Regularized Principal Manifolds

By Alex Smola, Sebastian Mika, Bernhard Schölkopf, Robert C. Williamson. 1999

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Citations in the corpus

Kernel Adaptive Metropolis-Hastings (2013, methodology): “While these strategies are beneficial for distributions that show high anisotropy (e.g., by ensuring the proposal uses the right scaling in all principal directions), they may still suffer from low acceptance probability and slow mixing when the target distributions are strongly nonlinear, and the…”
Distributed optimization of deeply nested systems (2012, background): “Various nonlinear versions of this approach exist where f is a spline (LeBlanc and Tibshirani, 1994), single-layer neural net (Tan and Mavrovouniotis, 1995), radial basis function net (Smola et al., 2001), kernel regression (Meinicke et al., 2005) or Gaussian process (Lawrence, 2005).”
Bibliography of Self-Organizing Map SOM) Papers: 1998-2001 Addendum (2003, unknown): “entropy [69, 212, 444, 684, 750, 784, 778, 780, 1069, 1162, 1281, 1357, 1377, 1070, 1707, 1847, 1862, 1927, 2005, 2040, 2067] entropy maximization [1707, 2005] environmental [194, 209, 494, 523, 756, 805, 929, 1279, 1287, 1304, 1384, 1629, 1764, 1765, 1873, 1912, 1913, 2052] epileptic [723, 724, 725”
Nonlinear Dimensionality Reduction with Local Spline Embedding (2009, methodology): “…structures, algorithms that can generate nonlinear mappings [40] have also been developed [31], such as self-organizing map [29], autoencoder neural network [1], generative topographic map [4], principal curve and manifold [24], [49], [28], [46], [21], [15], [20], and kernel PCA [42], [35].”
Model Selection: Beyond the Bayesian/Frequentist Divide (2010, methodology): “This last idea has been taken one step further in the method of structural risk minimization(Vapnik, 1979), by introducing new hyper-parameters in learning problems, w hich initially did not have any. Consider for instance the class of linear models f (x) = ∑i=1wixi . It is possible to introduce hyp”
Unsupervised and transfer learning challenge (2011, background): “Important progress has also been made in purely unsupervised learning [15], [16], [17], [18], [19].”
Incremental nonlinear dimensionality reduction by manifold learning (2006, background): “Over time, different nonlinear mapping techniques have been proposed, such as selforganizing maps (SOM) [29], principal curve and its extensions [24], [28], [45], [47], auto-encoder neural networks [1], [15], generative topographic maps (GTM) [6], and kernel principal component analysis [42].”
Riemannian Manifold Learning (2008, background): “The purpose of dimensionality reduction is to map a set of highdimensional data into a low-dimensional space while preserving the intrinsic structure in the data.”
Charting a Manifold (2002, background): “Smola and colleagues [10] analyzed the NLDR problem in the broader framework of regularized quantization methods.”
New Support Vector Algorithms (2000, background): “In Smola, Williamson, Mika, & Sch¨olkopf (1999), it is shown that one can specify invariance requirements, which imply that the regularizers act on the output dimensions separately and identically (i.e., in a scalar fashion).”
Advances in Large Margin Classifiers (2000, methodology): “, one effectively has to apply the operator Ad (in a similar fashion to [Smola et al., 1999]) with Multiple Scaling Ad : `2(`2)→ `2(`2) with Ad := A×A× .”
Sample Complexity of Testing the Manifold Hypothesis (2010, background): “Generalizations of principal curves to parameterized principal manifolds in certain regularized settings have been studied in [18].”
A survey of dimensionality reduction techniques (2014, background): “Radial basis functions such as the Gaussian are common kernels (   ( , ) i χ χ With this expansion, the regularization term becomes [Smola1999]”
Testing the Manifold Hypothesis (2013, unknown): “underlyinghypothesisasthe"manifoldhypothesis."ManifoldLearninghasbeenanareaof intenseactivityoverthepasttwodecades. Werefertheinterestedreadertoalimitedsetofpapersassociated withthisﬁeld;see[1,4,5,6,9,14,16,17,26,27,28,30,32,34,38]andthereferencestherein. Thegoalofthispaperistodevelopanalg”
Support Vector Method for Novelty Detection (1999, unknown): “Another kernel-based unsupervised learning technique, regularized principal manifolds [6], computes functions which give a mapping onto a lower-dimensional manifold minimizing a regularized quantization error.”
Distance Metric Learning: A Comprehensive Survey (2006, methodology): “• Generative methods This group of methods adopts generative topology models [44, 45, 46], and hypothesizes that observed data are generated from the evenly spaced low dimensional latent nodes.”