1292:). However, in the GPLVM the mapping is from the embedded(latent) space to the data space (like density networks and GTM) whereas in kernel PCA it is in the opposite direction. It was originally proposed for visualization of high dimensional data but has been extended to construct a shared manifold model between two observation spaces. GPLVM and its many variants have been proposed specially for human motion modeling, e.g., back constrained GPLVM, GP dynamic model (GPDM), balanced GPDM (B-GPDM) and topologically constrained GPDM. To capture the coupling effect of the pose and gait manifolds in the gait analysis, a multi-layer joint gait-pose manifolds was proposed.
3199:-nearest neighbors and tries to seek an embedding that preserves relationships in local neighborhoods. It slowly scales variance out of higher dimensions, while simultaneously adjusting points in lower dimensions to preserve those relationships. If the rate of scaling is small, it can find very precise embeddings. It boasts higher empirical accuracy than other algorithms with several problems. It can also be used to refine the results from other manifold learning algorithms. It struggles to unfold some manifolds, however, unless a very slow scaling rate is used. It has no model.
3219:(TCIE) is an algorithm based on approximating geodesic distances after filtering geodesics inconsistent with the Euclidean metric. Aimed at correcting the distortions caused when Isomap is used to map intrinsically non-convex data, TCIE uses weight least-squares MDS in order to obtain a more accurate mapping. The TCIE algorithm first detects possible boundary points in the data, and during computation of the geodesic length marks inconsistent geodesics, to be given a small weight in the weighted
1240:-nearest neighbors of every point. It then seeks to solve the problem of maximizing the distance between all non-neighboring points, constrained such that the distances between neighboring points are preserved. The primary contribution of this algorithm is a technique for casting this problem as a semidefinite programming problem. Unfortunately, semidefinite programming solvers have a high computational cost. Like Locally Linear Embedding, it has no internal model.
1288:(GPLVM) are probabilistic dimensionality reduction methods that use Gaussian Processes (GPs) to find a lower dimensional non-linear embedding of high dimensional data. They are an extension of the Probabilistic formulation of PCA. The model is defined probabilistically and the latent variables are then marginalized and parameters are obtained by maximizing the likelihood. Like kernel PCA they use a kernel function to form a non linear mapping (in the form of a
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learn to encode the vector into a small number of dimensions and then decode it back into the original space. Thus, the first half of the network is a model which maps from high to low-dimensional space, and the second half maps from low to high-dimensional space. Although the idea of autoencoders is quite old, training of deep autoencoders has only recently become possible through the use of
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95:(the letter 'A') and recover only the varying information (rotation and scale). The image to the right shows sample images from this dataset (to save space, not all input images are shown), and a plot of the two-dimensional points that results from using a NLDR algorithm (in this case, Manifold Sculpting was used) to reduce the data into just two dimensions.
629:). The graph thus generated can be considered as a discrete approximation of the low-dimensional manifold in the high-dimensional space. Minimization of a cost function based on the graph ensures that points close to each other on the manifold are mapped close to each other in the low-dimensional space, preserving local distances. The eigenfunctions of the
657:(MDS). Classic MDS takes a matrix of pair-wise distances between all points and computes a position for each point. Isomap assumes that the pair-wise distances are only known between neighboring points, and uses the Floyd–Warshall algorithm to compute the pair-wise distances between all other points. This effectively estimates the full matrix of pair-wise
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technique to find the low-dimensional embedding of points, such that each point is still described with the same linear combination of its neighbors. LLE tends to handle non-uniform sample densities poorly because there is no fixed unit to prevent the weights from drifting as various regions differ in sample densities. LLE has no internal model.
124:. In particular, if there is an attracting invariant manifold in the phase space, nearby trajectories will converge onto it and stay on it indefinitely, rendering it a candidate for dimensionality reduction of the dynamical system. While such manifolds are not guaranteed to exist in general, the theory of
50:, 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
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RankVisu is designed to preserve rank of neighborhood rather than distance. RankVisu is especially useful on difficult tasks (when the preservation of distance cannot be achieved satisfyingly). Indeed, the rank of neighborhood is less informative than distance (ranks can be deduced from distances but
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of the original weights produced by LLE. The creators of this regularised variant are also the authors of Local
Tangent Space Alignment (LTSA), which is implicit in the MLLE formulation when realising that the global optimisation of the orthogonal projections of each weight vector, in-essence, aligns
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defines a random walk on the data set which means that the kernel captures some local geometry of data set. The Markov chain defines fast and slow directions of propagation through the kernel values. As the walk propagates forward in time, the local geometry information aggregates in the same way as
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which is trained to approximate the identity function. That is, it is trained to map from a vector of values to the same vector. When used for dimensionality reduction purposes, one of the hidden layers in the network is limited to contain only a small number of network units. Thus, the network must
110:, which is a linear dimensionality reduction algorithm, is used to reduce this same dataset into two dimensions, the resulting values are not so well organized. This demonstrates that the high-dimensional vectors (each representing a letter 'A') that sample this manifold vary in a non-linear manner.
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is two, because two variables (rotation and scale) were varied in order to produce the data. Information about the shape or look of a letter 'A' is not part of the intrinsic variables because it is the same in every instance. Nonlinear dimensionality reduction will discard the correlated information
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algorithm. The algorithm finds a configuration of data points on a manifold by simulating a multi-particle dynamic system on a closed manifold, where data points are mapped to particles and distances (or dissimilarity) between data points represent a repulsive force. As the manifold gradually grows
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Traditional techniques like principal component analysis do not consider the intrinsic geometry of the data. Laplacian eigenmaps builds a graph from neighborhood information of the data set. Each data point serves as a node on the graph and connectivity between nodes is governed by the proximity of
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High dimensional data can be hard for machines to work with, requiring significant time and space for analysis. It also presents a challenge for humans, since it's hard to visualize or understand data in more than three dimensions. Reducing the dimensionality of a data set, while keep its essential
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It should be noticed that CCA, as an iterative learning algorithm, actually starts with focus on large distances (like the Sammon algorithm), then gradually change focus to small distances. The small distance information will overwrite the large distance information, if compromises between the two
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on the unit circle manifold). Attempts to place
Laplacian eigenmaps on solid theoretical ground have met with some success, as under certain nonrestrictive assumptions, the graph Laplacian matrix has been shown to converge to the Laplace–Beltrami operator as the number of points goes to infinity.
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learns a smooth diffeomorphic mapping which transports the data onto a lower-dimensional linear subspace. The methods solves for a smooth time indexed vector field such that flows along the field which start at the data points will end at a lower-dimensional linear subspace, thereby attempting to
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takes advantage of the assumption that disparate data sets produced by similar generating processes will share a similar underlying manifold representation. By learning projections from each original space to the shared manifold, correspondences are recovered and knowledge from one domain can be
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in his 1984 thesis, which he formally introduced in 1989. This idea has been explored further by many authors. How to define the "simplicity" of the manifold is problem-dependent, however, it is commonly measured by the intrinsic dimensionality and/or the smoothness of the manifold. Usually, the
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algorithms, and better results with many problems. LLE also begins by finding a set of the nearest neighbors of each point. It then computes a set of weights for each point that best describes the point as a linear combination of its neighbors. Finally, it uses an eigenvector-based optimization
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It should be apparent, therefore, that NLDR has several applications in the field of computer-vision. For example, consider a robot that uses a camera to navigate in a closed static environment. The images obtained by that camera can be considered to be samples on a manifold in high-dimensional
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The reduced-dimensional representations of data are often referred to as "intrinsic variables". This description implies that these are the values from which the data was produced. For example, consider a dataset that contains images of a letter 'A', which has been scaled and rotated by varying
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to train a multi-layer perceptron (MLP) to fit to a manifold. Unlike typical MLP training, which only updates the weights, NLPCA updates both the weights and the inputs. That is, both the weights and inputs are treated as latent values. After training, the latent inputs are a low-dimensional
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must be chosen such that it has a known corresponding kernel. Unfortunately, it is not trivial to find a good kernel for a given problem, so KPCA does not yield good results with some problems when using standard kernels. For example, it is known to perform poorly with these kernels on the
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give the natural geometric framework for nonlinear dimensionality reduction and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold, and by encoding using standard geometric projection onto the manifold. This approach was originally proposed by
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Laplacian eigenmaps uses spectral techniques to perform dimensionality reduction. This technique relies on the basic assumption that the data lies in a low-dimensional manifold in a high-dimensional space. This algorithm cannot embed out-of-sample points, but techniques based on
1303:(t-SNE) is widely used. It is one of a family of stochastic neighbor embedding methods. The algorithm computes the probability that pairs of datapoints in the high-dimensional space are related, and then chooses low-dimensional embeddings which produce a similar distribution.
