  Diffusion maps are a nonlinear manifold learning technique based on harmonic
analysis of a diffusion process over the data. Out-of-sample extensions with
computational complexity $\mathcal{O}(N)$, where $N$ is the number of points
comprising the manifold, frustrate applications to online learning applications
requiring rapid embedding of high-dimensional data streams. We propose landmark
diffusion maps (L-dMaps) to reduce the complexity to $\mathcal{O}(M)$, where $M
\ll N$ is the number of landmark points selected using pruned spanning trees or
k-medoids. Offering $(N/M)$ speedups in out-of-sample extension, L-dMaps
enables the application of diffusion maps to high-volume and/or high-velocity
streaming data. We illustrate our approach on three datasets: the Swiss roll,
molecular simulations of a C$_{24}$H$_{50}$ polymer chain, and biomolecular
simulations of alanine dipeptide. We demonstrate up to 50-fold speedups in
out-of-sample extension for the molecular systems with less than 4% errors in
manifold reconstruction fidelity relative to calculations over the full
dataset.
