Accelerate Vector Diffusion Maps by Landmarks
Sing-Yuan Yeh, Yi-An Wu, Hau-Tieng Wu, Mao-Pei Tsui

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.
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Taxonomy
TopicsAdvanced Graph Neural Networks · Topological and Geometric Data Analysis · Generative Adversarial Networks and Image Synthesis