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principal manifold is defined as a solution to an optimization problem. The objective function includes a quality of data approximation and some penalty terms for the bending of the manifold. The popular initial approximations are generated by linear PCA and
Kohonen's SOM.
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is also based on sparse matrix techniques. It tends to yield results of a much higher quality than LLE. Unfortunately, it has a very costly computational complexity, so it is not well-suited for heavily sampled manifolds. It has no internal model.
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gives conditions for the existence of unique attracting invariant objects in a broad class of dynamical systems. Active research in NLDR seeks to unfold the observation manifolds associated with dynamical systems to develop modeling techniques.
1236:, Isomap and Locally Linear Embedding share a common intuition relying on the notion that if a manifold is properly unfolded, then variance over the points is maximized. Its initial step, like Isomap and Locally Linear Embedding, is finding the
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Modified LLE (MLLE) is another LLE variant which uses multiple weights in each neighborhood to address the local weight matrix conditioning problem which leads to distortions in LLE maps. Loosely speaking the multiple weights are the local
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that is embedded inside of a higher-dimensional vector space. The main intuition behind MVU is to exploit the local linearity of manifolds and create a mapping that preserves local neighbourhoods at every point of the underlying manifold.
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between all of the points. Isomap then uses classic MDS to compute the reduced-dimensional positions of all the points. Landmark-Isomap is a variant of this algorithm that uses landmarks to increase speed, at the cost of some accuracy.
1519:); an analogy is drawn between the diffusion operator on a manifold and a Markov transition matrix operating on functions defined on the graph whose nodes were sampled from the manifold. In particular, let a data set be represented by
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and curvilinear component analysis except that (1) it simultaneously penalizes false neighborhoods and tears by focusing on small distances in both original and output space, and that (2) it accounts for
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manifold. However, one can view certain other methods that perform well in such settings (e.g., Laplacian
Eigenmaps, LLE) as special cases of kernel PCA by constructing a data-dependent kernel matrix.
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on the manifold serve as the embedding dimensions, since under mild conditions this operator has a countable spectrum that is a basis for square integrable functions on the manifold (compare to
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1445:(CCA) looks for the configuration of points in the output space that preserves original distances as much as possible while focusing on small distances in the output space (conversely to
27:) with a rectangular hole in the middle. Top-right: the original 2D manifold used to generate the 3D dataset. Bottom left and right: 2D recoveries of the manifold respectively using the
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S. Lespinats, M. Verleysen, A. Giron, B. Fertil, DD-HDS: a tool for visualization and exploration of high-dimensional data, IEEE Transactions on Neural
Networks 18 (5) (2007) 1265–1279.
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Plot of the two-dimensional points that results from using a NLDR algorithm. In this case, Manifold
Sculpting is used to reduce the data into just two dimensions (rotation and scale).
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amounts. Each image has 32Ă—32 pixels. Each image can be represented as a vector of 1024 pixel values. Each row is a sample on a two-dimensional manifold in 1024-dimensional space (a
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681:(LLE) was presented at approximately the same time as Isomap. It has several advantages over Isomap, including faster optimization when implemented to take advantage of
4732:, In Platt, J.C. and Koller, D. and Singer, Y. and Roweis, S., editor, Advances in Neural Information Processing Systems 20, pp. 513–520, MIT Press, Cambridge, MA, 2008
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is based on the intuition that when a manifold is correctly unfolded, all of the tangent hyperplanes to the manifold will become aligned. It begins by computing the
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2342:. The former means that very little diffusion has taken place while the latter implies that the diffusion process is nearly complete. Different strategies to choose
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local transitions (defined by differential equations) of the dynamical system. The metaphor of diffusion arises from the definition of a family diffusion distance
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as defining some sort of affinity on that graph. The graph is symmetric by construction since the kernel is symmetric. It is easy to see here that from the tuple (
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2132:. Since the exact structure of the manifold is not available, for the nearest neighbors the geodesic distance is approximated by euclidean distance. The choice
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eigenvectors of that matrix. By comparison, KPCA begins by computing the covariance matrix of the data after being transformed into a higher-dimensional space,
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the local tangent spaces of every data point. The theoretical and empirical implications from the correct application of this algorithm are far-reaching.
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PCA (a linear dimensionality reduction algorithm) is used to reduce this same dataset into two dimensions, the resulting values are not so well organized.
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transferred to another. Most manifold alignment techniques consider only two data sets, but the concept extends to arbitrarily many initial data sets.
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Relational perspective map was inspired by a physical model in which positively charged particles move freely on the surface of a ball. Guided by the
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2020:{\displaystyle K_{ij}={\begin{cases}e^{-\|x_{i}-x_{j}\|_{2}^{2}/\sigma ^{2}}&{\text{if }}x_{i}\sim x_{j}\\0&{\text{otherwise}}\end{cases}}}
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in size the multi-particle system cools down gradually and converges to a configuration that reflects the distance information of the data points.
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regularization exist for adding this capability. Such techniques can be applied to other nonlinear dimensionality reduction algorithms as well.
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takes into account all the relation between points x and y while calculating the distance and serves as a better notion of proximity than just
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Haller, George; Ponsioen, Sten (2016). "Nonlinear normal modes and spectral submanifolds: Existence, uniqueness and use in model reduction".
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representation of the observed vectors, and the MLP maps from that low-dimensional representation to the high-dimensional observation space.
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between particles, the minimal energy configuration of the particles will reflect the strength of repulsive forces between the particles.
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in its embedding. It is based on
Curvilinear Component Analysis (which extended Sammon's mapping), but uses geodesic distances instead.
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McInnes, Leland; Healy, John; Melville, James (2018-12-07). "Uniform manifold approximation and projection for dimension reduction".
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Uniform manifold approximation and projection (UMAP) is a nonlinear dimensionality reduction technique. Visually, it is similar to
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is that only local features of the data are considered in diffusion maps as opposed to taking correlations of the entire data set.
1273:(see above) to learn a non-linear mapping from the high-dimensional to the embedded space. The mappings in NeuroScale are based on
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is an open source C++ library containing implementations of LLE, Manifold
Sculpting, and some other manifold learning algorithms.
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Zhang, Zhenyue; Hongyuan Zha (2005). "Principal
Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment".
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Ham, Jihun; Lee, Daniel D.; Mika, Sebastian; Schölkopf, Bernhard. "A kernel view of the dimensionality reduction of manifolds".
1225:-first principal components in each local neighborhood. It then optimizes to find an embedding that aligns the tangent spaces.
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Gorban, A. N.; Zinovyev, A. (2010). "Principal manifolds and graphs in practice: from molecular biology to dynamical systems".
1613:. The underlying assumption of diffusion map is that the high-dimensional data lies on a low-dimensional manifold of dimension
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based on a non-linear mapping from the embedded space to the high-dimensional space. These techniques are related to work on
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KPCA has an internal model, so it can be used to map points onto its embedding that were not available at training time.
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Bengio, Yoshua; Paiement, Jean-Francois; Vincent, Pascal; Delalleau, Olivier; Le Roux, Nicolas; Ouimet, Marie (2004).
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features relatively intact, can make algorithms more efficient and allow analysts to visualize trends and patterns.
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nonzero eigen vectors provide an orthogonal set of coordinates. Generally the data points are reconstructed from
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incidence. Different forms and colors correspond to various geographical locations. Red bold line represents the
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to factor away much of the computation, such that the entire process can be performed without actually computing
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469:{\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\Phi (\mathbf {x} _{i})\Phi (\mathbf {x} _{i})^{\mathsf {T}}}.}
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1124:{\displaystyle C(Y)=\sum _{i}\left|\mathbf {Y} _{i}-\sum _{j}{\mathbf {W} _{ij}\mathbf {Y} _{j}}\right|^{2}}
833:{\displaystyle E(W)=\sum _{i}\left|\mathbf {X} _{i}-\sum _{j}{\mathbf {W} _{ij}\mathbf {X} _{j}}\right|^{2}}
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Taylor, D.; Klimm, F.; Harrington, H. A.; Kramár, M.; Mischaikow, K.; Porter, M. A.; Mucha, P. J. (2015).
