Accelerate Vector Diffusion Maps by Landmarks

2026-03-22

AI Summary

This work addresses the computational inefficiency and challenges in modeling connection structures inherent in Vector Diffusion Maps (VDM) when applied to non-uniformly sampled data. To overcome these limitations, the authors propose an anchor-guided LA-VDM algorithm that operates within the Graph Connection Laplacian (GCL) framework. By incorporating anchor-based constraints and a two-stage normalization mechanism, LA-VDM efficiently approximates the parallel transport operator over the manifold frame bundle while ensuring asymptotic convergence to the continuous connection Laplacian. Both theoretical analysis and empirical evaluations demonstrate that LA-VDM substantially improves computational efficiency on synthetic datasets and non-local image denoising tasks, without compromising the accuracy relative to the original VDM formulation.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Vector Diffusion Maps
• Graph Connection Laplacian
• Nonuniform Sampling
• Manifold Learning
• Landmark-based Acceleration

Innovation

• Vector Diffusion Maps
• Landmark Acceleration
• Graph Connection Laplacian
• Frame Bundle
• Nonuniform Sampling