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	Physical and Biological Computing Group: Nonlinear Dimensionality Reduction for the Analyis of Protein Motion
		Nonlinear Dimensionality Reduction for the Analysis of Protein Motion
		Recently, the use of linear dimensionality reduction techniques such as Principal Components Analysis (PCA) has been applied to simulation data to produce a low-dimensional scatterplot of the simulation points that can be used to study the main geometric Degrees of Freedom (DOFs) of the system and use these coordinates to compute, for example, free-energy surfaces.
		No dimensionality reduction technique in the past has produced such good results for a folding reaction.
	www.cs.rice.edu /CS/Robotics/bioinformatics/dimred.html (1066 words)

	NationMaster - Encyclopedia: Dimensionality reduction (Site not responding. Last check: 2007-10-05)
		In statistics, dimensionality reduction is mapping a multidimensional space into a space of fewer dimensions.
		Dimensionality reduction without loss of information is possible if the data in question fall exactly on a smooth, locally flat subspace; then the reduced dimensions are just coordinates in the subspace.
		Dimensionality reduction is effective if the loss of information due to mapping to a lower-dimensional space is less than the gain due simplifying the problem.
	www.nationmaster.com /encyclopedia/Dimensionality-reduction (595 words)

	Tools cut data down to size TRN 031401
		Two research teams are pushing the field forward with algorithms that reduce the dimensionality of nonlinear data.
		Dimensionality refers to the number of measurements available for each item in a data set, according to Sam Roweis, a senior research fellow at University College London's Gatsby Computational Neuroscience Unit.
		This mapping could be used to find the low dimensionality data for new data items, to interpolate, or morph new data items between existing items and to extrapolate new data beyond the range of the original data set, according to Roweis.
	www.trnmag.com /Stories/031401/Tools_cut_data_down_to_size_031401.html (1565 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-10-05)
		In statistics, dimensionality reduction is the process of reducing the number of random variables under consideration, and can be divided into feature selection and feature extraction.
		In Physics, dimensionality reduction is a widely discussed phenomenon, whereby a physical system exists in three dimensions, but its properties behave like those of a lower-dimensional system.
		Dimensional reduction at a quantum critical point (realisation of dimensional reduction in a magnet)
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Dimensionality_reduction (524 words)

	Erion Plaku: Nonlinear Dimensionality Reduction using Approximate Nearest Neighbors
		Dimensionality reduction facilitates the analysis of large amounts of high-dimensional data by extracting low-dimensional embeddings that effectively characterize the input data.
		Consequently, nonlinear dimensionality reduction methods that rely on the nearest-neighbors graph are rendered computationally inefficient for the characterization of considerably large and high-dimensional data sets.
		The low-dimensional embeddings extracted by nonlinear dimensionality reduction methods using the exact nearest-neighbors graph or using the approximate nearest-neighbors graph computed by hcDPES are practically indistinguishable.
	www.cs.rice.edu /~plakue/ResearchHCDPES.html (1754 words)

	keywords: dimension reduction
		Low dimensionality is especially important for learning, as the number of examples required for attaining a given level of performance grows exponentially with the dimensionality of the underlying representation space.
		Following a survey of computational techniques for dimensionality reduction, we show that it is possible to learn a low-dimensional representation that captures the intrinsic low-dimensional nature of certain classes of visual objects, thereby facilitating further learning of tasks...
		Nonlinear dimensionality reduction is formulated here as the problem of trying to find a Euclidean feature-space embedding of a set of observations that preserves as closely as possible their intrinsic metric structure the distances between points on the observation manifold as measured along geodesic paths.
	nivea.psycho.univ-paris5.fr /~philipona/betabiblio/keys2086.html (661 words)

	Nonlinear Dimensionality Reduction in Climate Data, Antonio Gamez (AGNLD) (Site not responding. Last check: 2007-10-05)
		Abstract: Linear methods of dimensionality reduction, such as PCA, are useful techniques when the high dimensionality of the data difficults the physical explanation of the processes that created them.
		However, if nonlinear relations between different variables are present, the projection onto linear subspaces could be confusing and, consequently, mislead the interpretation of the results.
		In this talk, a nonlinear method for dimensionality reduction, Isomap, is applied to data describing El Niño phenomenon and compared to the results offered by linear methods.
	www.pik-potsdam.de /~hrust/agnldws/agnldws/node6.html (115 words)