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A method based on proximity matrices is one where the data is presented to the algorithm in the form of a
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Contagion maps use multiple contagions on a network to map the nodes as a point cloud. In the case of the
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space, and the intrinsic variables of that manifold will represent the robot's position and orientation.
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46:, is any of various related techniques that aim to project high-dimensional data onto lower-dimensional
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4819:"UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction — umap 0.3 documentation"
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4445:"Curvilinear Component Analysis: A Self-Organizing Neural Network for Nonlinear Mapping of Data Sets"
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Roweis, S. T.; Saul, L. K. (2000). "Nonlinear
Dimensionality Reduction by Locally Linear Embedding".
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3397:"A unifying probabilistic perspective for spectral dimensionality reduction: insights and new models"
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4237:"Probabilistic Non-linear Principal Component Analysis with Gaussian Process Latent Variable Models"
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defines a distance between any two points of the data set based on path connectivity: the value of
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NIPS'06: Proceedings of the 19th International Conference on Neural Information Processing Systems
3517:. Proceedings of the International Joint Conference on Neural Networks IJCNN'11. pp. 1959–66.
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1221:-nearest neighbors of every point. It computes the tangent space at every point by computing the
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349:{\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\mathbf {x} _{i}\mathbf {x} _{i}^{\mathsf {T}}}.}
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Application of principal curves: Nonlinear quality of life index. Points represent data of the
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3552:. Lecture Notes in Computer Science and Engineering. Vol. 58. Springer. pp. 68–95.
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3266:. The variations tend to be differences in how the proximity data is computed; for example,
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dimensional space. A neighborhood preserving map is created based on this idea. Each point X
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4375:"Topological data analysis of contagion maps for examining spreading processes on networks"
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is presented by a blue straight line. Data points are the small grey circles. For PCA, the
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3282:(which is not in fact a mapping) are examples of metric multidimensional scaling methods.
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707:. The original point is reconstructed by a linear combination, given by the weight matrix
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Proceedings of the 21st International Conference on Machine Learning, Banff, Canada, 2004
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to optimize the coordinates. This minimization problem can be solved by solving a sparse
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Thus one can think of the individual data points as the nodes of a graph and the kernel
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CDA trains a self-organizing neural network to fit the manifold and seeks to preserve
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The Relational perspective map was introduced in. The algorithm firstly used the flat
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4757:"Nonlinear Dimensionality Reduction by Topologically Constrained Isometric Embedding"
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In manifold learning, the input data is assumed to be sampled from a low dimensional
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in local regions, and then uses convex optimization to fit all the pieces together.
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4100:"Locally Linear Embedding and fMRI feature selection in psychiatric classification"
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3881:"Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering"
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dimensional space and the goal of the algorithm is to reduce the dimensionality to
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4022:"Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data"
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distances cannot be deduced from ranks) and its preservation is thus easier.
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The stress function of CCA is related to a sum of right Bregman divergences.
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phenomenon by adapting the weighting function to the distance distribution.
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171 countries in 4-dimensional space formed by the values of 4 indicators:
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DeMers, D.; Cottrell, G.W. (1993). "Non-linear dimensionality reduction".
3813:"Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering"
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preserve pairwise differences under both the forward and inverse mapping.
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dimensional space will be used to reconstruct the same point in the lower
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4538:. The 25th International Conference on Machine Learning. pp. 1120–7.
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3577:(1998). "Nonlinear Component Analysis as a Kernel Eigenvalue Problem".
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1169:. For such an implementation the algorithm has only one free parameter
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2954:{\displaystyle D_{t}^{2}(x,y)=\|p_{t}(x,\cdot )-p_{t}(y,\cdot )\|^{2}}
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Top-left: a 3D dataset of 1000 points in a spiraling band (a.k.a. the
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Iterative Non-linear Dimensionality Reduction with Manifold Sculpting
4260:"Multilayer Joint Gait-Pose Manifolds for Human Gait Motion Modeling"
3913:"A Global Geometric Framework for Nonlinear Dimensionality Reduction"
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the speed of the spread can be adjusted with the threshold parameter
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Data-driven high-dimensional scaling (DD-HDS) is closely related to
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Perhaps the most widely used algorithm for dimensional reduction is
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4333:"Visualizing high-dimensional data with relational perspective map"
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3546:"3. Learning Nonlinear Principal Manifolds by Self-Organising Maps"
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and stacked denoising autoencoders. Related to autoencoders is the
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Scholz, M.; Kaplan, F.; Guy, C. L.; Kopka, J.; Selbig, J. (2005).
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Matlab code for Laplacian Eigenmaps can be found in algorithms at
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Principal Manifolds for Data Visualisation and Dimension Reduction
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3550:
Principal Manifolds for Data Visualization and Dimension Reduction
3415:
3291:
2578:
in one time step. Similarly the probability of transitioning from
1338:
as the image manifold, then it has been extended (in the software
4755:
Rosman, G.; Bronstein, M.M.; Bronstein, A.M.; Kimmel, R. (2010).
3110:
is much more robust to noise in the data than geodesic distance.
1176:
196:(GTM) use a point representation in the embedded space to form a
35:
algorithms as implemented by the Modular Data Processing toolkit.
4104:
IEEE Journal of Translational Engineering in Health and Medicine
4081:"MLLE: Modified Locally Linear Embedding Using Multiple Weights"
3720:(PhD). Stanford Linear Accelerator Center, Stanford University.
3548:. In Gorban, A.N.; KĂ©gl, B.; Wunsch, D.C.; Zinovyev, A. (eds.).
3232:
3226:
3211:
1606:{\displaystyle \mathbf {X} =\in \Omega \subset \mathbf {R^{D}} }
4632:
Venna, J.; Kaski, S. (2006). "Local multidimensional scaling".
3531:"Principal Component Analysis and Self-Organizing Maps: applet"
3320:
3267:
1431:
1343:
646:
4754:
3878:
1295:
98:
3235:, but it assumes that the data is uniformly distributed on a
3195:
to find an embedding. Like other algorithms, it computes the
2128:
distance should be used to actually measure distances on the
1335:
1134:
In this cost function, unlike the previous one, the weights W
906:
and (b) The sum of every row of the weight matrix equals 1.
3864:(PhD). Department of Mathematics, The University of Chicago.
204:, which also are based around the same probabilistic model.
4372:
2013:
1189:
1138:
are kept fixed and the minimization is done on the points Y
2766:
constitutes some notion of local geometry of the data set
1667:
which represents some notion of affinity of the points in
1280:
155:
3910:
1471:
1339:
879:
is reconstructed only from its neighbors, thus enforcing
4920:
4494:
European Symposium on Artificial Neural Networks (Esann)
4487:"Curvilinear component analysis and Bregman divergences"
3776:
483:
eigenvectors of that matrix, just like PCA. It uses the
4745:, Neurocomputing, vol. 72 (13–15), pp. 2964–2978, 2009.
3246:
is locally constant or approximately locally constant.
3169:
2152:
modulates our notion of proximity in the sense that if
216:. PCA begins by computing the covariance matrix of the
4741:
Lespinats S., Fertil B., Villemain P. and Herault J.,
4528:
3911:
Tenenbaum, J B.; de Silva, V.; Langford, J.C. (2000).
3780:; KĂ©gl, B.; Wunsch, D. C.; Zinovyev, A., eds. (2007).
3249:
207:
159:
Approximation of a principal curve by one-dimensional
152:
is one of the first and most popular NLDR techniques.
3116:
3089:
3047:
2997:
2970:
2851:
2803:
2782:
2752:
2725:
2698:
2642:
2611:
2584:
2557:
2530:
2481:
2461:
2417:
2394:
2371:
2348:
2312:
2253:
2217:
2158:
2138:
2103:
2076:
2036:
1875:
1772:
1704:
1677:
1645:
1619:
1525:
1410:
1372:
1024:
915:
733:
524:
493:
479:
It then projects the transformed data onto the first
372:
272:
248:
222:
120:
are of general interest for model order reduction in
4796:
4666:
4304:
3572:
3262:. These methods all fall under the broader class of
689:
LLE computes the barycentric coordinates of a point
3300:implements the method for the programming language
1449:which focus on small distances in original space).
1265:algorithm, which uses stress functions inspired by
16:
Projection of data onto lower-dimensional manifolds
3527:The illustration is prepared using free software:
3129:
3102:
3075:
3025:
2983:
2953:
2835:
2788:
2770:. The major difference between diffusion maps and
2758:
2731:
2711:
2684:
2624:
2597:
2570:
2543:
2516:
2467:
2445:
2400:
2377:
2365:In order to faithfully represent a Markov matrix,
2354:
2334:
2298:
2239:
2203:
2144:
2116:
2089:
2062:
2019:
1818:
1749:
1687:
1651:
1627:
1605:
1422:
1396:
1123:
1015:dimensional space by minimizing the cost function
951:
832:
530:
510:
468:
348:
256:
234:
4573:Coifman, Ronald R.; Lafon, Stephane (July 2006).
