Charting a Manifold

By Matthew Brand. Neural Information Processing Systems, 2002

533 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/c291edf0bb5ead2e45ee9fefb1683e2438158121

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Citations in the corpus

Unsupervised Feature Learning and Deep Learning: A Review and New Perspectives (2012, methodology): “This is most explicitly seen in Manifold Parzen Windows (Vincent and Bengio, 2003) and manifold Charting (Brand, 2003).”
The Curse of Highly Variable Functions for Local Kernel Machines (2005, background): “This class of learning algorithms therefore includes most of the kernel machine algorithms (Schölkopf, Burges and Smola, 1999), such as Support Vector Machines (SVMs) (Boser, Guyon and Vapnik, 1992; Cortes and Vapnik, 1995) or Gaussian processes (Williams and Rasmussen, 1996), but also unsupervised ”
Scaling learning algorithms towards AI (2007, background): “, 1998], Laplacian Eigenmaps [Belkin and Niyogi, 2003], Manifold Charting [Brand, 2003], and spectral clustering algorithms (see Weiss [1999] for a review).”
Nonlinear Dimensionality Reduction with Local Spline Embedding (2009, methodology): “Later, different manifold learning algorithms have been proposed, such as manifold charting [5], Laplacian Eigenmap (LE) [2], Hessian LLE (HLLE) [13], Local Tangent Space Alignment (LTSA) [61], semidefinite embedding [54], conformal eigenmap [45], and other extensive work [10], [22], [47], [38], [59”
Maximum Likelihood Estimation of Intrinsic Dimension (2004, methodology): “The charting algorithm, a recent LLE variant [7], uses a heuristic estimate of dimension which is essentially equivalent to the regression estimator of [8] discussed below.”
Nonlocal Estimation of Manifold Structure (2006, methodology): “For example one could use (Brand, 2003), and the local covariance matrix around x could be of the form F ( x ) 0 diag ( (cid:27) 2 ( x )) F ( x ), where (cid:27) 2 i ( x ) should estimate V ar ( w i ) around x .”
Feature Mining for Hyperspectral Image Classification (2013, background): “Manifold learning is widely used in machine learning for finding a low-dimensional space which approximately maintains the local structure of the original space [79], [117]–[119], and has recently received attention for analysis of hyperspectral data.”
Semisupervised alignment of manifolds (2005, background): “As an illustration of this semisupervised setting, we show how to learn mappings between different data sets of images that are parameterized by the same underlying modes of variability (e.g., pose and viewing angle).”
Riemannian Manifold Learning (2008, background): “Among the family of dimensionality reduction approaches, manifold learning1 algorithms have attracted extensive attention recently due to their nonlinear nature, geometric intuition, and computational feasibility.”
Dimensionality Reduction: A Comparative Review (2008, methodology): “Manifold charting Similar to LLC, manifold charting constructs a lowdimensional data representation by aligning a MoFA or MoPPCA model [20].”
Non-Local Manifold Parzen Windows (2005, methodology): “Like mixtures of factor analysis models (Roweis, Saul and Hinton, 2002) and other low-rank Gaussian mixtures (Brand, 2003), this model works well when the data locally span a smooth low-dimensional manifold.”
Large-scale kernel machines (2007, background): “…structure of the data, such as Locally Linear Embedding [Roweis and Saul, 2000], Isomap [Tenenbaum et al., 2000], kernel PCA [Schölkopf et al., 1998], Laplacian Eigenmaps [Belkin and Niyogi, 2003], Manifold Charting [Brand, 2003], and spectral clustering algorithms (see Weiss [1999] for a review).”
Distance Metric Learning: A Comprehensive Survey (2006, methodology): “Related work includes stochastic nearest neighborhood (SNE) [40] and manifold charting [41].”
The Intrinsic Dimension of Images and Its Impact on Learning (2021, background): “…been conducted clearly supporting the low-dimensional manifold hypothesis for many image datasets (Rud-erman, 1994; Sch ¨ olkopf et al., 1998; Roweis & Saul, 2000; Tenenbaum et al., 2000; Brand, 2003); see also (Fefferman et al., 2016) for principled algorithms on verifying the manifold hypothesis.”
The Manifold Tangent Classifier (2011, background): “Most of these algorithms are non-parametric and local, i.e., explicitly parametrizing the tangent plane around each training point (with a separate set of parameters for each, or derived mostly from the set of training examples in every neighborhood), as most explicitly seen in Manifold Parzen Windo”
Unsupervised Learning of Image Manifolds by Semidefinite Programming (2004, methodology): “These algorithms—including Isomap [19], locally linear embedding (LLE) [14, 15], hessian LLE [8], Laplacian eigenmaps [1], and others [5]—can reveal low dimensional manifolds that are not detected by classical linear methods, such as principal component analysis (PCA) [11].”
Think Globally, Fit Locally: Unsupervised Learning of Low Dimensional Manifold (2003, background): “Recently, robust variations on this basic idea have been developed to estimate the intrinsic dimensionality of finite data sets with noise and several degrees of freedom (Brand, 2003; Kegl, 2003).”
Random Projections of Smooth Manifolds (2009, methodology): “Example algorithms include ISOMAP [38], Hessian Eigenmaps (HLLE) [23], and Maximum Variance Unfolding (MVU) [44], which attempt to learn isometric embeddings of the manifold (preserving pairwise geodesic distances); Locally Linear Embedding (LLE) [34], which attempts to preserve local linear neighbo”
Testing the Manifold Hypothesis (2013, methodology): “The alignment of local coordinate mappings also underlies some other methods such as “local linear coordinates” [46] and “manifold charting” [7].”