	Nonlinear dimensionality reduction
		Below we summarise some important algorithms in the history of manifold learning and nonlinear dimensionality reduction.
		Many of these non-linear dimensionality reduction methods are related to linear methods which we list below.
		Those which actually provide a mapping (either from the high dimensional space to the low dimensional embedding or vice versa) and those that just give a visualisation.
	www.xasa.com /wiki/en/wikipedia/n/no/nonlinear_dimensionality_reduction.html (417 words)

	CiteULike: Low-dimensional, free-energy landscapes of protein-folding reactions by nonlinear dimensionality reduction (Site not responding. Last check: 2007-10-05)
		Low-dimensional, free-energy landscapes of protein-folding reactions by nonlinear dimensionality reduction
		We propose a general approach to this problem to obtain a few collective coordinates by using nonlinear dimensionality reduction.
		We validate the usefulness of this method by characterizing the folding landscape associated with a coarse-grained protein model of src homology 3 as sampled by molecular dynamics simulations.
	www.citeulike.org /user/samjlord/article/936867 (431 words)

	[No title] (Site not responding. Last check: 2007-10-05)
		The problem of dimensionality reduction arises in many fields of information processing, including machine learning, data compression, scientific visualization, pattern recognition, and neural computation.
		The mapping is derived from the symmetries of locally linear reconstructions, and the actual computation of the embedding reduces to a sparse eigenvalue problem.
		These examples are used to provide extensive illustrations of the algorithm's performance---both successes and failures---and to relate the algorithm to previous and ongoing work in nonlinear dimensionality reduction.
	jmlr.csail.mit.edu /papers/v4/saul03a.html (204 words)

	NIPS 2002 Workshop on Spectral Methods (Site not responding. Last check: 2007-10-05)
		Traditionally, this spectral approach was thought to be applicable only to learning problems with an essentially linear structure, such as principal component analysis or linear discriminant analysis.
		These problems include nonlinear dimensionality reduction, nonparameteric clustering, and nonlinear classification with fully or partially labeled data.
		Spectral methods applied to nonlinear or nonparametric problems often include one or more free numerical parameters that set a local scale over which linearity or some simple parameteric assumption can be assumed to hold.
	www-psych.stanford.edu /~jbt/spectral-workshop.html (526 words)

	NPG - Abstract - Nonlinear dimensionality reduction in climate data
		Linear methods of dimensionality reduction are useful tools for handling and interpreting high dimensional data.
		However, the cumulative variance explained by each of the subspaces in which the data space is decomposed may show a slow convergence that makes the selection of a proper minimum number of subspaces for successfully representing the variability of the process ambiguous.
		In this article, a nonlinear method for dimensionality reduction, Isomap, is applied to the sea surface temperature and thermocline data in the tropical Pacific Ocean, where the El Niño-Southern Oscillation (ENSO) phenomenon and the annual cycle phenomena interact.
	www.nonlin-processes-geophys.net /11/393/2004/npg-11-393-2004.html (179 words)

	CiteULike: A global geometric framework for nonlinear dimensionality reduction. (Site not responding. Last check: 2007-10-05)
		Here we describe an approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set.
		In contrast to previous algorithms for nonlinear dimensionality reduction, ours efficiently computes a globally optimal solution, and, for an important class of data manifolds, is guaranteed to converge asymptotically to the true structure.
		In contrast to previous algorithms for nonlinear dimensionality reduction, ours efficiently computes a globally optimal solution, and, for an important class of data manifolds, is guaranteed to converge asymptotically to the true structure.}, address = {Department of Psychology, Stanford University, Stanford, CA 94305, USA.
	www.citeulike.org /user/sdvillal/article/266187 (426 words)

	in-cites - An Interview With Dr. Lawrence Saul
		There are many linear approaches to this problem—to determine, for example, if a set of high-dimensional points are concentrated along a (one-dimensional) line, near a (two-dimensional) plane, or more generally in a low-dimensional subspace.
		LLE is a nonlinear approach to this problem: it can be used even more generally to analyze high-dimensional points that lie on or near a low dimensional manifold.
		At the time that LLE was published, this was fairly surprising: most researchers did not believe that optimizations for nonlinear dimensionality reduction could be anywhere near as tractable as optimizations for linear dimensionality reduction.
	www.in-cites.com /scientists/LawrenceSaul.html (749 words)