4516:Advances in Neural Information Processing Systems
4485:Sun, Jigang; Crowe, Malcolm; Fyfe, Colin (2010).
4442:
4204:Advances in neural information processing systems
4164:
3888:Advances in Neural Information Processing Systems
3820:Advances in Neural Information Processing Systems
3144:
1659:represent the distribution of the data points on
1437:
1342:to use other types of closed manifolds, like the
549:
4927:
4743:Rankvisu: Mapping from the neighbourhood network
1819:{\displaystyle k(x,y)\geq 0\qquad \forall x,y,k}
1459:
1203:
4507:Walder, Christian; Schölkopf, Bernhard (2009).
4506:
4307:"Visualizing High-Dimensional Data Using t-SNE"
3751:Journal of the American Statistical Association
3615:
3507:
1157:being the number of data points), whose bottom
175:in this example is 23.23%, for SOM it is 6.86%.
4726:Gashler, M. and Ventura, D. and Martinez, T.,
4201:
3737:
3638:
3462:
3083:involves a sum over of all paths of length t,
2299:{\displaystyle \|x_{i}-x_{j}\|_{2}\ll \sigma }
2204:{\displaystyle \|x_{i}-x_{j}\|_{2}\gg \sigma }
1866:) can be constructed using a Gaussian kernel.
1177:Hessian locally-linear embedding (Hessian LLE)
952:{\displaystyle \sum _{j}{\mathbf {W} _{ij}}=1}
852:refer to the amount of contribution the point
4568:
4566:
4529:Wang, Chang; Mahadevan, Sridhar (July 2008).
4305:van der Maaten, L.J.P.; Hinton, G.E. (2008).
3227:Uniform manifold approximation and projection
3217:Topologically constrained isometric embedding
3212:Topologically constrained isometric embedding
1311:
1228:
28:
4921:Nonlinear PCA by autoencoder neural networks
4572:
4532:Manifold Alignment using Procrustes Analysis
4518:. Vol. 22. MIT Press. pp. 1713–20.
4484:
4019:
3806:
3437:
3033:will be smaller the more paths that connect
2942:
2885:
2818:
2804:
2281:
2254:
2186:
2159:
1935:
1908:
962:The original data points are collected in a
4582:Applied and Computational Harmonic Analysis
3697:- Multidimensional Data Visualization Tool
3511:Temporal Nonlinear Dimensionality Reduction
1301:t-distributed stochastic neighbor embedding
1296:t-distributed stochastic neighbor embedding
673:
4631:
4563:
4091:
3968:
1007:dimensional space is mapped onto a point Y
4802:
4694:
4684:
4669:"Non-linear PCA: a missing data approach"
4618:Diffusion Maps: Applications and Analysis
4438:
4436:
4416:
4390:
4178:
4133:
4115:
4078:
4055:
4045:
3853:
3851:
3654:
3476:
3414:
2524:is the probability of transitioning from
1173:which can be chosen by cross validation.
359:It then projects the data onto the first
4764:International Journal of Computer Vision
4679:(20). Oxford University Press: 3887–95.
4509:"Diffeomorphic Dimensionality Reduction"
4234:
3438:Lee, John A.; Verleysen, Michel (2007).
3394:
3149:Local Multidimensional Scaling performs
2385:must be normalized by the corresponding
1507:leverages the relationship between heat
1190:Modified Locally-Linear Embedding (MLLE)
553:
154:
97:
76:
18:
4257:
3642:International Journal of Neural Systems
1430:the contagion map is equivalent to the
1286:Gaussian process latent variable models
1281:Gaussian process latent variable models
167:with red squares, 20 nodes). The first
65:
4928:
4891:Gaussian Process Latent Variable Model
4845:
4552:Diffusion Maps and Geometric Harmonics
4496:. d-side publications. pp. 81–86.
4478:
4433:
3857:
3848:
3710:
3528:
1472:Diffeomorphic dimensionality reduction
605:
456:
336:
179:
4548:
4097:
3186:
3041:and vice versa. Because the quantity
1487:
139:
4452:IEEE Transactions on Neural Networks
4443:Demartines, P.; HĂ©rault, J. (1997).
4314:Journal of Machine Learning Research
4241:Journal of Machine Learning Research
4167:SIAM Journal on Scientific Computing
3727:from the original on August 2, 2019.
3402:Journal of Machine Learning Research
3170:Data-driven high-dimensional scaling
4614:
3543:
3250:Methods based on proximity matrices
1306:
861:has while reconstructing the point
625:neighboring points (using e.g. the
511:{\displaystyle \Phi (\mathbf {x} )}
208:Kernel principal component analysis
144:
13:
4839:
4330:
3508:Gashler, M.; Martinez, T. (2011).
3440:Nonlinear Dimensionality Reduction
2836:{\displaystyle \{D_{t}\}_{t\in N}}
2685:{\displaystyle P^{t}(x_{i},x_{j})}
1798:
1680:
1585:
1165:nearest neighbors, as measured by
525:
494:
432:
411:
134:nonlinear dimensionality reduction
40:Nonlinear dimensionality reduction
14:
4947:
4869:
3861:Problems of Learning on Manifolds
1847:) one can construct a reversible
1499:
1357:
4621:(Masters). University of Oxford.
4264:IEEE Transactions on Cybernetics
3156:
1621:
1597:
1593:
1527:
1316:Relational perspective map is a
1099:
1084:
1058:
929:
808:
793:
767:
619:Reproducing kernel Hilbert space
501:
440:
419:
325:
313:
250:
192:) and its probabilistic variant
173:Fraction of variance unexplained
4854:. MIT Press. pp. 682–699.
4811:
4790:
4748:
4735:
4720:
4711:
4660:
4625:
4608:
4542:
4522:
4500:
4366:
4324:
4298:
4251:
4228:
4206:. Vol. 5. pp. 580–7.
4195:
4158:
4072:
4020:Donoho, D.; Grimes, C. (2003).
4013:
3962:
3904:
3872:
3858:Belkin, Mikhail (August 2003).
3800:
3770:
3731:
3704:
3687:
3264:metric multidimensional scaling
2475:now represents a Markov chain.
2063:{\displaystyle x_{i}\sim x_{j}}
1797:
1243:
897:is not a neighbor of the point
4916:Short review of Diffusion Maps
4881:Generative Topographic Mapping
4852:Probabilistic Machine Learning
4696:11858/00-001M-0000-0014-2B1F-2
3764:10.1080/01621459.1989.10478797
3632:
3609:
3566:
3537:
3521:
3501:
3456:
3431:
3388:
3145:Local multidimensional scaling
3070:
3058:
3020:
3008:
2938:
2926:
2910:
2898:
2879:
2867:
2679:
2653:
2517:{\displaystyle P(x_{i},x_{j})}
2511:
2485:
1788:
1776:
1750:{\displaystyle k(x,y)=k(y,x),}
1741:
1729:
1720:
1708:
1579:
1534:
1443:Curvilinear component analysis
1438:Curvilinear component analysis
1391:
1379:
1275:radial basis function networks
1034:
1028:
743:
737:
550:Principal curves and manifolds
505:
497:
451:
435:
429:
414:
194:generative topographic mapping
1:
4686:10.1093/bioinformatics/bti634
3991:10.1126/science.290.5500.2323
3940:10.1126/science.290.5500.2319
3714:Principal Curves and Surfaces
3381:
1695:has the following properties
1688:{\displaystyle {\mathit {k}}}
1460:Curvilinear distance analysis
1259:restricted Boltzmann machines
1210:Local tangent space alignment
1204:Local tangent space alignment
136:techniques are listed below.
4646:10.1016/j.neunet.2006.05.014
4549:Lafon, Stephane (May 2004).
4352:10.1057/palgrave.ivs.9500051
4079:Zhang, Z.; Wang, J. (2006).
3741:; Stuetzle, W. (June 1989).
3711:Hastie, T. (November 1984).
2772:principal component analysis
1628:{\displaystyle \mathbf {d} }
1479:Dimensionality Reduction or
627:k-nearest neighbor algorithm
257:{\displaystyle \mathbf {X} }
108:principal component analysis
60:principal component analysis
56:singular value decomposition
7:
3890:. Vol. 16. MIT Press.
