A recursive method of approximation of the inverse of genomic relationships matrix

Genomic evaluations by some procedures such as genomic BLUP (GBLUP) or single-step GBLUP (ssGBLUP) use the inverse of the genomic relationship matrix (G). The cost to create such an inverse is cubic and becomes prohibitively expensive after 30–100k genotypes. The purpose of this study was to develop methodologies, which eventually could compute a good approximation of G−1 at reduced cost. A
recursive approximation of the inverse is based on a decomposition similar to that for the pedigree-based relationship: G−1 = (T−1)’D−1T−1, where T is a triangular and D a diagonal matrix. In the first step, animals are processed from the oldest to the youngest. For each animal, a subset of ancestors is selected with coefficients of genomic relationship to that animal greater than a threshold. A system of equations is
created where the coefficients of G for the selected ancestors are in the left hand side and the coefficients of G for the given animal corresponding to the ancestors are the right hand side. The solution to that
system of equation is stored in one line of T. Then, D is computed as diagonal elements of T−1G(T−1)’. If off-diagonals of D are too large, the approximation to G can be improved by repeated applications of
G−1 = (T−1)’D−1T−1 and D = T−1G(T−1)’. After n rounds, the approximation of G inverse is a product of 2n triangular matrices and one diagonal matrix. This recursive method has been assessed on a sample
of 1,718 genotyped dairy bulls. The correlation between GEBV using the complete or approximated G were 0.54 in the first round, 0.96 in the second, and 0.99 in the third. The cost of the proposed method depends on the population structure. It is likely to be high for closely related animals but lower for populations where few animals are strongly related. Additional research is needed to identify near-sparsity in T and D to eliminate unimportant operations.