Exploiting Low-Rank Structure in Genomic Relationship Matrices for Computationally Efficient Optimum Contribution Selection

Motivation: Optimum contribution selection (OCS) maximizes genetic gain under a constraint on relatedness. Genomic relationship matrices are dense, so the quadratic term costs O(n^3) operations, and existing OCS software assumes pedigree sparsity. Results: We replace the relationship matrix by its leading eigenpairs, computed by randomized SVD, and solve OCS in factorized form. On Norway spruce population (n=5,525), ranks 10--30 kept the exact objective within 1% of the optimum at 110--330-fold speedups; on the QTL-MAS 2010 (n=3,226), ranks up to 100 were needed (24-211 fold). The required rank rises as the diversity constraint loosens. Gain alone is a misleading accuracy check, since coancestry exceeded the optimum by up to 28%, so we choose the rank on the exact objective, which can be checked without a dense solve. The speedup comes from computing only the leading eigenpairs. We extend the factorization to robust OCS (18-fold). Availability and implementation: Julia implementation, and examples are available at https://github.com/jonhar97/fastocs under the MIT license (archived at Zenodo, DOI 10.5281/zenodo.23214804).