Phenotypic variation across related individuals is often correlated: a high or low value of one phenotype tends to be associated with a high or low value of others. This may reflect lowerdimensional structure in the genotype-phenotype map, such that genotype affects a relatively small set of unobserved "core" processes that in turn determine the observed phenotypes. Identifying low-dimensional structure in high-dimensional genotype-phenotype data thus offers the promise of inferring the identity and genetic basis of core biological processes, as well as the way in which core processes determine each observed phenotype. However, inferring this lower-dimensional structure requires appropriate biologically motivated constraints, even with high-throughput genotypephenotype measurements. Here, we show that several recent empirical genotype-phenotype data sets exhibit evidence of sparse structure, and that a sparsity-favoring matrix decomposition approach can accurately recover latent processes if each genetic perturbation affects few core processes or if each phenotype is affected by few core processes. Motivated by this, we develop a generally applicable framework based on penalized matrix decomposition for sparse structure discovery (SSD) and apply it to three empirical datasets spanning adaptive mutations in yeast, genotoxin robustness assay in human cell lines, and genetic loci identified from a yeast cross. More generally, we propose sparsity as a guiding prior for resolving latent structure in empirical genotype-phenotype maps.
