Finding median eigenvalue with sparse matrix in r

Question

I am working with SVD on a matrix $$Y_{m,n} = T_{m,m} \Sigma D^T_{n,n} $$
where $T$ and $D$ describe the row and the column entities of Y, respectively.

The truncated SVD takes the first $r$ eigenvalues and reduces as well the dimensionality of the problem: $$ \hat{Y}{m,n} = T{m,r} S_{r,r} D^T_{r,n} $$

Instead of looking at the scree plot and get the 90% of Variance https://arxiv.org/pdf/1305.5870.pdf sets out a (approximate) rule to pick r optimally (eq. 5). However, the approximation depends on $\sigma_{med}$ that is the "median empirical singular value" of the matrix $\Sigma$.

Problem is that Y is a sparse matrix 150000*400000 and I don't know it's rank and the number of eigenvalues. I'd like to run svd on the matrix get the diagonal matrix and find the optimal $\tau$ (truncation threshold under which the $\sigma_{i}$ is not considered), however I cannot compute the full svd because the problem is too large. I could think of few options:

1. run svds with very large $r$ and then computing $s_{med}$ as an approximation
2. given second answer Full SVD of a large sparse matrix (where only the eigenvalues are required) look for eigs of square 150k matrix (by $YY^T$)
3. switch to matlab or other languages?

what I tried: > lambdas = eigs(Y %*% Matrix::t(Y),symmetric=TRUE,only.values=TRUE)

getting error: Error in eigs_real_sym(A, nrow(A), k, which, sigma, opts, mattype = "sym_dgCMatrix", : argument "k" is missing, with no default

Any shortcut that exploits linear algebra properties of the matrix and doesn't require computing the whole decomposition?