Algorithm 1: MRF-MSC algorithm.

Input: cancer multi-omics data X = {X1, X2, ⋯, Xt}, the number of cancer subtypes k, the maximum number of iterations MaxIter, K is the number of neighbors in KNN, hyperparameters α, β and λ. Output: smooth representation of each omics data Zv, fused similarity graph S, eigenvectors F.

Initialize S = I, εv = 1/t. Repeat Update Zv according to Eq. 13, Set $z_{ij}^v=\max (z_{ij}^v,0)$ for every element $z_{ij}^v$ in Zv, Update S according to Eq. 16, Update F by optimizing Eq. 17 Update εv according to Eq. 7, Until meeting stop condition Stop condition: the maximum number of iterations MaxIter is reached or the relative change of S is less than 10–3.