3369:Growing self-organizing map
3327:
3285:
3202:
3161:Nonlinear PCA (NLPCA) uses
3141:or even geodesic distance.
1639:represent the data set and
132:Some of the more prominent
126:spectral submanifolds (SSM)
10:
4952:
4901:Relational Perspective Map
4594:10.1016/j.acha.2006.04.006
4258:Ding, M.; Fan, G. (2015).
4126:10.1109/JTEHM.2019.2936348
3595:10.1162/089976698300017467
3573:Schölkopf, B.; Smola, A.;
3533:. University of Leicester.
3276:maximum variance unfolding
3076:{\displaystyle D_{t}(x,y)}
3026:{\displaystyle D_{t}(x,y)}
2446:{\displaystyle P=D^{-1}K.}
1312:Relational perspective map
1234:Maximum Variance Unfolding
1229:Maximum variance unfolding
1207:
609:
4846:Murphy, Kevin P. (2022).
4823:umap-learn.readthedocs.io
4776:10.1007/s11263-010-0322-1
4340:Information Visualization
4276:10.1109/TCYB.2014.2373393
4189:10.1137/s1064827502419154
3665:10.1142/S0129065710002383
3487:10.1007/s11071-016-2974-z
3395:Lawrence, Neil D (2012).
3349:Whitney embedding theorem
3272:locally linear embeddings
2097:is a nearest neighbor of
1832:is positivity preserving
641:
631:Laplace–Beltrami operator
235:{\displaystyle m\times n}
4896:Locally Linear Embedding
4026:Proc Natl Acad Sci U S A
3191:Manifold Sculpting uses
3181:concentration of measure
3151:multidimensional scaling
2335:{\displaystyle K_{ij}=1}
2240:{\displaystyle K_{ij}=0}
1318:multidimensional scaling
1267:multidimensional scaling
679:Locally-linear Embedding
674:Locally-linear embedding
655:Multidimensional Scaling
651:Floyd–Warshall algorithm
649:is a combination of the
564:gross product per capita
92:intrinsic dimensionality
52:dimensionality reduction
4047:10.1073/pnas.1031596100
3626:10.1145/1015330.1015417
2636:time steps is given by
2355:{\displaystyle \sigma }
2145:{\displaystyle \sigma }
2030:In the above equation,
1854:For example, the graph
698:based on its neighbors
612:Manifold regularization
3826:. MIT Press: 586–691.
3193:graduated optimization
3131:
3104:
3077:
3027:
2985:
2955:
2837:
2790:
2760:
2733:
2713:
2686:
2626:
2599:
2572:
2545:
2518:
2469:
2447:
2402:
2379:
2356:
2336:
2300:
2241:
2205:
2146:
2118:
2091:
2064:
2021:
1820:
1751:
1689:
1653:
1629:
1607:
1424:
1398:
1354:, as image manifolds.
1125:
987:that reconstructs the
953:
834:
587:
532:
512:
470:
409:
350:
309:
258:
236:
176:
103:
82:
36:
4886:Mike Tipping's Thesis
4379:Nature Communications
4331:Li, James X. (2004).
4235:Lawrence, N. (2005).
4098:Sidhu, Gagan (2019).
3529:Mirkes, E.M. (2011).
3354:Discriminant analysis
3132:
3130:{\displaystyle D_{t}}
3105:
3103:{\displaystyle D_{t}}
3078:
3028:
2986:
2984:{\displaystyle D_{t}}
2956:
2838:
2791:
2761:
2739:multiplied by itself
2734:
2714:
2712:{\displaystyle P^{t}}
2687:
2627:
2625:{\displaystyle x_{j}}
2600:
2598:{\displaystyle x_{i}}
2573:
2571:{\displaystyle x_{j}}
2546:
2544:{\displaystyle x_{i}}
2519:
2470:
2448:
2403:
2380:
2357:
2337:
2301:
2242:
2206:
2147:
2119:
2117:{\displaystyle x_{j}}
2092:
2090:{\displaystyle x_{i}}
2065:
2022:
1821:
1752:
1690:
1654:
1630:
1608:
1425:
1399:
1397:{\displaystyle t\in }
1364:Global cascades model
1197:orthogonal projection
1126:
991:th data point in the
954:
835:
557:
533:
531:{\displaystyle \Phi }
513:
471:
389:
351:
289:
259:
237:
198:latent variable model
158:
101:
80:
22:
3339:Spectral submanifold
3307:The method has also
3114:
3087:
3045:
2995:
2968:
2849:
2801:
2780:
2750:
2723:
2696:
2640:
2609:
2582:
2555:
2528:
2479:
2459:
2415:
2392:
2369:
2346:
2310:
2251:
2215:
2156:
2136:
2101:
2074:
2034:
1873:
1770:
1702:
1675:
1663:. Further, define a
1652:{\displaystyle \mu }
1643:
1617:
1523:
1408:
1370:
1022:
913:
888:to be zero if point
731:
522:
491:
370:
270:
246:
220:
66:Applications of NLDR
4936:Dimension reduction
4848:"Manifold Learning"
4401:2015NatCo...6.7723T
4038:2003PNAS..100.5591D
3983:2000Sci...290.2323R
3932:2000Sci...290.2319T
3544:Yin, Hujun (2007).
3425:2010arXiv1010.4830L
3375:Self-organizing map
3334:Manifold hypothesis
3240:Riemannian manifold
3221:stress majorization
2866:
1948:
1423:{\displaystyle t=0}
1151:eigen value problem
978:. The same weights
606:Laplacian eigenmaps
341:
186:self-organizing map
180:Self-organizing map
169:principal component
118:Invariant manifolds
4409:10.1038/ncomms8723
3743:"Principal Curves"
3579:Neural Computation
3465:Nonlinear Dynamics
3187:Manifold sculpting
3139:Euclidean distance
3127:
3100:
3073:
3023:
2981:
2951:
2852:
2833:
2786:
2756:
2746:The Markov matrix
2729:
2709:
2682:
2622:
2595:
2568:
2541:
2514:
2465:
2443:
2398:
2375:
2352:
2332:
2296:
2237:
2201:
2142:
2114:
2087:
2060:
2017:
2012:
1934:
1816:
1747:
1685:
1649:
1625:
1603:
1493:Manifold alignment
1488:Manifold alignment
1466:geodesic distances
1420:
1394:
1252:is a feed-forward
1167:Euclidean distance
1121:
1080:
1049:
949:
925:
830:
789:
758:
659:geodesic distances
588:
528:
508:
466:
346:
323:
254:
232:
188:(SOM, also called
177:
140:Important concepts
106:By comparison, if
104:
83:
37:
4911:RankVisu homepage
4861:978-0-262-04682-4
4464:10.1109/72.554199
3926:(5500): 2319–23.
3807:Belkin, Mikhail;
3793:978-3-540-73749-0
3559:978-3-540-73749-0
3449:978-0-387-39350-6
3256:similarity matrix
3244:Riemannian metric
3237:locally connected
2789:{\displaystyle K}
2759:{\displaystyle P}
2732:{\displaystyle P}
2468:{\displaystyle P}
2401:{\displaystyle D}
2378:{\displaystyle K}
2362:can be found in.
2008:
1971:
1453:have to be made.
1071:
1040:
916:
780:
749:
387:
287:
122:dynamical systems
44:manifold learning
4943:
4865:
4833:
4832:
4830:
4829:
4815:
4809:
4808:
4806:
4794:
4788:
4787:
4761:
4752:
4746:
4739:
4733:
4724:
4718:
4715:
4709:
4708:
4698:
4688:
4664:
4658:
4657:
4640:(6–7): 889–899.
4629:
4623:
4622:
4615:Bah, B. (2008).
4612:
4606:
4605:
4579:
4575:"Diffusion Maps"
4570:
4561:
4560:
4546:
4540:
4539:
4537:
4526:
4520:
4519:
4513:
4504:
4498:
4497:
4491:
4482:
4476:
4475:
4449:
4440:
4431:
4430:
4420:
4394:
4370:
4364:
4363:
4337:
4328:
4322:
4321:
4311:
4302:
4296:
4295:
4255:
4249:
4248:
4232:
4226:
4225:
4199:
4193:
4192:
4182:
4162:
4156:
4155:
4137:
4119:
4095:
4089:
4088:
4076:
4070:
4069:
4059:
4049:
4017:
4011:
4010:
3977:(5500): 2323–6.
