Empirical spectral properties, i.e., eigenvalues and eigenvectors, of embedding matrices as a function of parameter k. (a) Top 101 eigenvalues of the lazy Markov operator with autotuned bandwidths, for ﹩k={2}^{1}﹩ (red) to ﹩k={2}^{11}﹩ (yellow). This illustrates faster eigenvalue decay as k increases, meaning that for small k there is more heterogeneous structure not well approximated by a low-rank space. (b) Max-to-median ratio of eigenvector norms, as a function of the embedding dimension for the lazy Markov operator with autotuned bandwidths, for ﹩k={2}^{1}﹩ (red) to ﹩k={2}^{11}﹩ (yellow). This illustrates more smooth eigenvectors as k increases, meaning that for small k there is more heterogeneous local structure.