Convergence of Laplacian Eigenmaps

By Mikhail Belkin, Partha Niyogi. Neural Information Processing Systems, 2006

333 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/708c3b1ded68ea1c386c32f2d3c1902da7c5b072

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In Computational Mechanics

In Computational Theory and Mathematics

Citations in the corpus

The spectrum of kernel random matrices (2010, background): “In light of recent results on manifold learning (see Belkin and Niyogi (2008)) where these matrices play a key role, it is natural to ask what happens to them in our context.”
$\Gamma$-convergence of graph Ginzburg-Landau functionals (2012, neutral): “spired by [5] specically. We like to point out that there is also a substantial literature on the convergence of graph Laplacians and their eigenvalues and eigenvectors to continuum limits. See e.g. [40, 36, 12, 41, 13, 57, 43] and references therein. The techniques used and the kind of results obta”
DeepSphere: a graph-based spherical CNN (2020, background): “…of the graph Laplacian to approximate the spherical harmonics, a stronger type of convergence of the graph Laplacian would be preferable, i.e., spectral convergence (that is proved for a full graph in the case of random sampling for a class of Lipschitz functions in (Belkin & Niyogi, 2007)).”
Large-scale manifold learning (2008, background): “In the limit of infinite samples, one can recover the true underlying manifold for certain classes of manifolds [ 22][4][8].”
DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications (2018, background): “Indeed, [34] showed that the graph Laplacian converges to the Laplace-Beltrami when the number of pixels goes to in-ﬁnity providing uniform sampling of the manifold and a fully connected graph built with exponentially decaying weights.”
Geometry on Probability Spaces (2009, background): “Some related analysis can be found in [3, 6, 13, 16, 22].”
Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems (2008, background): “In [39, 12, 25, 48, 4] (and references therein) it is shown that for points xi randomly sampled from a probability density φ0(x) = Cβe−βU(x), as the number of points N → ∞, and as ε → 0, the discrete operator (M − I)/ε converges (in probability) to the backward FP operator (2.”
Efficient Computation of k-Nearest Neighbour Graphs for Large High-Dimensional Data Sets on GPU Clusters (2013, background): “[5] Manifold embedding measures the similarities or nonlinear correlations between clouds of snapshots by tracking their response to an operator and finding the orthonormal coordinates needed to describe the manifold.”
Vector Diffusion Maps and the Connection Laplacian (2011, background): “In the case of uniform sampling, Belkin and Niyogi [4] have shown that the eigenvectors of the graph Laplacian converge to the eigenfunctions of the Laplace-Beltrami operator on the manifold, which is stronger than the pointwise convergence given in (5.1).”
A class of Laplacian multiwavelets bases for high-dimensional data (2015, background): “[7] Belkin, M.”
On Learning with Integral Operators (2010, methodology): “Finally, convergence of eigenvalues and eigenfunctions for the case of the uniform distribution was shown in Belkin and Niyogi (2007).”