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Nonlinear dimensionality reduction

Index Nonlinear dimensionality reduction

High-dimensional data, meaning data that requires more than two or three dimensions to represent, can be difficult to interpret. [1]

74 relations: Aleksandr Gorban, Autoencoder, Backpropagation, Charles-Augustin de Coulomb, Concentration of measure, Coulomb's law, Curse of dimensionality, Degree matrix, Diffusion map, Dimension, Dimensionality reduction, Distance, Distance (graph theory), Distance matrix, Dynamic time warping, Dynamical system, Eigendecomposition of a matrix, Elastic map, Embedding, Euclidean distance, Expectation–maximization algorithm, Factor analysis, Feature extraction, Feature learning, Floyd–Warshall algorithm, Fourier series, Gaussian process, Generative topographic map, Geodesic, Global cascades model, Graduated optimization, Growing self-organizing map, Hamming space, Independent component analysis, Isomap, Journal of Machine Learning Research, K-nearest neighbors algorithm, Kernel method, Kernel principal component analysis, Klein bottle, Laplace–Beltrami operator, Latent variable model, Linear discriminant analysis, Local tangent space alignment, Machine learning, Manifold, Manifold alignment, Markov chain, MIT Press, Multidimensional scaling, ..., Neural network, Nonlinear dimensionality reduction, Ohio State University, Partha Niyogi, Principal component analysis, Projective space, Radial basis function network, Random walk, Reproducing kernel Hilbert space, Restricted Boltzmann machine, Sammon mapping, Self-organizing map, Semidefinite embedding, Similarity measure, Singular-value decomposition, Sparse matrix, Sphere, Stress majorization, Swiss roll, T-distributed stochastic neighbor embedding, Torus, Trevor Hastie, University of Chicago, Yale University. Expand index (24 more) »

Aleksandr Gorban

Alexander Nikolaevich Gorban (Александр Николаевич Горба́нь.) is a scientist of Soviet origin, working in the United Kingdom.

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Autoencoder

An autoencoder is a type of artificial neural network used to learn efficient data codings in an unsupervised manner.

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Backpropagation

Backpropagation is a method used in artificial neural networks to calculate a gradient that is needed in the calculation of the weights to be used in the network.

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Charles-Augustin de Coulomb

Charles-Augustin de Coulomb (14 June 1736 – 23 August 1806) was a French military engineer and physicist.

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Concentration of measure

In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory.

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Coulomb's law

Coulomb's law, or Coulomb's inverse-square law, is a law of physics for quantifying the amount of force with which stationary electrically charged particles repel or attract each other.

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Curse of dimensionality

The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces (often with hundreds or thousands of dimensions) that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience.

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Degree matrix

In the mathematical field of graph theory the degree matrix is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edges attached to each vertex.

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Diffusion map

Diffusion maps is a dimensionality reduction or feature extraction algorithm introduced by Coifman and Lafon which computes a family of embeddings of a data set into Euclidean space (often low-dimensional) whose coordinates can be computed from the eigenvectors and eigenvalues of a diffusion operator on the data.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Dimensionality reduction

In statistics, machine learning, and information theory, dimensionality reduction or dimension reduction is the process of reducing the number of random variables under consideration by obtaining a set of principal variables.

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Distance

Distance is a numerical measurement of how far apart objects are.

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Distance (graph theory)

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them.

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Distance matrix

In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set.

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Dynamic time warping

In time series analysis, dynamic time warping (DTW) is one of the algorithms for measuring similarity between two temporal sequences, which may vary in speed.

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Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.

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Eigendecomposition of a matrix

In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.

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Elastic map

Elastic maps provide a tool for nonlinear dimensionality reduction.

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Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

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Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

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Expectation–maximization algorithm

In statistics, an expectation–maximization (EM) algorithm is an iterative method to find maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables.

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Factor analysis

Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors.

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Feature extraction

In machine learning, pattern recognition and in image processing, feature extraction starts from an initial set of measured data and builds derived values (features) intended to be informative and non-redundant, facilitating the subsequent learning and generalization steps, and in some cases leading to better human interpretations.

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Feature learning

In machine learning, feature learning or representation learning is a set of techniques that allows a system to automatically discover the representations needed for feature detection or classification from raw data.

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Floyd–Warshall algorithm

In computer science, the Floyd–Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights (but with no negative cycles).

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Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

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Gaussian process

In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed.

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Generative topographic map

Generative topographic map (GTM) is a machine learning method that is a probabilistic counterpart of the self-organizing map (SOM), is probably convergent and does not require a shrinking neighborhood or a decreasing step size.

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Geodesic

In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

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Global cascades model

Global cascades models are a class of models aiming to model large and rare cascades that are triggered by exogenous perturbations which are relatively small compared with the size of the system.

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Graduated optimization

Graduated optimization is a global optimization technique that attempts to solve a difficult optimization problem by initially solving a greatly simplified problem, and progressively transforming that problem (while optimizing) until it is equivalent to the difficult optimization problem.

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Growing self-organizing map

A growing self-organizing map (GSOM) is a growing variant of a self-organizing map (SOM).

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Hamming space

In statistics and coding theory, a Hamming space is usually the set of all 2^N binary strings of length N. It is used in the theory of coding signals and transmission.

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Independent component analysis

In signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents.

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Isomap

Isomap is a nonlinear dimensionality reduction method.

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Journal of Machine Learning Research

The Journal of Machine Learning Research is a peer-reviewed open access scientific journal covering machine learning.