	[No title] (Site not responding. Last check: 2007-10-05)
		Among various nonlinear dimensionality reduction techniques, the principal curve, which is a nonlinear generalization of principal components, was found to be the most intuitive, interpretable, and theoretically elegant.
		However, the original principal curve formulation suffers from a number of problems: it is non-parametric, biased, inefficient, and not guaranteed to converge.
		In addition, a generalized expectation maximization algorithm with guaranteed convergence is derived for the PPS.
	www.lans.ece.utexas.edu /~kuiyu/paper/kuiyu2000phd.bib (306 words)

	Olga Kayo. Locally linear embedding algorithm. ISBN 951-42-8041-5
		Since the nature of real-world data is often nonlinear, linear dimensionality reduction techniques, such as principal component analysis (PCA), fail to preserve a structure and relationships in a highdimensional space when data are mapped into a low-dimensional space.
		This means that nonlinear dimensionality reduction methods are in demand in this case.
		The benefits of dimensionality reduction are commonly used in obtaining compact data representation before applying a classifier.
	herkules.oulu.fi /isbn9514280415 (430 words)

	Manifold Learning Resource Page
		A Spatio-temporal Extension to Isomap Nonlinear Dimension Reduction.
		Grouping and dimensionality reduction by locally linear embedding.
		Comparison of dimensionality reduction methods for wood surface inspection.
	www.cse.msu.edu /~lawhiu/manifold (2060 words)

	Nonlinear dimensionality reduction Details, Meaning Nonlinear dimensionality reduction Article and Explanation Guide
		If that manifold is of low enough dimension then the data can be visualised in the low dimensional spadce.
		The variations tend to be differences in how the proximity data is computed, for example Isomap, locally linear embeddings are examples of metric multidimensional scaling and the Sammon mapping (which is not in fact a mapping) is an example of a non-metric multidimensional scaling method.
		This is an Article on Nonlinear dimensionality reduction.
	www.e-paranoids.com /n/no/nonlinear_dimensionality_reduction.html (453 words)

	Dimensionality Reduction
		Dimensionality reduction allows us to visualize, categorize, or simplify large datasets.
		Tenenbaum, V. de Silva, and J. Langford.A global geometric framework for nonlinear dimensionality reduction.Science, 290(5500):2319-2323, December 2000.
		Roweis and L. Saul.Nonlinear dimensionality reduction by locally linear embedding.Science, 290(5500):2323-2326, December 2000.
	www.cs.ubc.ca /~mwill/dimreductGroup.htm (157 words)

	[No title] (Site not responding. Last check: 2007-10-05)
		If the data is mainly confined to a low dimensional subspace, then simple linear methods can be used to discover the subspace and estimate its dimensionality.
		More generally, though, if the data lies on (or near) a low dimensional submanifold, then its structure may be highly nonlinear,and linear methods are bound to fail.
		These methods are able to reveal low dimensional structure in high dimensional data from the top or bottom eigenvectors of specially constructed matrices.
	www.ipam.ucla.edu /abstract.aspx?tid=5531 (189 words)

	Mark Moll :: publications classified by publication type (Site not responding. Last check: 2007-10-05)
		We propose a general approach to this problem to obtain a few collective coordinates by using nonlinear dimensionality reduction.
		We validate the usefulness of this method by characterizing the folding landscape associated with a coarse-grained protein model of src homology 3 as sampled by molecular dynamics simulations.
		In this paper we show that the 3D case is fundamentally different from the planar case, due to increased tangent dimensionality.
	www.isi.edu /~moll/publications/index.html (4041 words)

