Accelerate Vector Diffusion Maps by Landmarks

TL;DR

This paper introduces LA-VDM, a landmark-based algorithm that accelerates Vector Diffusion Maps by addressing nonuniform sampling, enabling accurate recovery of geometric structures and improving efficiency in complex data analysis.

Contribution

The paper presents a novel landmark-constrained normalization method for VDM, improving computational efficiency and accuracy in manifold learning with nonuniform data.

Findings

LA-VDM accurately recovers parallel transport from point clouds.

LA-VDM converges to the connection Laplacian asymptotically.

Demonstrated improved performance in image denoising applications.

Abstract

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.