Implicit Hierarchical Tensor Decomposition of Single-Cell Four-Omics Data Reveals Cell-Type-Associated Enhancer--Promoter Regulatory Programs

Single-cell multi-omics provides complementary views of gene regulation, but integrating modalities with natural features comprising enhancers, genes, and three-dimensional (3D) genomic loci remains challenging. This study developed an implicit hierarchical tensor decomposition framework and applied it to the CHARM single-cell four-omics mouse brain data (GSE303006) by jointly analyzing ATAC, H3K27ac, RNA, and reconstructed 3D enhancer--promoter (E--P) distances. After quality control, 730,969 E--P pairs, 42,669 enhancers, 16,239 genes, 391,435 20-kb bin pairs, and 4,258 cells were retained. Each modality was independently reduced to 20 components and mapped to an implicit 730,969x4, 258x20x4 tensor, which was decomposed without materializing the full array. All 12 distinct cell components represented among the 100 largest Tucker core elements showed greater cell-type effects than replicate effects (P=2.44 x 10-4, sign test). A representative core linked the Inh_Ndnf/Lamp5--Ex_L3/4_IT cell axis to contrasting presynaptic-transmission and developmental/morphogenetic E--P programs. Backprojection showed reproducible ATAC, H3K27ac, and 3D effects across three biological replicates, whereas module-level RNA effects were weaker. The framework reveals the regulatory structure that is shared across molecular modalities while remaining distinct from any single molecular modality.