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K-nearest neighbors algorithm

In pattern recognition, the k-nearest neighbors algorithm (k-NN) is a non-parametric method used for classification and regression.

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Kernel method

In machine learning, kernel methods are a class of algorithms for pattern analysis, whose best known member is the support vector machine (SVM).

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Kernel principal component analysis

In the field of multivariate statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods.

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Klein bottle

In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.

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Laplace–Beltrami operator

In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds.

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Latent variable model

A latent variable model is a statistical model that relates a set of observable variables (so-called manifest variables) to a set of latent variables.

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Linear discriminant analysis

Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics, pattern recognition and machine learning to find a linear combination of features that characterizes or separates two or more classes of objects or events.

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Local tangent space alignment

Local tangent space alignment (LTSA) is a method for manifold learning, which can efficiently learn a nonlinear embedding into low-dimensional coordinates from high-dimensional data, and can also reconstruct high-dimensional coordinates from embedding coordinates.

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Machine learning

Machine learning is a subset of artificial intelligence in the field of computer science that often uses statistical techniques to give computers the ability to "learn" (i.e., progressively improve performance on a specific task) with data, without being explicitly programmed.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Manifold alignment

Manifold alignment is a class of machine learning algorithms that produce projections between sets of data, given that the original data sets lie on a common manifold.

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Markov chain

A Markov chain is "a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event".

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MIT Press

The MIT Press is a university press affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts (United States).

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Multidimensional scaling

Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset.

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Neural network

The term neural network was traditionally used to refer to a network or circuit of neurons.

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Nonlinear dimensionality reduction

High-dimensional data, meaning data that requires more than two or three dimensions to represent, can be difficult to interpret.

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Ohio State University

The Ohio State University, commonly referred to as Ohio State or OSU, is a large, primarily residential, public university in Columbus, Ohio.

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Partha Niyogi

Partha Niyogi (July 31, 1967 – October 1, 2010) was the Louis Block Professor in Computer Science and Statistics at the University of Chicago.

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Principal component analysis

Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.

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Projective space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.

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Radial basis function network

In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions.

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Random walk

A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.

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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.

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Restricted Boltzmann machine

A restricted Boltzmann machine (RBM) is a generative stochastic artificial neural network that can learn a probability distribution over its set of inputs.

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Sammon mapping

Sammon mapping or Sammon projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality (see multidimensional scaling) by trying to preserve the structure of inter-point distances in high-dimensional space in the lower-dimension projection.

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Self-organizing map

A self-organizing map (SOM) or self-organizing feature map (SOFM) is a type of artificial neural network (ANN) that is trained using unsupervised learning to produce a low-dimensional (typically two-dimensional), discretized representation of the input space of the training samples, called a map, and is therefore a method to do dimensionality reduction.

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Semidefinite embedding

Semidefinite embedding (SDE) or maximum variance unfolding (MVU) is an algorithm in computer science that uses semidefinite programming to perform non-linear dimensionality reduction of high-dimensional vectorial input data.

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Similarity measure

In statistics and related fields, a similarity measure or similarity function is a real-valued function that quantifies the similarity between two objects.

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Singular-value decomposition

In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix.

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Sparse matrix

In numerical analysis and computer science, a sparse matrix or sparse array is a matrix in which most of the elements are zero.

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Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

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Stress majorization

Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of n m-dimensional data items, a configuration X of n points in r(\sigma(X). Usually r is 2 or 3, i.e. the (n x r) matrix X lists points in 2- or 3-dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function \sigma is a cost or loss function that measures the squared differences between ideal (m-dimensional) distances and actual distances in r-dimensional space. It is defined as: where w_\ge 0 is a weight for the measurement between a pair of points (i,j), d_(X) is the euclidean distance between i and j and \delta_ is the ideal distance between the points (their separation) in the m-dimensional data space. Note that w_ can be used to specify a degree of confidence in the similarity between points (e.g. 0 can be specified if there is no information for a particular pair). A configuration X which minimizes \sigma(X) gives a plot in which points that are close together correspond to points that are also close together in the original m-dimensional data space. There are many ways that \sigma(X) could be minimized. For example, Kruskal recommended an iterative steepest descent approach. However, a significantly better (in terms of guarantees on, and rate of, convergence) method for minimizing stress was introduced by Jan de Leeuw.. De Leeuw's iterative majorization method at each step minimizes a simple convex function which both bounds \sigma from above and touches the surface of \sigma at a point Z, called the supporting point. In convex analysis such a function is called a majorizing function. This iterative majorization process is also referred to as the SMACOF algorithm ("Scaling by MAjorizing a COmplicated Function").

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Swiss roll

A Swiss roll, jelly roll, or cream roll is a type of sponge cake roll filled with whipped cream, jam, or icing.

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T-distributed stochastic neighbor embedding

T-distributed Stochastic Neighbor Embedding (t-SNE) is a machine learning algorithm for visualization developed by Laurens van der Maaten and Geoffrey Hinton.

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Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

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Trevor Hastie

Trevor John Hastie (born 27 June 1953) is a South African and American statistician and computer scientist.

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University of Chicago

The University of Chicago (UChicago, U of C, or Chicago) is a private, non-profit research university in Chicago, Illinois.

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Yale University

Yale University is an American private Ivy League research university in New Haven, Connecticut.

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Local linear embedding, Locally Linear Embedding, Locally linear embedding, Locally linear embeddings, Manifold learning, Non-linear dimensionality reduction.

References

[1] https://en.wikipedia.org/wiki/Nonlinear_dimensionality_reduction

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