parts of the framework. The training loss dynamics are shown in Fig. 4 (A), and the final classification performance comparison is presented in Fig. 4 (B). The proposed Triad-LMF framework achieves the best performance in terms of the ACC metric, confirming its overall effectiveness. In contrast, removing any Local Pairwise Fusion module from Triad-LMF causes a significant drop in Accuracy, indicating that pairwise cross-modal interactions are crucial for enhancing classification performance and that the model effectively extracts valuable discriminative features from the fusion of modal pairs. Notably, when the Global Triadic Fusion is removed and replaced with simple concatenation, the model’s performance falls short of the full Triad-LMF and underperforms compared to the baseline model. This result highlights that simply obtaining pairwise fused features is insufficient; the subsequently designed low-rank tri-modal fusion mechanism is essential for effectively integrating information from all three pairwise interactions and capturing higher-order complex relationships rather than merely summing the information. Through systematic ablation experiments, we confirm that each core component of the Triad-LMF model contributes positively to the final performance. Removing any single component leads to a performance decline. This fully validates the rationality and effectiveness of our proposed hierarchical, two-stage fusion architecture, demonstrating that Triad-LMF can fully leverage both pairwise cross-modal interactions and higher-order tri-modal integrated information to achieve superior cancer subtype classification performance. Fig. 4. Open in a new tab A BRCA ablation experiment accuracy bar chart ( B ) BRCA ablation experiment loss function line chart (local pairwise fusion1: removed the fusion of modalities 1 and 2 from the baseline. Local Pairwise Fusion2: removed the fusion of modalities 2 and 3 from the baseline. Local pairwise fusion3: removed the fusion of modalities 2 and 3 from the baseline. No global triadic fusion: removed Global Triadic Fusion from the baseline, replaced with simple concatenation) Feature importance analysis and biological interpretation based on SHAP To explain the cancer subtype classification mechanism of Triad-LMF and explore its biological significance, we employed the SHapley Additive exPlanations (SHAP) method [ 23 ] for a systematic analysis of feature importance in Triad-LMF. SHAP is a model interpretability tool based on game theory that quantifies the marginal contribution of each input feature to the model output, providing both global and local assessments of feature importance. We calculated SHAP values for all features in the test set of the BRCA dataset, with a focus on the features with the highest overall contributions and those that have a significant impact on the classification of specific cancer subtypes. Using SHAP’s TreeExplainer, we analyzed the prediction results of the Triad-LMF model, computed SHAP values for each feature, and ranked the features based on their mean absolute SHAP values. Table 6 lists the top 60 ranked features, encompassing key proteins, phosphorylation sites, and other biomarkers. These features play important roles in the molecular mechanisms of breast cancer, reflecting the model’s sensitivity to biologically relevant signals. Additionally, as shown in Fig. 5 (C), we present the sensitivity of the four BRCA subtype categories to the significant features. Table 6. Top 60 features ranked by SHAP importance scores for the Triad-LMF model on the BRCA dataset, highlighting the most influential features contributing to cancer subtype prediction Rank Feature 1 S6 2 ERALPHA 3 PR 4 P27_pT198 5 BCL2 6 P90RSK_pT359S363 7 CAVEOLIN1 8 EEF2 9 EGFR 10 IRS1 11 SYK 12 CASPASE7CLEAVEDD198 13 PAXILLIN 14 HER3 15 ASNS 16 RICTOR_pT1135 17 MYOSINIIA 18 ACETYLATUBULINLYS40 19 HER2_pY1248 20 GSK3ALPHABETA Rank Feature 21 EGFR_pY1173 22 S6_pS235S236 23 SRC_pY527 24 G6PD7 25 CYCLINB1 26 DVL3 27 TUBERIN_pT1462 28 ERALPHA_pS118 29 JAK2 30 LCK 31 AMPKALPHA_pT172 32 ANNEXINVII 33 EGFR_pY1068 34 XRCC1 35 RBM15 36 IGFBP2 37 AR 38 PARPCLEAVED 39 PDL1 40 STAT3_pY705 Rank Feature 41 PDK1 42 PTEN 43 CYCLINE1 44 MYH11 45 GATA3 46 TFRC 47 PI3KP110ALPHA 48 ANNEXIN1 49 MAPK_pT202Y204 50 BIM 51 CASPASE3 52 SCD1 53 AMPKALPHA 54 RICTOR 