EpiTensor

The EpiTensor model is based on high-order tensor decomposition (see Supplementary Materials for details). Let D_mnk be a third order tensor, where m is the cell type, n is the assay index and k the genomic locus index. Applying tensor decomposition to D, we obtain D=S × ₁U^cell × ₂U^assay × U^locus, where U^cell is the cell type subspace, U^assay is the assay subspace, U^locus is the genomic locus subspace and S is the core tensor that governs the interactions among the three subspaces. In this study, we focused on analysing U^locus, which encodes the spatial association among distal loci. Each eigenlocus vector in U^locusrepresents one epigenomic pattern. Dimension reduction in tensor decomposition was obtained by computing for D, a best rank-(R₁,R₂,…,R_N)approximation⁵⁷ , which minimizes the error function, subject to (U^cell)^T U^cell=I, (U^assay)^T U^assay=I, and (U^locus)^T U^locus=I (ref. ⁵⁸). The three constraints are to ensure orthonormality of the three subspaces. In practice, we used full rank in the cell and assay subspaces because we focus on the locus subspace. In the locus subspace, we chose a rank that keeps at least 95% of the original energy .