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Accelerate Vector Diffusion Maps by Landmarks
stat.MLcs.LGmath.DG Sing-Yuan Yeh, Yi-An Wu, Hau-Tieng Wu et al. · Mar 22, 2026

Vector Diffusion Maps (VDM) capture pairwise connection relationships in complex datasets via the Graph Connection Laplacian, but eigenvalue decomposition costs $O(n^{2.81})$, prohibiting large-scale applications. This paper proposes LA-VDM (Landmark Accelerated VDM), which constrains diffusion through landmark points and introduces a novel two-stage normalization scheme with parameters $\alpha$ and $\beta$ to handle non-uniform sampling densities in both data and landmarks. Under a manifold model with the frame bundle structure, the authors prove that LA-VDM asymptotically converges to the connection Laplacian while reducing complexity to $O(nm^2)$, enabling applications to datasets with millions of points.

We propose a landmark-constrained algorithm, LA-VDM (Landmark Accelerated Vector Diffusion Maps), to accelerate the Vector Diffusion Maps (VDM) framework built upon the Graph Connection Laplacian (GCL), which captures pairwise connection relationships within complex datasets. LA-VDM introduces a novel two-stage normalization that effectively address nonuniform sampling densities in both the data and the landmark sets. Under a manifold model with the frame bundle structure, we show that we can accurately recover the parallel transport with landmark-constrained diffusion from a point cloud, and hence asymptotically LA-VDM converges to the connection Laplacian. The performance and accuracy of LA-VDM are demonstrated through experiments on simulated datasets and an application to nonlocal image denoising.
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