55 MYOSINIIA_pS1943 56 GATA6 57 HER2 58 RAB11 59 PCADHERIN 60 CHK2 Open in a new tab Fig. 5. Open in a new tab Biological interpretation of important features identified by Triad-LMF on the BRCA dataset: A GO functional enrichment analysis of the top 60 SHAP-ranked features, highlighting enrichment in apoptosis- and cell death–related processes; B average absolute SHAP values illustrating the relative importance of the top 60 features; and C SHAP-based sensitivity analysis showing the contributions of significant features across four BRCA molecular subtypes As shown in Table 6 , the top-ranked features show the precision of the Triad-LMF model in capturing the key nodes of breast cancer biology. Ribosomal protein S6 is a core downstream effector of the mTOR signaling pathway, regulating protein synthesis and cell proliferation. Its significant ranking suggests that Triad-LMF is highly sensitive to mTOR-driven tumor proliferation. Aberrant activation of the mTOR signaling pathway is widespread in breast cancer and is closely associated with tumor progression and treatment resistance [ 24 ]. The primacy of S6 in the SHAP analysis is consistent with the clinical efficacy of mTOR inhibitors (e.g., everolimus), which studies have shown to significantly improve the prognosis of patients with hormone receptor-positive breast cancer [ 25 , 26 ]. ERALPHA and PR are signature molecules of the LumA and LumB subtypes that regulate hormone-dependent gene expression and tumor cell proliferation. The high ranking reflects the model’s prioritized focus on hormonal signaling pathways, which is highly consistent with breast cancer molecular typing studies [ 27 ]. ERALPHA expression level is a key predictor of the efficacy of endocrine therapies (e.g., tamoxifen), and PR expression further refines prognostic assessment of the Luminal subtype [ 28 , 29 ]. The ability of Triad-LMF to recognize both underscores its Precision in the classification of hormone receptor-positive breast cancer.P27_pT198 and BCL2 are involved in cell cycle regulation and anti-apoptotic mechanisms, respectively.P27, through the phosphorylation site T198 regulates nucleoplasmic localization and cell cycle progression, and its high ranking suggests that the model focuses on cell cycle dysregulation in breast cancer progression [ 30 ]. It has been shown that P27 phosphorylation correlates with poor prognosis, reflecting the active state of tumor proliferation [ 31 ].BCL2, as an anti-apoptotic protein, inhibits programmed cell death and promotes tumor survival. It is often highly expressed, especially in Luminal subtypes. The prominent ranking of BCL2 in the SHAP analysis suggests that the model effectively captures the tumor survival mechanisms, consistent with its role in breast cancer prognosis and treatment response [ 32 ]. eGFR_pY1068, HER2_pY1248, and MAPK_pT202Y204 mark the activation status of EGFR, HER2, and MAPK/ERK signaling pathways, respectively. These features are particularly prominent in HER2+ subtypes, driving cell proliferation and survival.HER2_pY1248 is a key target for targeted therapies (e.g., trastuzumab), whereas MAPK_pT202Y204 is associated with tumor invasion and metastasis [ 33 ]. The high ranking of EGFR_pY1068 is further suggestive of the model’s identification of the EGFR signaling pathway in breast cancer heterogeneity [ 34 ]. The outstanding contribution of these features suggests that the model Triad-LMF accurately captures the dynamics of the signaling network, consistent with the complexity of breast cancer molecular mechanisms [ 35 ]. In addition, PI3KP110ALPHA, PTEN, GATA3, and PDL1 regulate the PI3K/AKT pathway, differentiation characteristics of Luminal subtypes, and immune escape mechanisms in TNBC, respectively.PI3KP110ALPHA and PTEN co-regulate the PI3K signaling pathway, which is closely associated with tumor proliferation and treatment resistance [ 36 ]. GATA3, as a transcription factor, is associated with the hyper differentiation characteristics of the Lum subtype, and its ranking reflects the model’s focus on subtype-specific features [ 37 ]. PDL1, on the other hand, is a key molecule for immune escape in TNBC, which is correlated with the efficacy of immune checkpoint inhibitors, suggesting that the model integrates the tumor microcircuits with the tumor microcircuits–correlates, suggesting