Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data

By David L. Donoho, Carrie Grimes. 2003

1,829 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Radiology, Nuclear Medicine and Imaging, Representation & Self-Supervised Learning

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Citations in the corpus

Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization (2020, background): “Isomap (Tenenbaum et al., 2000), Local Linear Embedding (LLE) (Roweis and Saul, 2000), Hessian Local Linear Embedding (Donoho and Grimes, 2003), and Laplacian Eigenmaps (Belkin and Niyogi, 2001) all try to preserve local Euclidean distances from the original space when creating embeddings.”
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).”
Deep Learning on Lie Groups for Skeleton-Based Action Recognition (2016, methodology): “As well-known from classical manifold learning theory [37, 35, 4, 12, 20, 19], one can learn or preserve the original data structure to faithfully maintain geodesic distances for better classification.”
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].”
Topological Data Analysis (2012, background): “For example, a disc is contractible, but not homeomorphic, to a point.”
On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning (2005, methodology): “See, e.g., Isomap, local linear embedding, and graph Laplacian eigenmap [36, 32, 4] as well as Hessian eigenmaps and semidefinite embedding [ 11 , 37].”
Quality assessment of dimensionality reduction: Rank-based criteria (2009, background): “For the past 25 years, this field of research has deeply evolved and after some interest in neural approaches [14,16,26,7,24], the community has recently focused on spectral techniques [31,34,27,3,10,41].”
Sample Complexity of Testing the Manifold Hypothesis (2010, methodology): “It includes a number of extensively used algorithms such as Locally Linear Embedding [17], ISOMAP [19], Laplacian Eigenmaps [4] and Hessian Eigenmaps [8].”
Riemannian Manifold Learning (2008, background): “A large number of nonlinear manifold learning methods have been proposed recently, including isometric feature mapping (ISOMAP) [13], [14], [15], locally linear embedding (LLE) [16], [17], Laplacian eigenmaps [18], [19], Hessian eigenmaps [ 20 ], semidefinite embedding (SDE) [21], manifold charting ”
Dimensionality Reduction by Learning an Invariant Mapping (2006, methodology): “Laplacian Eigenmap [2] and Hessian LLE [ 8 ] do not require a meaningful metric in input space (they merely require a list of neighbors for every sample), but as with LLE, new points whose relationships with training samples are unknown cannot be processed.”
Unsupervised Learning of Image Manifolds by Semidefinite Programming (2004, methodology): “) Moreover, comparing the eigenvalue spectra of the algorithms, only SDE detects the correct underlying dimensionality of the inputs; Isomap is foiled by non-convexity, while the eigenvalue spectra of LLE and hLLE do not reveal this type of information [8, 15].”
Multiview Spectral Embedding (2010, methodology): “Representative algorithms include Isomap [7], LLE [8], LE [9], Hessian eigenmaps [10], local tangent space alignment [11], transductive component analysis [2], discriminative locality alignment [30], and DLLE [27].”
Computational microstructure characterization and reconstruction: Review of the state-of-the-art techniques (2018, background): “Locally linear embedding [206], Laplacian eigenmaps [207], and Hessian eigenmaps [208] are examples of this set of nonlinear embedding approaches.”