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Nonlinear Dimensionality Reduction
Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis. Applications of NLDR Consider a dataset represented as a matrix (or a database table), such that each row represents a set of attributes (or features or dimensions) that describe a particular instance of something. If the number of attributes is large, then the space of unique possible rows is exponentially large. Thus, the larger the dimensionality, the more difficult it becomes to sample the space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lle Hlle Swissroll
The Laboratory for Laser Energetics (LLE) is a scientific research facility which is part of the University of Rochester's south campus, located in Brighton, New York. The lab was established in 1970 and its operations since then have been funded jointly; mainly by the United States Department of Energy, the University of Rochester and the New York State government. The Laser Lab was commissioned to serve as a center for investigations of high-energy physics, specifically those involving the interaction of extremely intense laser radiation with matter. Many types of scientific experiments are performed at the facility with a strong emphasis on inertial confinement, direct drive, laser-induced fusion, fundamental plasma physics and astrophysics using the Omega Laser Facility. In June 1995, OMEGA became the world's highest-energy ultraviolet laser."World's Most Powerful Ultraviolet Laser Comes On-line" http://www.rochester.edu/news/show.php?id=641 The lab shares its building with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Networks
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically, density is defined as mass divided by volume: : \rho = \frac where ''ρ'' is the density, ''m'' is the mass, and ''V'' is the volume. In some cases (for instance, in the United States oil and gas industry), density is loosely defined as its weight per unit volume, although this is scientifically inaccurate – this quantity is more specifically called specific weight. For a pure substance the density has the same numerical value as its mass concentration. Different materials usually have different densities, and density may be relevant to buoyancy, purity and packaging. Osmium and iridium are the densest known elements at standard conditions for temperature and pressure. To simplify comparisons of density across different systems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomap
Isomap is a nonlinear dimensionality reduction method. It is one of several widely used low-dimensional embedding methods. Isomap is used for computing a quasi-isometric, low-dimensional embedding of a set of high-dimensional data points. The algorithm provides a simple method for estimating the intrinsic geometry of a data manifold based on a rough estimate of each data point’s neighbors on the manifold. Isomap is highly efficient and generally applicable to a broad range of data sources and dimensionalities. Introduction Isomap is one representative of isometric mapping methods, and extends metric multidimensional scaling (MDS) by incorporating the geodesic distances imposed by a weighted graph. To be specific, the classical scaling of metric MDS performs low-dimensional embedding based on the pairwise distance between data points, which is generally measured using straight-line Euclidean distance. Isomap is distinguished by its use of the geodesic distance induced by a nei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace–Beltrami Operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami. For any twice-differentiable real-valued function ''f'' defined on Euclidean space R''n'', the Laplace operator (also known as the ''Laplacian'') takes ''f'' to the divergence of its gradient vector field, which is the sum of the ''n'' pure second derivatives of ''f'' with respect to each vector of an orthonormal basis for R''n''. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-nearest Neighbor Algorithm
In statistics, the ''k''-nearest neighbors algorithm (''k''-NN) is a non-parametric supervised learning method first developed by Evelyn Fix and Joseph Hodges in 1951, and later expanded by Thomas Cover. It is used for classification and regression. In both cases, the input consists of the ''k'' closest training examples in a data set. The output depends on whether ''k''-NN is used for classification or regression: :* In ''k-NN classification'', the output is a class membership. An object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its ''k'' nearest neighbors (''k'' is a positive integer, typically small). If ''k'' = 1, then the object is simply assigned to the class of that single nearest neighbor. :* In ''k-NN regression'', the output is the property value for the object. This value is the average of the values of ''k'' nearest neighbors. If ''k'' = 1, then the output is simply assigned to the v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reproducing Kernel Hilbert Space
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g in the RKHS are close in norm, i.e., \, f-g\, is small, then f and g are also pointwise close, i.e., , f(x)-g(x), is small for all x. The converse does not need to be true. Informally, this can be shown by looking at the supremum norm: the sequence of functions \sin^n (x) converges pointwise, but do not converge uniformly i.e. do not converge with respect to the supremum norm (note that this is not a counterexample because the supremum norm does not arise from any inner product due to not satisfying the parallelogram law). It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Some examples, however, have been found. Note that ''L''2 spaces are not Hilbert spaces of functions (and hence not RKH ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Statistical Association
The ''Journal of the American Statistical Association (JASA)'' is the primary journal published by the American Statistical Association, the main professional body for statisticians in the United States. It is published four times a year in March, June, September and December by Taylor & Francis, Ltd on behalf of the American Statistical Association. As a statistics journal it publishes articles primarily focused on the application of statistics, statistical theory and methods in economic, social, physical, engineering, and health sciences. The journal also includes reviews of academic books which are important to the advancement of the field. It had an impact factor of 2.063 in 2010, tenth highest in the "Statistics and Probability" category of ''Journal Citation Reports''. In a 2003 survey of statisticians, the ''Journal of the American Statistical Association'' was ranked first, among all journals, for "Applications of Statistics" and second (after ''Annals of Statistics'') f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trevor Hastie
Trevor John Hastie (born 27 June 1953) is an American statistician and computer scientist. He is currently serving as the John A. Overdeck Professor of Mathematical Sciences and Professor of Statistics at Stanford University. Hastie is known for his contributions to applied statistics, especially in the field of machine learning, data mining, and bioinformatics. He has authored several popular books in statistical learning, including ''The Elements of Statistical Learning: Data Mining, Inference, and Prediction''. Hastie has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge. Education and career Hastie was born on 27 June 1953 in South Africa. He received his B.S. in statistics from the Rhodes University in 1976 and master's degree from University of Cape Town in 1979. Hastie joined the doctoral program at Stanford University in 1980 and received his Ph.D. in 1984 under the supervision of Werner Stuetzle. His dissertation was "Principal Curves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Curve
Principal may refer to: Title or rank * Principal (academia), the chief executive of a university ** Principal (education), the office holder/ or boss in any school * Principal (civil service) or principal officer, the senior management level in the UK Civil Service * Principal dancer, the top rank in ballet * Principal (music), the top rank in an orchestra Law * Principal (commercial law), the person who authorizes an agent ** Principal (architecture), licensed professional(s) with ownership of the firm * Principal (criminal law), the primary actor in a criminal offense * Principal (Catholic Church), an honorific used in the See of Lisbon Places * Principal, Cape Verde, a village * Principal, Ecuador, a parish Media * ''The Principal'' (TV series), a 2015 Australian drama series * ''The Principal'', a 1987 action film * Principal (music), the lead musician in a section of an orchestra * Principal photography, the first phase of movie production * "The Principal", a song on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Swiss Roll
A Swiss roll, jelly roll (United States), roll cake, cream roll, roulade or Swiss log is a type of rolled sponge cake filled with whipped cream, jam, or icing. The origins of the term are unclear; in spite of the name "Swiss roll", the cake is believed to have originated elsewhere in Central Europe, possibly Austria or Slovenia. It appears to have been invented in the nineteenth century, along with Battenberg cake, doughnuts, and Victoria sponge. In the U.S., commercial versions of the cake are sold with the brand names of Ho Hos, Yodels, Swiss Cake Rolls, and others. A type of roll cake called Yule log is traditionally served at Christmas. The spiral layered shape of the Swiss roll has inspired usage as a descriptive term in other fields, such as the jelly roll fold, a protein fold, the "Swiss roll" metamaterial in optics, and the term ''jelly roll'' in science, quilting and other fields. History The earliest published reference for a rolled cake spread with jelly wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