that the model integrates tumor microenvironmental features [ 38 ]. Identifying these features further validates the model’s ability to integrate signal transduction, molecular differentiation, and immune regulation. The top 60 features identified by SHAP are highly consistent with the known molecular mechanisms of breast cancer, encompassing mTOR, hormone receptors, PI3K/AKT, EGFR/HER2, and immune-related pathways. This validates the Triad-LMF model’s biological interpretability and highlights its potential application value in Precision medicine, providing an important basis for biomarker validation and targeted therapy research. Next, we conducted a Gene Ontology (GO) on the top 60 high-ranking features, focusing on the Biological Process category. The results showed significant enrichment ( ) in key pathways, including (1) negative regulation of programmed cell death involving BCL2, CASPASE7CLEAVEDD198, CASPASE3, and BIM, underscoring tumor anti-apoptotic mechanisms closely related to breast cancer cell survival and therapy resistance [ 32 , 39 ]; (2) regulation of DNA recombination, including XRCC1 and CHK2, reflecting the role of DNA repair and genome stability in BRCA1/2 mutant subtypes [ 40 , 41 ]; (3) phosphatidylinositol-mediated signaling, covering IRS1, PDK1, and PI3KP110ALPHA, pointing to the PI3K/AKT/mTOR pathway, which is associated with tumor proliferation and resistance to targeted therapies [ 25 , 36 ]. These pathways are highly consistent with the molecular mechanisms of breast cancer, for example, the PI3K/AKT pathway driving cell survival [ 42 ] and DNA repair influencing genome instability subtypes [ 35 ]. Figure 5 (B) visually illustrates the associations between these pathways and genes, while Fig. 5 (C) further quantifies pathway significance and gene counts. We confirmed that the Triad-LMF model can achieve interpretable subtype classification by capturing key biological processes. Validating effectiveness using other BRCA datasets To evaluate the robustness and generalization ability of the Triad-LMF framework, we applied it to the BRCA_2 dataset. This dataset features the same cancer types and subtypes as BRCA but utilizes a different combination of omics modalities. We trained and evaluated the Triad-LMF model on the BRCA_2 dataset using the same hyperparameter settings applied to the original BRCA dataset. Triad-LMF also demonstrated strong performance on the BRCA_2 dataset, achieving an accuracy of 0.9221, an F1 score of 0.9150, a precision of 0.9082, and a recall of 0.9219. As shown in Fig. 6 , both the training and testing accuracy and loss curves remain closely aligned throughout the learning process, indicating stable convergence and minimal overfitting. These results are comparable to those obtained on the BRCA dataset, confirming consistent performance across both datasets. This indicates that the proposed Triad-LMF framework is not limited to specific combinations of omics data types but effectively adapts to different modalities while maintaining robust classification performance for the same biological subtypes. Overall, these findings further support the universality and robustness of the proposed hierarchical fusion strategy for multi-omics cancer data integration. Fig. 6. Open in a new tab Training and testing curves of Triad-LMF on the BRCA_2 dataset across training epochs, including the loss curves (left) and accuracy curves (right), which illustrate the model’s learning dynamics during training Discussion In cancer research, multi-omics data integration plays a crucial role in revealing the complexity of diseases and improving diagnostic Accuracy. Our proposed Triad-LMF model features a hierarchical fusion architecture that integrates progressively from local modality interactions to global information consolidation through the Two-Feature LMF module and Three-way LMF modules. This effectively captures the complex interactions among multi-omics data, demonstrating promising application potential. However, although the Triad-LMF model performs well overall, it was found that tri-modal fusion may introduce noise in specific tasks, leading to performance that is inferior to single-modal or dual-modal combinations, this may occur when modal redundancy occurs, or some modalities may carry more noise than signal. Including