Laplacian Eigenmaps for Dimensionality Reduction and Data Representation

By Mikhail Belkin, Partha Niyogi. 2003

8,581 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Representation & Self-Supervised Learning

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Citations in the corpus

Generative Modeling by Estimating Gradients of the Data Distribution (2019, background): “This hypothesis empirically holds for many datasets, and has become the foundation of manifold learning [3, 47].”
Geometric Deep Learning on Graphs and Manifolds Using Mixture Model CNNs (2016, methodology): “Modelling data as low-dimensional manifolds is the core of manifold learning techniques such as Laplacian eigenmaps [3] for non-linear dimensionality reduction, spectral clustering [34] or spectral hashing [57].”
UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction (2018, methodology): “Algorithms such as PCA [8], MDS [9], and Sammon mapping [20] fall into the former category while t-SNE [15] [26], Isomap [24], LargeVis [22], Laplacian eigenmaps [1] [2], diffusion maps [4], NeRV [28], and JSE [12] all fall into the latter category.”
MURAL: Multimodal, Multitask Retrieval Across Languages (2021, methodology): “Using SVCCA scores computed for 100 languages, we plot a 2dimensional visualization using Laplacian Eigenmaps (Belkin and Niyogi, 2003).”
Representation Learning: A Review and New Perspectives (2012, methodology): “…a non-parametric approach, based on a training set nearest neighbor graph (Schölkopf et al., 1998; Roweis and Saul, 2000; Tenenbaum et al., 2000; Brand, 2003; Belkin and Niyogi, 2003; Donoho and Grimes, 2003; Weinberger and Saul, 2004; Hinton and Roweis, 2003; van der Maaten and Hinton, 2008).”
Deep learning via semi-supervised embedding (2008, methodology): “One then uses those distances to embed using conventional MDS. Laplacian Eigenmaps (Belkin & Niyogi, 2003) learn manifold structure by emphasizing the preservation of local distances.”
Learning Interpretable Anatomical Features Through Deep Generative Models: Application to Cardiac Remodeling (2018, methodology): “2, the latent 64-dimensional representation µ of the training set segmentations together with the latent representations µ t obtained at each iteration t were reduced for visualization purposes to a bi-dimensional space using Laplacian Eigenmaps [19].”
A practical tutorial on autoencoders for nonlinear feature fusion: Taxonomy, models, software and guidelines (2018, background): “Laplacian Eigenmaps [105] is a framework aiming to retain local properties as well.”
Relational autoencoder for feature extraction (2017, background): “Laplacian Eigenmaps (LE) [18] minimizes the pairwise distances in the projected space by weighting the corresponding distances in the original space.”
k-NN Regression Adapts to Local Intrinsic Dimension (2011, background): “Fortunately it often occurs that high-dimensional data has low intrinsic dimension: typical examples are data lying near low-dimensional manifolds [3, 4, 5].”
Nonparametric Divergence Estimation with Applications to Machine Learning on Distributions (2011, background): “…embedding and manifold learning (Roweis and Saul, 2000) are well studied problems; several diﬀerent algorithms have been proposed for this problem (Borg and Groenen, 2005, Tenenbaum et al., 2000, Sun et al., 2010, Zhang and Zha, 2004, Belkin and Niyogi, 2003, Donoho and Grimes, 2003).”
Unsupervised Inductive Whole-Graph Embedding by Preserving Graph Proximity (2019, methodology): “As described in the main paper, there are many methods to obtain such node embeddings, such as: • Matrix Factorization: This category includes a vast amount of both early and recent works on network (node) embedding, such as LAPLACIAN EIGENMAPS (LLE) (Belkin & Niyogi, 2003), MNMF (Wang et al., 2017)”
Unsupervised Inductive Graph-Level Representation Learning via Graph-Graph Proximity (2019, background): “There has been a rich body of work [Belkin and Niyogi, 2003; Tang et al., 2015; Qiu et al., 2018] on node-level embeddings that turn each node in a graph into a vector preserving node-node proximity (similarity/distance).”
Geometric Deep Learning: Going beyond Euclidean data (2016, background): “Examples of manifold learning include different flavors of multidimensional scaling (MDS) [26], locally linear embedding (LLE) [27], stochastic neighbor embedding (t-SNE) [28], spectral embeddings such as Laplacian eigenmaps [29] and diffusion maps [30], and deep models [31].”
Deep Learning on Lie Groups for Skeleton-Based Action Recognition (2016, methodology): “In particular, inspired by the classical manifold learning theory [38, 36, 4, 12, 20, 19], we equip the new network structure with rotation mapping layers, with which the input Lie group features are transformed to new ones with better alignment.”
Learning deep dynamical models from image pixels (2014, methodology): “ture, there exists a vast number of other well studied nonlinear dimensionality reduction methods such as the Gaussian process latent variable model (GP-LVM) [7], kernel PCA [14], Laplacian Eigenmaps [1] and Locally Linear Embedding [12]. However, all of them provide at most one of the two mappings ”
Smooth Neighbors on Teacher Graphs for Semi-Supervised Learning (2017, methodology): “Traditional choices include multidimensional scaling [14], ISOMAP [47] and Laplacian eigenmaps [3].”