3966:
3960:
3959:
3917:
3908:
3902:
3901:
3885:
3876:
3870:
3865:
3855:
3846:
3845:
3817:
3804:
3798:
3797:
3774:
3768:
3767:
3747:
3735:
3729:
3728:
3726:
3719:
3708:
3702:
3691:
3685:
3684:
3658:
3636:
3630:
3629:
3613:
3607:
3606:
3570:
3564:
3563:
3541:
3535:
3534:
3525:
3519:
3518:
3516:
3505:
3499:
3498:
3480:
3471:(3): 1493–1534.
3460:
3454:
3453:
3435:
3429:
3428:
3418:
3409:(May): 1609–38.
3392:
3364:Feature learning
3309:been implemented
3176:Sammon's mapping
3136:
3134:
3133:
3128:
3126:
3125:
3109:
3107:
3106:
3101:
3099:
3098:
3082:
3080:
3079:
3074:
3057:
3056:
3032:
3030:
3029:
3024:
3007:
3006:
2990:
2988:
2987:
2982:
2980:
2979:
2960:
2958:
2957:
2952:
2950:
2949:
2925:
2924:
2897:
2896:
2865:
2860:
2842:
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2834:
2832:
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2816:
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2795:
2793:
2792:
2787:
2765:
2763:
2762:
2757:
2738:
2736:
2735:
2730:
2718:
2716:
2715:
2710:
2708:
2707:
2691:
2689:
2688:
2683:
2678:
2677:
2665:
2664:
2652:
2651:
2631:
2629:
2628:
2623:
2621:
2620:
2604:
2602:
2601:
2596:
2594:
2593:
2577:
2575:
2574:
2569:
2567:
2566:
2550:
2548:
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2542:
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2539:
2523:
2521:
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2515:
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2509:
2497:
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2474:
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2466:
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2407:
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2399:
2384:
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2376:
2361:
2359:
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2341:
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2325:
2324:
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2297:
2289:
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2279:
2278:
2266:
2265:
2246:
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2243:
2238:
2230:
2229:
2210:
2208:
2207:
2202:
2194:
2193:
2184:
2183:
2171:
2170:
2151:
2149:
2148:
2143:
2123:
2121:
2120:
2115:
2113:
2112:
2096:
2094:
2093:
2088:
2086:
2085:
2069:
2067:
2066:
2061:
2059:
2058:
2046:
2045:
2026:
2024:
2023:
2018:
2016:
2015:
2009:
2006:
1995:
1994:
1982:
1981:
1972:
1969:
1965:
1964:
1963:
1962:
1953:
1947:
1942:
1933:
1932:
1920:
1919:
1888:
1887:
1825:
1823:
1822:
1817:
1756:
1754:
1753:
1748:
1694:
1692:
1691:
1686:
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1658:
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1634:
1632:
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1626:
1624:
1612:
1610:
1609:
1604:
1602:
1601:
1600:
1578:
1577:
1559:
1558:
1546:
1545:
1530:
1447:Sammon's mapping
1429:
1427:
1426:
1421:
1403:
1401:
1400:
1395:
1348:projective space
1307:Other algorithms
1290:Gaussian process
1130:
1128:
1127:
1122:
1120:
1119:
1114:
1110:
1109:
1108:
1107:
1102:
1096:
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831:
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823:
819:
818:
817:
816:
811:
805:
804:
796:
788:
776:
775:
770:
757:
592:Principal curves
572:infant mortality
537:
535:
534:
529:
517:
515:
514:
509:
504:
475:
473:
472:
467:
462:
461:
460:
459:
449:
448:
443:
428:
427:
422:
408:
403:
388:
380:
355:
353:
352:
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342:
340:
339:
333:
328:
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321:
316:
308:
303:
288:
280:
263:
261:
260:
255:
253:
241:
239:
238:
233:
202:density networks
150:Sammon's mapping
145:Sammon's mapping
48:latent manifolds
42:, also known as
4951:
4950:
4946:
4945:
4944:
4942:
4941:
4940:
4926:
4925:
4906:DD-HDS homepage
4872:
4862:
4842:
4840:Further reading
4837:
4836:
4827:
4825:
4817:
4816:
4812:
4795:
4791:
4759:
4753:
4749:
4740:
4736:
4725:
4721:
4716:
4712:
4665:
4661:
4634:Neural Networks
4630:
4626:
4613:
4609:
4577:
4571:
4564:
4557:Yale University
4547:
4543:
4535:
4527:
4523:
4511:
4505:
4501:
4489:
4483:
4479:
4447:
4441:
4434:
4371:
4367:
4335:
4329:
4325:
4309:
4303:
4299:
4270:(11): 2413–24.
4256:
4252:
4233:
4229:
4214:
4200:
4196:
4180:10.1.1.211.9957
4163:
4159:
4096:
4092:
4077:
4073:
4018:
4014:
3967:
3963:
3915:
3909:
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3883:
3877:
3873:
3856:
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3775:
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3692:
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3571:
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3560:
3542:
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3526:
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3514:
3506:
3502:
3461:
3457:
3450:
3436:
3432:
3393:
3389:
3384:
3344:Taken's theorem
3330:
3288:
3260:distance matrix
3252:
3229:
3214:
3205:
3189:
3172:
3163:backpropagation
3159:
3147:
3121:
3117:
3115:
3112:
3111:
3094:
3090:
3088:
3085:
3084:
3052:
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3002:
2998:
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2975:
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2920:
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2892:
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2807:
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2393:
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2389:
2370:
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2344:
2343:
2317:
2313:
2311:
2308:
2307:
2284:
2280:
2274:
2270:
2261:
2257:
2252:
2249:
2248:
2222:
2218:
2216:
2213:
2212:
2189:
2185:
2179:
2175:
2166:
2162:
2157:
2154:
2153:
2137:
2134:
2133:
2108:
2104:
2102:
2099:
2098:
2081:
2077:
2075:
2072:
2071:
2054:
2050:
2041:
2037:
2035:
2032:
2031:
2011:
2010:
2005:
2003:
1997:
1996:
1990:
1986:
1977:
1973:
1968:
1966:
1958:
1954:
1949:
1943:
1938:
1928:
1924:
1915:
1911:
1904:
1900:
1893:
1892:
1880:
1876:
1874:
1871:
1870:
1771:
1768:
1767:
1703:
1700:
1699:
1679:
1678:
1676:
1673:
1672:
1644:
1641:
1640:
1620:
1618:
1615:
1614:
1596:
1592:
1591:
1573:
1569:
1554:
1550:
1541:
1537:
1526:
1524:
1521:
1520:
1502:
1490:
1474:
1462:
1440:
1409:
1406:
1405:
1371:
1368:
1367:
1360:
1314:
1309:
1298:
1283:
1271:Sammon mappings
1246:
1231:
1212:
1206:
1192:
1179:
1141:
1137:
1115:
1103:
1098:
1097:
1088:
1083:
1082:
1081:
1075:
1062:
1057:
1056:
1055:
1051:
1050:
1044:
1023:
1020:
1019:
1010:
1002:
986:
933:
928:
927:
926:
920:
914:
911:
910:
905:
896:
887:
878:
869:
860:
851:
824:
812:
807:
806:
797:
792:
791:
790:
784:
771:
766:
765:
764:
760:
759:
753:
732:
729:
728:
715:
706:
697:
676:
644:
614:
608:
580:principal curve
568:life expectancy
552:
523:
520:
519:
500:
492:
489:
488:
455:
454:
450:
444:
439:
438:
423:
418:
417:
410:
404:
393:
379:
371:
368:
367:
335:
334:
329:
324:
317:
312:
311:
310:
304:
293:
279:
271:
268:
267:
249:
247:
244:
243:
221:
218:
217:
210:
182:
147:
142:
74:
68:
17:
12:
11:
5:
4949:
4939:
4938:
4924:
4923:
4918:
4913:
4908:
4903:
4898:
4893:
4888:
4883:
4878:
4871:
4870:External links
4868:
4867:
4866:
4860:
4841:
4838:
4835:
4834:
4810:
4789:
4747:
4734:
4719:
4710:
4673:Bioinformatics
4659:
4624:
4607:
4562:
4541:
4521:
4499:
4477:
4458:(1): 148–154.
4432:
4365:
4323:
4297:
4250:
4227:
4212:
4194:
4173:(1): 313–338.
4157:
4090:
4071:
4032:(10): 5591–6.
