The combinatorial effect of genetic variants is often assumed to be additive. Although genetic variation can clearly interact non-additively, methods to uncover epistatic relationships remain in their infancy. We develop low-signal signed iterative random forests to elucidate the complex genetic architecture of cardiac hypertrophy. We derive deep learning-based estimates of left ventricular mass from the cardiac MRI scans of 29,661 individuals enrolled in the UK Biobank. We report epistatic genetic variation including variants close to CCDC141, IGF1R, TTN, and TNKS. Several loci not prioritized by univariate genome-wide association analysis are identified. Functional genomic and integrative enrichment analyses reveal a complex gene regulatory network in which genes mapped from these loci share biological processes and myogenic regulatory factors. Through a network analysis of transcriptomic data from 313 explanted human hearts, we show that these interactions are preserved at the level of the cardiac transcriptome. We assess causality of epistatic effects via RNA silencing of gene-gene interactions in human induced pluripotent stem cell-derived cardiomyocytes. Finally, single-cell morphology analysis using a novel high-throughput microfluidic system shows that cardiomyocyte hypertrophy is non-additively modifiable by specific pairwise interactions between CCDC141 and both TTN and IGF1R. Our results expand the scope of genetic regulation of cardiac structure to epistasis.