用关键点加速向量扩散映射，提升复杂数据处理效率
Accelerate Vector Diffusion Maps by Landmarks
- 通过关键点约束实现两阶段归一化，缓解采样密度不均问题
- 在流形框架下可准确恢复平行传输，渐近收敛于连接拉普拉斯算子
- 实验验证其在模拟数据与非局部图像去噪中的高效与精准
我们提出一种基于关键点约束的算法 LA-VDM（Landmark Accelerated Vector Diffusion Maps），用于加速基于图连接拉普拉斯算子（GCL）的向量扩散映射（VDM）框架，该框架能捕捉复杂数据集中的成对连接关系。LA-VDM 引入一种新颖的两阶段归一化机制，有效应对数据与关键点集中的非均匀采样密度问题。在具有框架丛结构的流形模型下，我们证明可通过关键点约束扩散从点云中准确恢复平行传输，因此在渐近意义下 LA-VDM 收敛于连接拉普拉斯算子。实验在模拟数据集上以及非局部图像去噪应用中验证了 LA-VDM 的性能与准确性。
原文摘要 · Abstract (English)
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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