In our sparse version of the MBPLS problem, we searched the sparse representations of loading vectors whose non-zero elements can form a multi-dimensional module.
The maximization of covariance between t and u can reveal the associations between from and Y, which lead to the discovery of a multi-dimensional module.
Specifically, we want to identify a multi-dimensional module whose input variables have the maximum covariance with response variables across a subset of samples.
From each multi-dimensional module, we formed a set consisting of genes involved in the GE dimension, CNV-harbored genes, methylation-adjacent genes and microRNAs.
The non-zero elements of converged loading vectors and latent variable (t) identify a multi-dimensional module that contains subsets of input and response variables and a subset of samples.
Although the solution to this objective can identify a multi-dimensional module by selecting those variables and samples with large absolute values from, such module may not be the most distinct.
The multi-dimensional module would appear in the left-top and right-bottom corners (whose corresponding variables and samples have large absolute values of weights) in the reordered blocks.
For example, the reordered blocks by MBPLS as shown in Figure S1(B -panel2 (in Supplementary material) are oB -panel2o have no clear modular submatrineSupplementary while a materialmensionaremobservedn be observed in reordered blocks by sMBPLS in Figure S1(B)-panel3 and zoomed outoin Figure S1(C).
Given three input blocks and a response block Y, a multi-dimensional module is defined by satisfying the criterion "the profiles extracted from columns across k rows of (i = 1, 2, 3) has strong association with (or has similar and coherent pattern with) those from m columns across the same k rows of Y" (Fig. 1).
We then test the transcriptional homogeneity for each dimension of the identified multi-dimensional modules.
In this article, we further expanded the MBPLS method by imposing sparse constraints to identify multi-dimensional modules.
