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 $100k\times100k$.

翻译：随着人工智能的兴起，数值算法正朝着更近似的方法发展。对于主成分分析或扩散映射等方法，需要计算大型矩阵的特征值，该矩阵可能因核函数的不同而呈现稠密特性。全局方法，即同时需要所有数据点的方法，其规模随数据维度N而非内在维度d变化；精确稠密特征分解的复杂度导致$\mathcal{O}(N^{3})$。我们将两个框架——$\mathsf{datafold}$和$\mathsf{GOFMM}$——进行了结合。第一个框架计算扩散映射，其计算瓶颈在于特征分解，而通过第二个框架，我们可在迭代Lanczos方法中近似计算特征分解。层级近似方法的运行时复杂度大致为$\mathcal{O}(Nlog(N))$，而经典方法为$\mathcal{O}(N^{3})$。我们在两个基准数据集——scurve和MNIST——上评估了该方法，使用OpenMP和MPI对最大规模为$100k\times100k$的稠密矩阵进行了强扩展和弱扩展测试。