Vector Diffusion Maps and the Connection Laplacian

By Amit Singer, Hau-Tieng Wu. Communications on Pure and Applied Mathematics, 2011

341 citations recorded by the source corpus.

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high‐dimensional data sets, images, and shapes. VDMis a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low‐dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on a low‐dimensional manifold ${\cal M}^d$ embedded in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^p$, we prove the relation between VDM and the connection Laplacian operator for vector fields over the manifold. © 2012 Wiley Periodicals, Inc.

Subfields: Biophysics, Computational Theory and Mathematics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/b69d58691a413086c05e9bf432aa59e759156108

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Citations in the corpus

Nonasymptotic rates for manifold, tangent space and curvature estimation (2017, methodology): “When k ≥ 3, the procedure (2) outperforms PCA-based estimators of [25] and [27], where convergence rates of the form (1/n)β with 1/d < β < 2/d are obtained.”
Local Kernels and the Geometric Structure of Data (2014, neutral): “In Section 2 we summarize the relevant developments and techniques related to diffusion maps as found in [2, 6, 19, 12, 21, 4].”
Estimating the reach of a manifold (2017, methodology): “point cloud Xn, asymptotic and nonasymptotic rates of tangent space estimation derived in C3-like models can be found in [2, 14, 36], yielding bounds on sin θ of order (logn/n) 1/d .”
Image Co-segmentation via Consistent Functional Maps (2013, background): “The consistency property of functional maps is also related to diffusion maps [6] and vector diffusion maps [22].”
Weisfeiler and Lehman Go Cellular: CW Networks (2021, background): “D.2 Convolutional Operators One can use the general sheaf Laplacian to define linear, local diffusion operators which, in the GNN literature, are broadly addressed as ‘convolutional’.”
Embeddings of Riemannian Manifolds with Heat Kernels and Eigenfunctions (2013, methodology): “Singer and Wu [37] introduced a data analysis algorihm similar to the eigenmaps and diffusion maps algorithms that is however based on the connection Laplacian for vector fields rather than the Laplace-Beltrami operator for functions.”
THE SPECTRUM OF RANDOM INNER-PRODUCT KERNEL MATRICES (2012, methodology): “Similar kernels with discontinuity have also been used for dimensionality reduction [17] and sensor network localization [8].”