	Penn Dimensionality Reduction Reading Group (Site not responding. Last check: 2007-10-05)
		This Reading group provides interested CIS graduate students with the possibility to read and discuss recent developments and publications in the field of dimensionality reduction.
		A Global Geometric Framework for Nonlinear Dimensionality reduction, Science vol.
		Nonlinear Component Analysis as a Kernel Eigenvalue Problem (1996)
	www.seas.upenn.edu /~kilianw/workpage/drg (140 words)

	P
		Isomap is a nonlinear dimensionality reduction method proposed and based on the theory of manifold.
		By distinguishing the intrinsic dimensionality of high-dimensional data space from the manifold dimensionality, and it is proved that the intrinsic dimensionality is the upper bound of the manifold dimensionality in the high-dimensional space in which there is a toroidal manifold.
		Study on the low-dimensional embedding and the embedding dimensionality of manifold of high-dimensional data.
	www.jos.org.cn /1000-9825/16/1423.htm (358 words)

	Learning a kernel matrix for nonlinear dimensionality reduction
		We investigate how to learn a kernel matrix for high dimensional data that lies on or near a low dimensional manifold.
		Noting that the kernel matrix implicitly maps the data into a nonlinear feature space, we show how to discover a mapping that unfolds the underlying manifold from which the data was sampled.
		The kernel matrix is constructed by maximizing the variance in feature space subject to local constraints that preserve the angles and distances between nearest neighbors.
	repository.upenn.edu /cis_papers/2 (210 words)

	Using Nonlinear Dimensionality Reduction in 3D Figure Animation (Site not responding. Last check: 2007-10-05)
		This paper demonstrates that through nonlinear dimensionality reduction and frame re-sequencing, visually compelling motion can be produced from a set of motion capture data.
		The technique presented uses Isomap and ST-Isomap to reduce the dimensionality of the data set.
		Two distance metrics for nonlinear dimensionality reduction are compared as well as the effect of global degrees of freedom on the visual appeal of the newly generated motion.
	www.vuse.vanderbilt.edu /~bobbyb/pubs/und05.html (121 words)

	Payel Das-Graduate research
		Good reaction coordinates should be able to accurately predict the transition state ensemble(s) and free energy barrier height(s), and correctly identify maxima and minima on a low dimensional landscape.
		We have recently proposed a new and general approach to this problem, based on nonlinear dimensionality reduction techniques [5].
		We have shown that nonlinear dimensionality reduction captures the intrinsic dimensionality of the folding process and identify the optimal low dimensional projection of the folding landscape [5].
	www.ruf.rice.edu /~payel/researchinterest.html (1213 words)

	UCL/MLG - Nonlinear Dimensionality Reduction
		Methods of dimensionality reduction are innovative and important tools in the fields of data analysis, data mining and machine learning.
		Until recently, very few methods were able to reduce the data dimensionality in a nonlinear way.
		The purpose of the book is to summarize clear facts and ideas about well-known methods as well as recent developments in the topic of nonlinear dimensionality reduction.
	www.dice.ucl.ac.be /mlg/index.php?page=NLDR (437 words)

	Laplacian Eigenmaps for Dimensionality Reduction and Data Representation -- Belkin and Niyogi 15 (6): 1373 -- Neural ...
		Laplacian Eigenmaps for Dimensionality Reduction and Data Representation -- Belkin and Niyogi 15 (6): 1373 -- Neural Computation
		Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
		reduction that has locality-preserving properties and a natural
	neco.mitpress.org /cgi/content/abstract/15/6/1373 (285 words)

	Local Context Finder (LCF) reveals multidimensional relationships among mRNA expression profiles of Arabidopsis ... (Site not responding. Last check: 2007-10-05)
		dimensionality reduction (7, 8) is an improvement in both of
		dimensionality reduction, the k value for LCF was chosen to
		are generally superior to linear dimensionality reduction methods
	www.pnas.org /cgi/content/full/100/19/10842 (4591 words)

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