  With the emergence of Artificial Intelligence, numerical algorithms are
moving towards more approximate approaches. For methods such as PCA or
diffusion maps, it is necessary to compute eigenvalues of a large matrix, which
may also be dense depending on the kernel. A global method, i.e. a method that
requires all data points simultaneously, scales with the data dimension N and
not with the intrinsic dimension d; the complexity for an exact dense
eigendecomposition leads to $\mathcal{O}(N^{3})$. We have combined the two
frameworks, $\mathsf{datafold}$ and $\mathsf{GOFMM}$. The first framework
computes diffusion maps, where the computational bottleneck is the
eigendecomposition while with the second framework we compute the
eigendecomposition approximately within the iterative Lanczos method. A
hierarchical approximation approach scales roughly with a runtime complexity of
$\mathcal{O}(Nlog(N))$ vs. $\mathcal{O}(N^{3})$ for a classic approach. We
evaluate the approach on two benchmark datasets -- scurve and MNIST -- with
strong and weak scaling using OpenMP and MPI on dense matrices with maximum
size of $100k\times100k$.