them can dilute the predictive power of strong modalities. This indicates that balancing information integration and noise control during the fusion process is a key challenge. In future work, we plan to investigate more adaptive fusion mechanisms that dynamically optimize modality selection and fusion weights [ 43 ], or Redundancy-adaptive multimodal learning [ 44 ] to mitigate the impact of redundant information, thereby enhancing model robustness and Accuracy. Additionally, given the complexity and diversity of cancer research, we intend to expand the model to incorporate more types of omics data. Considering the efficiency and generalizability of Triad-LMF, we believe that through these improvements and extensions, the model will play a more critical role in pan-cancer subtype classification, biomarker discovery, and precision medicine research, providing a more powerful tool to advance multi-omics-driven cancer studies and accelerate the individualization and precision of cancer diagnosis and treatment. Moreover, in the absence of considering label noises, which some subtypes might be mislabeled [ 45 ], future work could focus on developing robust methods against label noises. Conclusion This study proposes a deep learning framework called Triad-LMF based on low-rank multimodal fusion, aiming to address the challenges of integrating multi-omics data for cancer subtype classification. Through extensive experiments on BRCA, STAD, and BRCA_2 datasets, we demonstrate the significant advantages of Triad-LMF in classification performance, robustness, and generalization ability. Experimental results show that Triad-LMF achieved an F1 score of 0.8613 on the BRCA dataset, 0.8330 on the STAD dataset, and an Accuracy of 0.9237 on the BRCA_2 dataset, surpassing traditional machine learning baselines and advanced multi-omics fusion methods such as MOGONET. UMAP visualization validated the critical role of Global Triadic Fusion in enhancing the latent space’s ability to distinguish subtypes, manifested by more compact subtype clusters and clearer boundaries between clusters. Ablation studies further confirmed the necessity of both Local Pairwise Fusion and Global Triadic Fusion; removing either component led to performance degradation, highlighting the synergistic effect of the hierarchical fusion architecture. SHAP-based feature importance analysis These findings provide important evidence for identifying subtype-specific biomarkers and developing targeted therapeutic strategies. Acknowledgements The authors would also like to thank Mr. Xiongtao Xiao for valuable discussion and thoughtful comments on the manuscript and language revision. Author contributions XC and LZ conceptualized the project, XT and XC wrote and edited the original manuscript. XT and XC performed coding. XT conducted experiments. RT visualized the results. MJ and QC curated the data. DY and LZ edited the manuscript, LZ supervised the study. All authors read and approved the final manuscript. Funding This work was supported by the National Natural Science Foundation of China under Grants (62276034, 62306052), Group Building Scientific Innovation Project for Universities in Chongqing (CXQT21021), Joint Training Base Construction Project for Graduate Students in Chongqing (JDLHPYJD2021016), Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202100712), and Chongqing Jiaotong University Internal Science Research Fund Project 2014kjc-A002. This work was supported by the Key Laboratory of Complex Systems Optimization and Intelligent Control of Chongqing Municipal Education Commission, Chongqing Jiaotong University. Data availability No datasets were generated or analysed during the current study. Declarations Ethics approval and consent to participate Not applicable. Consent for publication Not applicable. Competing interests The authors declare no Conflict of interest. Footnotes Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Xingyue Tan and Xiran Chen have contributed equally to this work. References 1. Chen X, Wang Y, Zhang Z, Zhang J, Zhang Y. Deep-learning approach to identifying cancer subtypes using high-dimensional omics data. Bioinformatics. 2020;36(5):1476–83. 10.1093/bioinformatics/btz769. [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] 2. 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