# DiffRS-net: A Novel Framework for Classifying Breast Cancer Subtypes on Multi-Omics Data
## 2 Materials and Methods
### Multi-Way Association Analysis  
A comprehensive integrated analysis of mRNA, miRNA, and DNA methylation data is essential for uncovering the molecular basis of cancer development and progression. The complex interactions between these different omics play a key role in determining the phenotypic complexity of breast cancer subtypes. Although Sparse Canonical Correlation Analysis methods have been leading the way in regard to uncovering these complex biomolecular interactions and selecting significant features [35; 36], traditional SCCA approaches typically only identify bivariate relationships between two variables [37]. It may not fully capture the subtle relationships present in a network of associations that link mRNA, miRNA, and DNA methylation. To delve into these complex interactions and classify breast cancer subtypes with higher precision, we employ a robustness-aware adaptive SMCCA model(rAdaSMCCA). This model is able to perform adaptive balancing among SCCA models, thus effectively solving the problem of gradient domination when integrating multiple SCCA models. Suppose we have \(n\) samples with \(p\) DE-mRNAs, \(d\) DE-miRNAs, and \(q\) DE-DNAms. Then, let \(\mathbf{X_{1}}\in\ R^{n\times p}\) denote the DE-miRNA expression data, \(\mathbf{X_{2}}\in R^{n\times d}\) denote the DE-miRNA expression data, and \(\mathbf{X_{3}}\in\ R^{n\times q}\) denote the DE-DNAm expression data. The rAdaSMCCA model integrates three paired SCCA models related to three types of omics, aiming to maximize a comprehensive objective on a global scale. Specifically, the formal definition of rAdaSMCCA can be defined as follows [38]:  
\[ \begin{array}{l}\min_{\mathbf{w_{1}},\mathbf{w_{2}},\cdots,\mathbf{w_{T}}} \sum_{i<j}\parallel\mathbf{X_{i}}\mathbf{w_{i}}-\mathbf{X_{j}}\mathbf{w_{j}} \parallel_{2}+\sum_{t}\Omega(\mathbf{w_{t}})\\ \text{s.t.}\parallel\mathbf{X_{t}}\mathbf{w_{t}}\parallel_{2}^{2}=1,\forall t =1,2,\cdots,T\end{array} \, \tag{1}\]  
where \(\mathbf{w_{t}}\) is the canonical weight for each kind of omics, and \(\Omega(\mathbf{w_{t}})\) is a regularization term used to introduce sparsity, thereby selecting the omics features of interest. Generally, mRNA does not directly influence the levels of miRNA and DNA methylation. However, the expression patterns of mRNA can be indicative of the regulatory effects imposed by miRNA and DNA methylation, as both have the capacity to affect the mRNA levels of specific genes [39; 40]. To identify the isolated impact of a specific feature, the model employs the \(\ell\)1-norm to regulate the weights of DE-mRNA data. At the same time, to discern the combined influence of multiple features, we utilize a fused pairwise group  
Figure 2: Volcano plots of differentially expressed mRNA, miRNA, and DNAm between the healthy group and the disease group of breast cancer patients. Red dots represent genes with high expression, while blue dots indicate genes with low expression. Genes with normal expression are shown as gray dots. Both highly expressed and lowly expressed genes are considered to be differentially expressed genes (DE-genes). lasso approach that penalizes them based on the genomic positions of each adjacent pair of features. Therefore, \(\Omega(\mathbf{w}_{1})\) for DE-mRNA data takes the following form:  
\[\Omega(\mathbf{w_{1}})=\gamma_{1}\beta\sum_{i=1}^{p-1}\sqrt{w_{1i}^{2}+w_{1(i+1) }^{2}}+\gamma_{1}(1-\beta)||\mathbf{w_{1}}|\,|_{1}, \tag{2}\]  
where \(\gamma_{1}\) and \(\beta\) are non-negative hyperparameters. \(\beta\) achieves a balance between joint feature selection and individual feature selection, while \(\gamma_{1}\) finds an equilibrium between overall penalties and the loss function. Therefore, this compound penalty method fosters a comprehensive and effective feature selection mechanism. Furthermore, given that not every miRNA/DNA methylation expression feature is related to the formation of breast cancer, it is necessary to apply sparse constraints to select the miRNA and DNA methylation features that are relevant.