Accelerate Vector Diffusion Maps by Landmarks
This work addresses computational bottlenecks in manifold learning for researchers in data analysis and image processing, offering an incremental improvement over existing VDM methods.
The paper tackles the problem of accelerating Vector Diffusion Maps (VDM) for analyzing complex datasets by proposing LA-VDM, a landmark-constrained algorithm that introduces a two-stage normalization to handle nonuniform sampling densities. It demonstrates accurate recovery of parallel transport and asymptotic convergence to the connection Laplacian, with performance validated on simulated datasets and nonlocal image denoising applications.
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.