Estimates \(d_{\mathrm{eff}}(\mu) = \sum_j \lambda_j / (\lambda_j + \mu)\) from the eigenvalues of a low-rank approximation, and the rank \(2 \lceil 1.5\, d_{\mathrm{eff}} \rceil + 1\) that Frangella, Tropp and Udell show is enough for a well-conditioned preconditioner.
Value
A list with d_eff, recommended_rank, and sufficient, which
is TRUE when approx already has at least the recommended rank.
Details
The estimate uses only the retained eigenvalues, so it can only
understate the true effective dimension. If the recommended rank exceeds
the rank of approx, build a larger approximation and ask again.
Examples
set.seed(1)
X <- matrix(rnorm(1000), ncol = 2)
effective_dim(nystrom(kernel_matrix(X), l = 60), mu = 1e-2)
#> $d_eff
#> [1] 25.95871
#>
#> $recommended_rank
#> [1] 79
#>
#> $sufficient
#> [1] FALSE
#>