4012:
3961:
3903:
3896:
3871:
3868:Ohio-state.edu
3847:
3832:
3809:Niyogi, Partha
3799:
3792:
3769:
3758:(406): 502–6.
3730:
3703:
3699:Institut Curie
3686:
3649:(3): 219–232.
3631:
3608:
3565:
3558:
3536:
3520:
3500:
3455:
3448:
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3346:
3341:
3336:
3329:
3326:
3325:
3324:
3305:
3295:
3287:
3284:
3280:Sammon mapping
3251:
3248:
3228:
3225:
3223:that follows.
3213:
3210:
3204:
3201:
3188:
3185:
3171:
3168:
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3146:
3143:
3124:
3120:
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3019:
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2859:
2855:
2830:
2827:
2824:
2820:
2814:
2810:
2806:
2785:
2755:
2728:
2719:is the matrix
2706:
2702:
2681:
2676:
2672:
2668:
2663:
2659:
2655:
2650:
2646:
2619:
2615:
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2200:
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2182:
2178:
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2165:
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2141:
2111:
2107:
2084:
2080:
2057:
2053:
2049:
2044:
2040:
2028:
2027:
2014:
2004:
2002:
1999:
1998:
1993:
1989:
1985:
1980:
1976:
1967:
1961:
1957:
1952:
1946:
1941:
1937:
1931:
1927:
1923:
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1914:
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1568:
1565:
1562:
1557:
1553:
1549:
1544:
1540:
1536:
1533:
1529:
1505:Diffusion maps
1501:
1500:Diffusion maps
1498:
1489:
1486:
1473:
1470:
1461:
1458:
1439:
1436:
1419:
1416:
1413:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1359:
1358:Contagion maps
1356:
1313:
1310:
1308:
1305:
1297:
1294:
1282:
1279:
1254:neural network
1245:
1242:
1230:
1227:
1208:Main article:
1205:
1202:
1191:
1188:
1178:
1175:
1139:
1135:
1132:
1131:
1118:
1113:
1106:
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756:
752:
748:
745:
742:
739:
736:
711:
702:
693:
675:
672:
643:
640:
635:Fourier series
607:
604:
551:
548:
527:
507:
503:
499:
496:
477:
476:
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458:
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67:
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15:
9:
6:
4:
3:
2:
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4909:
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3907:
3899:
3897:0-262-20152-6
3893:
3889:
3882:
3875:
3869:
3863:
3862:
3854:
3852:
3843:
3839:
3835:
3833:0-262-27173-7
3829:
3825:
3821:
3814:
3810:
3803:
3795:
3789:
3785:
3784:
3779:
3778:Gorban, A. N.
3773:
3765:
3761:
3757:
3753:
3752:
3744:
3740:
3734:
3723:
3716:
3715:
3707:
3700:
3696:
3693:A. Zinovyev,
3690:
3682:
3678:
3674:
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3666:
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3657:
3652:
3648:
3644:
3643:
3635:
3627:
3623:
3619:
3612:
3604:
3600:
3596:
3592:
3589:: 1299–1319.
3588:
3584:
3580:
3576:
3575:MĂĽller, K.-R.
3569:
3561:
3555:
3551:
3547:
3540:
3532:
3524:
3513:
3512:
3504:
3496:
3492:
3488:
3484:
3479:
3474:
3470:
3466:
3459:
3451:
3445:
3441:
3434:
3426:
3422:
3417:
3412:
3408:
3404:
3403:
3398:
3391:
3387:
3376:
3373:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3331:
3322:
3318:
3314:
3310:
3306:
3303:
3299:
3296:
3293:
3290:
3289:
3283:
3281:
3277:
3273:
3269:
3265:
3261:
3257:
3247:
3245:
3242:and that the
3241:
3238:
3234:
3224:
3222:
3218:
3209:
3200:
3198:
3194:
3184:
3182:
3177:
3167:
3164:
3157:Nonlinear PCA
3154:
3152:
3142:
3140:
3122:
3118:
3095:
3091:
3067:
3064:
3061:
3053:
3049:
3040:
3036:
3017:
3014:
3011:
3003:
2999:
2976:
2972:
2964:For fixed t,
2946:
2935:
2932:
2929:
2921:
2917:
2913:
2907:
2904:
2901:
2893:
2889:
2882:
2876:
2873:
2870:
2862:
2857:
2853:
2845:
2844:
2843:
2828:
2825:
2822:
2812:
2808:
2783:
2775:
2773:
2769:
2753:
2744:
2742:
2726:
2704:
2700:
2674:
2670:
2666:
2661:
2657:
2648:
2644:
2635:
2617:
2613:
2590:
2586:
2563:
2559:
2536:
2532:
2506:
2502:
2498:
2493:
2489:
2482:
2462:
2440:
2437:
2432:
2429:
2425:
2421:
2418:
2411:
2410:
2409:
2395:
2388:
2387:degree matrix
2372:
2363:
2349:
2329:
2326:
2321:
2318:
2314:
2293:
2290:
2285:
2275:
2271:
2267:
2262:
2258:
2234:
2231:
2226:
2223:
2219:
2198:
2195:
2190:
2180:
2176:
2172:
2167:
2163:
2139:
2131:
2127:
2109:
2105:
2082:
2078:
2070:denotes that
2055:
2051:
2047:
2042:
2038:
2000:
1991:
1987:
1983:
1978:
1974:
1959:
1955:
1950:
1944:
1939:
1929:
1925:
1921:
1916:
1912:
1905:
1901:
1894:
1889:
1884:
1881:
1877:
1869:
1868:
1867:
1865:
1861:
1857:
1852:
1850:
1846:
1842:
1838:
1833:
1831:
1813:
1810:
1807:
1804:
1801:
1794:
1791:
1785:
1782:
1779:
1773:
1766:
1765:
1764:
1763:is symmetric
1762:
1744:
1738:
1735:
1732:
1726:
1723:
1717:
1714:
1711:
1705:
1698:
1697:
1696:
1671:. The kernel
1670:
1666:
1662:
1646:
1638:
1588:
1582:
1574:
1570:
1566:
1563:
1560:
1555:
1551:
1547:
1542:
1538:
1531:
1518:
1514:
1510:
1506:
1497:
1494:
1485:
1482:
1478:
1477:Diffeomorphic
1469:
1467:
1457:
1454:
1450:
1448:
1444:
1435:
1433:
1417:
1414:
1411:
1388:
1385:
1382:
1376:
1373:
1365:
1355:
1353:
1349:
1345:
1341:
1337:
1332:
1330:
1327:
1322:
1319:
1304:
1302:
1293:
1291:
1287:
1278:
1276:
1272:
1268:
1264:
1260:
1255:
1251:
1241:
1239:
1235:
1226:
1224:
1220:
1216:
1211:
1201:
1198:
1187:
1184:
1174:
1172:
1168:
1164:
1160:
1156:
1152:
1149:
1145:
1116:
1111:
1104:
1092:
1089:
1076:
1072:
1068:
1063:
1052:
1045:
1041:
1037:
1031:
1025:
1018:
1017:
1016:
1014:
1006:
998:
994:
990:
985:
981:
977:
973:
969:
965:
946:
943:
937:
934:
921:
917:
909:
908:
907:
904:
900:
895:
891:
886:
882:
877:
873:
868:
864:
859:
855:
850:
846:
825:
820:
813:
801:
798:
785:
781:
777:
772:
761:
754:
750:
746:
740:
734:
727:
726:
725:
723:
719:
714:
710:
705:
701:
696:
692:
687:
684:
683:sparse matrix
680:
671:
668:
663:
660:
656:
653:with classic
652:
648:
639:
636:
632:
628:
622:
620:
613:
603:
600:
599:Trevor Hastie
595:
594:and manifolds
593:
585:
581:
577:
573:
569:
565:
561:
556:
547:
544:
542:
486:
482:
463:
445:
424:
405:
400:
397:
394:
390:
384:
381:
376:
373:
366:
365:
364:
362:
343:
330:
318:
305:
300:
297:
294:
290:
284:
281:
276:
273:
266:
265:
264:
229:
226:
223:
215:
205:
203:
199:
195:
191:
187:
174:
170:
166:
162:
157:
153:
151:
137:
135:
130:
127:
123:
119:
115:
111:
109:
100:
96:
93:
89:
88:Hamming space
79:
75:
72:
63:
61:
57:
53:
49:
45:
41:
34:
30:
26:
21:
4851:
4826:. Retrieved
4822:
4813:
4792:
4770:(1): 56–68.
