A global geometric framework for nonlinear dimensionality reduction.

By Josh Tenenbaum, V. de Silva, Vin de Silva, John C. Langford. 2000

12,870 citations recorded by the source corpus.

Subfields: Computer Vision and Pattern Recognition, Media Technology, Representation & Self-Supervised Learning, Signal Processing

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Citations in the corpus

A Generative Process for Contractive Auto-Encoders (2012, background): “…discovers a linear manifold near which the density may concentrate, Non-Linear Manifold Learning algorithms (Sch¨olkopf et al., 1998; Roweis and Saul, 2000; Tenenbaum et al., 2000) attempt to discover the structure of manifolds (which may be non-linear) near which the true density concentrates.”
Generating Sentences by Editing Prototypes (2017, background): “More generally, results in manifold learning demonstrate that a weak metric such as lexical similarity can be used to extract semantic similarity through distributional statistics (Tenenbaum et al., 2000; Hashimoto et al., 2016).”
node2vec: Scalable Feature Learning for Networks (2016, methodology): “…and non-linear dimensionality reduction techniques such as Principal Component Analysis, Multi-Dimensional Scaling and their extensions [3, 27, 30, 35] optimize an objective that transforms a representative data matrix of the network such that it maximizes the variance of the data…”
LINE: Large-scale Information Network Embedding (2015, methodology): “Our work is related to classical methods of graph embedding or dimension reduction in general, such as multi-dimensional scaling (MDS) [4], IsoMap [20], LLE [18] and Laplacian Eigenmap [2].”
Predicting multicellular function through multi-layer tissue networks (2017, methodology): “The data matrix is projected to a low dimensional space using linear techniques based on SVD (Tang et al. , 2012), or non-linear techniques based on multi-dimensional scaling (Tenenbaum et al. , 2000; Belkin and Niyogi, 2001; Hou et al. , 2014).”
Measuring the Intrinsic Dimension of Objective Landscapes (2018, methodology): “Finally, note that although our approach is related to a rich body of work on estimating the “intrinsic dimension of a dataset” (Camastra & Vinciarelli, 2002; K ´ egl, 2003; Fukunaga & Olsen, 1971; Lev-ina & Bickel, 2005; Tenenbaum et al., 2000), it differs in a few respects.”
Learning Parameterized Skills (2012, methodology): “To analyze the geometry and topology of the policy space and estimate the number D of lower-dimensional surfaces on which skill policies lie we used the ISOMAP algorithm (Tenenbaum et al., 2000).”
Representation Learning: A Review and New Perspectives (2012, methodology): “The large majority of algorithms built on this geometric perspective adopt a nonparametric approach, based on a training set nearest neighbor graph [181], [172], [203], [38], [7], [62], [214], [91], [209].”
Deep learning via semi-supervised embedding (2008, background): “The authors also suggest to combine modi.ed ISOMAP with TSVMs rather than SVMs, and call it Low Density Separation (LDS).”
Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges (2021, 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 exact local Euclidean distances from the original space when creating low-dimensional”
SHINE: Signed Heterogeneous Information Network Embedding for Sentiment Link Prediction (2017, methodology): “Earlier works such as IsoMap [24] and Laplacian Eigenmap [1] first construct the affinity graph of data using the feature vectors and then embed the affinity graph into a low-dimension space.”
A practical tutorial on autoencoders for nonlinear feature fusion: Taxonomy, models, software and guidelines (2018, methodology): “Isomap [33] is a manifold learning method which extends MDS in order to ﬁnd coordinates that describe the actual degrees of freedom of the data while preserving distances among neighbors and geodesic distances between the rest of points.”
Relational autoencoder for feature extraction (2017, background): “ISOMAP [16] learns low-dimensional features by retaining the geodesic distances among data samples.”
Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization (2020, methodology): “More traditional methods that focus on local structure include LLE (Roweis and Saul, 2000), Isomap (Tenenbaum et al., 2000), and Laplacian Eigenmaps (Belkin and Niyogi, 2001), and more modern methods that focus on preserving local structure are t-SNE (van der Maaten and Hinton, 2008), LargeVis (Tang”
The Dangers of Post-hoc Interpretability: Unjustified Counterfactual Explanations (2019, background): “Using such a graph to approximate connectedness is also found in some manifold learning approaches (e.g. Isomap [Tenenbaum et al. , 2000]), where local neighborhoods help to approximate connectedness and thus reduce the dimension of the data.”
Interpretable Basis Decomposition for Visual Explanation (2018, methodology): “Other decomposition methods such as Bilinear models [23] and Isomap [22] are also used to discover meaningful subspaces and structure in the data.”
Visualizing Large-scale and High-dimensional Data (2016, methodology): “The non-linear methods such as Isomap [24], local linear embedding (LLE) [20], Laplacian Eigenmaps [2] are very eﬀective on laboratory data sets but do not perform really well on real-world high-dimensional data.”