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/*
 * Copyright (c) 2010-2021 Haifeng Li. All rights reserved.
 *
 * Smile is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * Smile is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with Smile.  If not, see .
 */

/**
 * Manifold learning finds a low-dimensional basis for describing
 * high-dimensional data. Manifold learning is a popular approach to nonlinear
 * dimensionality reduction. Algorithms for this task are based on the idea
 * that the dimensionality of many data sets is only artificially high; though
 * each data point consists of perhaps thousands of features, it may be
 * described as a function of only a few underlying parameters. That is, the
 * data points are actually samples from a low-dimensional manifold that is
 * embedded in a high-dimensional space. Manifold learning algorithms attempt
 * to uncover these parameters in order to find a low-dimensional representation
 * of the data.
 * 
 * Some prominent approaches are locally linear embedding
 * (LLE), Hessian LLE, Laplacian eigenmaps, and LTSA. These techniques
 * construct a low-dimensional data representation using a cost function
 * that retains local properties of the data, and can be viewed as defining
 * a graph-based kernel for Kernel PCA. More recently, techniques have been
 * proposed that, instead of defining a fixed kernel, try to learn the kernel
 * using semidefinite programming. The most prominent example of such a
 * technique is maximum variance unfolding (MVU). The central idea of MVU
 * is to exactly preserve all pairwise distances between nearest neighbors
 * (in the inner product space), while maximizing the distances between points
 * that are not nearest neighbors.
 * 

 * An alternative approach to neighborhood preservation is through the
 * minimization of a cost function that measures differences between
 * distances in the input and output spaces. Important examples of such
 * techniques include classical multidimensional scaling (which is identical
 * to PCA), Isomap (which uses geodesic distances in the data space), diffusion
 * maps (which uses diffusion distances in the data space), t-SNE (which
 * minimizes the divergence between distributions over pairs of points),
 * and curvilinear component analysis.
 * 

 * Multidimensional scaling is a set of related statistical techniques
 * often used in information visualization for exploring similarities or
 * dissimilarities in data. An MDS algorithm starts with a matrix of item-item
 * similarities, then assigns a location to each item in N-dimensional space.
 * For sufficiently small N, the resulting locations may be displayed in a
 * graph or 3D visualization.
 * 

 * The major types of MDS algorithms include:
 * 

 * Classical multidimensional scaling
 * takes an input matrix giving dissimilarities between pairs of items and
 * outputs a coordinate matrix whose configuration minimizes a loss function
 * called strain.
 * Metric multidimensional scaling
 * A superset of classical MDS that generalizes the optimization procedure
 * to a variety of loss functions and input matrices of known distances with
 * weights and so on. A useful loss function in this context is called stress
 * which is often minimized using a procedure called stress majorization.
 * Non-metric multidimensional scaling
 * In contrast to metric MDS, non-metric MDS finds both a non-parametric
 * monotonic relationship between the dissimilarities in the item-item matrix
 * and the Euclidean distances between items, and the location of each item in
 * the low-dimensional space. The relationship is typically found using isotonic
 * regression.
 * Generalized multidimensional scaling
 * An extension of metric multidimensional scaling, in which the target
 * space is an arbitrary smooth non-Euclidean space. In case when the
 * dissimilarities are distances on a surface and the target space is another
 * surface, GMDS allows finding the minimum-distortion embedding of one surface
 * into another.
 * 
 *
 * @author Haifeng Li
 */
package smile.manifold;