4767:
4763:
4750:
4737:
4727:
4722:
4713:
4676:
4672:
4662:
4637:
4633:
4627:
4617:
4610:
4585:
4581:
4551:
4544:
4531:
4524:
4515:
4502:
4493:
4480:
4455:
4451:
4382:
4378:
4368:
4343:
4339:
4326:
4320:: 2579–2605.
4317:
4313:
4300:
4267:
4263:
4253:
4247:: 1783–1816.
4244:
4240:
4230:
4203:
4197:
4170:
4166:
4160:
4107:
4103:
4093:
4087:: 1593–1600.
4084:
4074:
4029:
4025:
4015:
3974:
3970:
3964:
3923:
3919:
3906:
3887:
3874:
3860:
3823:
3819:
3802:
3782:
3772:
3755:
3749:
3733:
3713:
3706:
3689:
3646:
3640:
3634:
3617:
3611:
3582:
3578:
3568:
3549:
3539:
3523:
3510:
3503:
3468:
3464:
3458:
3442:. Springer.
3439:
3433:
3406:
3400:
3390:
3253:
3230:
3215:
3206:
3196:
3190:
3173:
3160:
3148:
3038:
3034:
2963:
2776:
2767:
2745:
2740:
2633:
2455:
2364:
2124:. Properly,
2029:
1863:
1859:
1855:
1853:
1849:Markov Chain
1844:
1840:
1836:
1834:
1829:
1828:
1760:
1759:
1668:
1664:
1660:
1636:
1517:Markov Chain
1503:
1491:
1480:
1475:
1463:
1455:
1451:
1441:
1361:
1352:Klein bottle
1333:
1323:
1315:
1299:
1284:
1247:
1244:Autoencoders
1237:
1232:
1222:
1218:
1213:
1193:
1180:
1170:
1162:
1158:
1154:
1147:
1143:
1133:
1012:
1004:
996:
992:
988:
983:
979:
975:
971:
967:
963:
961:
902:
898:
893:
889:
884:
880:
875:
871:
866:
862:
857:
853:
848:
844:
843:The weights
842:
721:
717:
712:
708:
703:
699:
694:
690:
688:
677:
664:
645:
623:
615:
590:
589:
579:
576:tuberculosis
545:
518:. Of course
485:kernel trick
480:
478:
360:
358:
211:
189:
183:
148:
133:
131:
116:
112:
105:
84:
73:
69:
43:
39:
38:
4588:(1): 5–30.
3359:Elastic map
1513:random walk
1434:algorithm.
1250:autoencoder
1183:Hessian LLE
584:elastic map
190:Kohonen map
165:broken line
33:Hessian LLE
4828:2019-05-04
4804:1802.03426
4213:1558600159
4117:1908.06319
3739:Hastie, T.
3695:ViDaExpert
3478:1602.00560
3382:References
1263:NeuroScale
1181:Like LLE,
970:such that
610:See also:
541:Swiss roll
214:kernel PCA
54:, such as
25:Swiss roll
4392:1408.1168
4346:: 49–59.
4222:928936290
4175:CiteSeerX
4152:201832756
3956:221338160
3656:1001.1122
3587:MIT Press
3416:1010.4830
3317:available
2943:‖
2936:⋅
2914:−
2908:⋅
2886:‖
2826:∈
2430:−
2350:σ
2294:σ
2291:≪
2282:‖
2268:−
2255:‖
2199:σ
2196:≫
2187:‖
2173:−
2160:‖
2140:σ
2048:∼
2007:otherwise
1984:∼
1956:σ
1936:‖
1922:−
1909:‖
1906:−
1799:∀
1792:≥
1647:μ
1589:⊂
1586:Ω
1583:∈
1564:…
1509:diffusion
1481:Diffeomap
1377:∈
1073:∑
1069:−
1042:∑
974:>>
918:∑
782:∑
778:−
751:∑
526:Φ
495:Φ
433:Φ
412:Φ
391:∑
291:∑
227:×
4930:Category
4705:16109748
4654:16787737
4602:17160669
4472:18255618
4427:26194875
4385:: 7723.
4292:15591304
4284:25532201
4144:31497410
4110:: 1–11.
4066:16576753
3999:11125150
3948:11125149
3842:52710683
3811:(2001).
3722:Archived
3701:, Paris.
3673:20556849
3495:44074026
3328:See also
3286:Software
3203:RankVisu
2130:manifold
2126:Geodesic
1970:if
667:manifold
4784:1365750
4555:(PhD).
4418:4566922
4397:Bibcode
4360:7566939
4135:6726465
4034:Bibcode
4007:5987139
3979:Bibcode
3971:Science
3928:Bibcode
3920:Science
3681:2170982
3603:6674407
3421:Bibcode
3298:UMAP.jl
3292:Waffles
2743:times.
2692:. Here
2247:and if
1340:VisuMap
1326:Coulomb
1011:in the
1003:in the
242:matrix
90:). The
4876:Isomap
4858:
4782:
4703:
4652:
4600:
4470:
4425:
4415:
4358:
4290:
4282:
4220:
4210:
4177:
4150:
4142:
4132:
4064:
4057:156245
4054:
4005:
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3954:
3946:
3894:
3840:
3830:
3790:
3679:
3671:
3601:
3556:
3493:
3446:
3371:(GSOM)
3321:GitHub
3315:(code
3313:Python
3278:, and
3268:isomap
1665:kernel
1635:. Let
1511:and a
1432:Isomap
1404:. For
1350:, and
1344:sphere
647:Isomap
642:Isomap
4799:arXiv
4780:S2CID
4760:(PDF)
4598:S2CID
4578:(PDF)
4536:(PDF)
4512:(PDF)
4490:(PDF)
4448:(PDF)
4387:arXiv
4356:S2CID
4336:(PDF)
4310:(PDF)
4288:S2CID
4148:S2CID
4112:arXiv
4003:S2CID
3952:S2CID
3916:(PDF)
3884:(PDF)
3816:(PDF)
3746:(PDF)
3725:(PDF)
3718:(PDF)
3677:S2CID
3651:arXiv
3599:S2CID
3585:(5).
3515:(PDF)
3491:S2CID
3473:arXiv
3411:arXiv
3377:(SOM)
3302:Julia
3258:or a
3233:t-SNE
2306:then
2211:then
1336:torus
1329:force
4856:ISBN
4701:PMID
4650:PMID
4468:PMID
4423:PMID
4280:PMID
4218:OCLC
4208:ISBN
4140:PMID
4062:PMID
3995:PMID
3944:PMID
3892:ISBN
3838:OCLC
3828:ISBN
3788:ISBN
3669:PMID
3554:ISBN
3444:ISBN
1269:and
1215:LTSA
184:The
58:and
31:and
4772:doi
4691:hdl
4681:doi
4642:doi
4590:doi
4460:doi
4413:PMC
4405:doi
4348:doi
4272:doi
4185:doi
4130:PMC
4122:doi
4052:PMC
4042:doi
4030:100
3987:doi
3975:290
3936:doi
3924:290
3760:doi
3661:doi
3622:doi
3591:doi
3483:doi
3319:on
3311:in
3037:to
2632:in
2605:to
2551:to
1858:= (
1248:An
724:).
163:(a
161:SOM
29:LLE
4932::
4850:.
4821:.
4778:.
4768:89
4766:.
4762:.
4699:.
4689:.
4677:21
4675:.
4671:.
4648:.
4638:19
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4596:.
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4580:.
4565:^
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1046:i
1038:=
1035:)
1032:Y
1029:(
1026:C
1013:d
1009:i
1005:D
1001:i
997:d
993:D
989:i
980:W
976:d
972:D
968:d
964:D
947:1
944:=
938:j
935:i
930:W
922:j
903:i
899:X
894:j
890:X
881:W
876:i
872:X
867:i
863:X
858:j
854:X
845:W
826:2
821:|
814:j
809:X
802:j
799:i
794:W
786:j
773:i
768:X
762:|
755:i
747:=
744:)
741:W
738:(
735:E
722:W
720:(
718:E
709:W
704:j
700:X
695:i
691:X
506:)
502:x
498:(
481:k
464:.
457:T
452:)
446:i
441:x
436:(
430:)
425:i
420:x
415:(
406:m
401:1
398:=
395:i
385:m
382:1
377:=
374:C
361:k
344:.
337:T
331:i
326:x
319:i
314:x
306:m
301:1
298:=
295:i
285:m
282:1
277:=
274:C
251:X
230:n
224:m
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