Skip to main content

Table 1 Time complexity of stochastic algorithms

From: Stochastic Lanczos estimation of genomic variance components for linear mixed-effects models

MethodOverheadObjective function evaluation
SLDF_REML\(\left \{\begin {array}{l}\text {with precomputed GRM}\\ \text {with genotype matrix}\end {array}\right.\)\(\mathcal {O}{\left (n^{2}\cdot (n_{\text {rand}}+c)\cdot n_\kappa \right)}\)\(\mathcal {O}{(n\cdot c\cdot n_\kappa)}\)
 \(\mathcal {O}(2m\cdot n\cdot (n_{\text {rand}}+c)\cdot n_\kappa)\)\(\mathcal {O}(n\cdot c\cdot n_\kappa)\)
L_FOMC_REML\(\mathcal {O}(4m\cdot n\cdot n_{\text {rand}}\cdot n_\kappa)\)\(\mathcal {O}(m\cdot n\cdot n_{\text {rand}})\)
BOLT_LMM\(\mathcal {O}\left (n\cdot c^{2}+m\cdot c\right)\)\(\mathcal {O}(4m\cdot n\cdot n_{\text {rand}}\cdot n_\kappa)\)
  1. n denotes the number of individuals, m the number of markers, and c the number of covariates. nrand indicates the number of random probing vectors and is fixed at 15 in all numerical experiments. nκ reflects the number of conjugate gradient iterations required to achieve convergence at a specified tolerance and can be bounded in terms of the spectral condition number of H0. As noted in [8], implicit preconditioning of H0 can be achieved by including the first few right singular vectors of the genotype matrix (or eigenvectors of the GRM) as covariates