A Global Geometric Framework for Nonlinear Dimensionality Reduction

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

Subfields: Representation & Self-Supervised Learning

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Citations in the corpus

UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction (2018, neutral): “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.”
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, positive): “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.”
Understanding Deep Features with Computer-Generated Imagery (2015, neutral): “Later work studied nonlinear embeddings, such as LLE [27] and IsoMap [32].”
High-Dimensional Probability Estimation with Deep Density Models (2013, methodology): “…way to implicitly characterize the density of data via a low-dimensional embedding, e.g., locally-linear embeddding [Roweis and Saul, 2000], IsoMap [Tenenbaum et al., 2000], the Gaussian process latent variable model [Lawrence, 2005], kernel PCA [Sch¨olkopf et al., 1998], and t-SNE [Van der Maaten…”
Unsupervised Feature Learning and Deep Learning: A Review and New Perspectives (2012, methodology): “Well-known manifold learning (“em-bedding”) algorithms include Kernel PCA (Sch¨olkopf et al. , 1998), LLE (Roweis and Saul, 2000), Isomap (Tenenbaum et al. , 2000), Laplacian Eigenmap (Belkin and Niyogi, 2003), Hessian Eigenmaps (Donoho and Grimes, 2003), Semideﬁnite Embedding (Weinberger and Saul,…”
LanczosNet: Multi-Scale Deep Graph Convolutional Networks (2019, unknown): “The non-linear dimensionality reduction methods, such as locally linear embedding (LLE) [47], ISOMAP [48], Hessian LLE [49], Laplacian eigenmaps [50], and diffusion maps [17], assume that the high-dimensional data lie on or close to a low dimensional manifold and use the local afﬁnities in the…”
CLUE: cluster-based retrieval of images by unsupervised learning (2005, methodology): “• CLUE may be combined with nonlinear dimensionality reduction techniques, such as the methods in [25] and [37], to provide a global visualization together with a local retrieval.”
Discriminative Autoencoders for Small Targets Detection (2014, methodology): “Regarding non linear methods, some of them are based on the conservation of geodesic distances such as Isomap [23].”
Discriminant Locally Linear Embedding With High-Order Tensor Data (2008, background): “4) As in other previous work [6], [10]–[12], [29], [37], [53], [54], several parameters need to be decided beforehand.”
Hierarchical Stochastic Neighbor Embedding (2016, methodology): “Algorithms such as Sammon Mapping [Sam69], LLE [RS00], ISOMAP [TDSL00] or tSNE [vdMH08] help during Exploratory Data Analysis by giving a meaningful representation of these high-dimensional spaces.”
Prefrontal cortex activity during flexible categorization (2010, unknown): “MDS was calculated using a nonlinear dimensionality reduction method (Tenenbaum et al., 2000) that aims to preserve any intrinsic geometry in the data (neighborhood function, K 14).”
Face recognition using Laplacianfaces (2005, unknown): “Recently, a number of research efforts have shown that the face images possibly reside on a nonlinear submanifold [7], [10], [18], [19], [21], [23], [27].”
Graph Embedding and Extensions: A General Framework for Dimensionality Reduction (2007, unknown): “We present a detailed discussion on the relationship between the graph embedding framework and some wellknown related works. and Out-of-Sample Extension [4] Ham et al. [13] proposed a kernel interpretation of KPCA, ISOMAP, LLE, and Laplacian Eigenmap and demonstrated that they share a common KPCA fo”
Under Consideration for Publication in Knowledge and Information Systems Improving Svm Classification on Imbalanced Time Series Data Sets with Ghost Points (unknown): “Thus, we assume that the local structure of any distance space X can be locally approximated by the metric space, which is also the assumption for embedding approaches [22,24].”
Discrete Signal Processing on Graphs (2012, unknown): “Extensive work is dedicated to discovering efficient data representations for large high-dimensional data [19]–[22].”