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Isomap is a nonlinear dimensionality reduction method. It is one of several widely used low-dimensional embedding methods. Isomap is used for computing a quasi-isometric, low-dimensional embedding of a set of high-dimensional data points. The algorithm provides a simple method for estimating the intrinsic geometry of a data
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
based on a rough estimate of each data point’s neighbors on the manifold. Isomap is highly efficient and generally applicable to a broad range of data sources and dimensionalities.


Introduction

Isomap is one representative of isometric mapping methods, and extends metric
multidimensional scaling Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset. MDS is used to translate "information about the pairwise 'distances' among a set of n objects or individuals" into a configurati ...
(MDS) by incorporating the geodesic distances imposed by a weighted graph. To be specific, the classical scaling of metric MDS performs low-dimensional embedding based on the pairwise distance between data points, which is generally measured using straight-line
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore o ...
. Isomap is distinguished by its use of the geodesic distance induced by a neighborhood graph embedded in the classical scaling. This is done to incorporate manifold structure in the resulting embedding. Isomap defines the geodesic distance to be the sum of edge weights along the
shortest path In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between t ...
between two nodes (computed using Dijkstra's algorithm, for example). The top ''n'' eigenvectors of the geodesic distance matrix, represent the coordinates in the new ''n''-dimensional Euclidean space.


Algorithm

A very high-level description of Isomap algorithm is given below. * Determine the neighbors of each point. ** All points in some fixed radius. ** ''K'' nearest neighbors. * Construct a neighborhood graph. ** Each point is connected to other if it is a ''K'' nearest neighbor. ** Edge length equal to Euclidean distance. * Compute shortest path between two nodes. ** Dijkstra's algorithm **
Floyd–Warshall algorithm In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding shortest paths in a directed weighted graph with ...
* Compute lower-dimensional embedding. **
Multidimensional scaling Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset. MDS is used to translate "information about the pairwise 'distances' among a set of n objects or individuals" into a configurati ...


Extensions of ISOMAP

* LandMark ISOMAP (L-ISOMAP): Landmark-Isomap is a variant of Isomap which is faster than Isomap. However, the accuracy of the manifold is compromised by a marginal factor. In this algorithm, n << N landmark points are used out of the total N data points and an nxN matrix of the geodesic distance between each data point to the landmark points is computed. Landmark-MDS (LMDS) is then applied on the matrix to find a Euclidean embedding of all the data points. * C Isomap : C-Isomap involves magnifying the regions of high density and shrink the regions of low density of data points in the manifold. Edge weights that are maximized in Multi-Dimensional Scaling(MDS) are modified, with everything else remaining unaffected. *Parallel Transport Unfolding : Replaces the Dijkstra path-based geodesic distance estimates with parallel transport based approximations instead, improving robustness to irregularity and voids in the sampling.


Possible issues

The connectivity of each data point in the neighborhood graph is defined as its nearest ''k'' Euclidean neighbors in the high-dimensional space. This step is vulnerable to "short-circuit errors" if ''k'' is too large with respect to the manifold structure or if noise in the data moves the points slightly off the manifold. Even a single short-circuit error can alter many entries in the geodesic distance matrix, which in turn can lead to a drastically different (and incorrect) low-dimensional embedding. Conversely, if ''k'' is too small, the neighborhood graph may become too sparse to approximate geodesic paths accurately. But improvements have been made to this algorithm to make it work better for sparse and noisy data sets.


Relationship with other methods

Following the connection between the classical scaling and PCA, metric MDS can be interpreted as
kernel PCA 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. Using a kernel, the originally linear operations of PCA are performed ...
. In a similar manner, the geodesic distance matrix in Isomap can be viewed as a
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
matrix. The doubly centered geodesic distance matrix ''K'' in Isomap is of the form : K = -\frac HD^2 H\, where D^2 = D^2_:=(D_)^2 is the elementwise square of the geodesic distance matrix ''D'' = 'D''''ij'' ''H'' is the centering matrix, given by : H = I_n-\frac e_N e^T_N, \quad\texte_N= \ \dots\ 1T \in \mathbb^N. However, the kernel matrix ''K'' is not always positive semidefinite. The main idea for kernel Isomap is to make this ''K'' as a
Mercer Mercer may refer to: Business * Mercer (car), a defunct American automobile manufacturer (1909–1925) * Mercer (consulting firm), a large human resources consulting firm headquartered in New York City * Mercer (occupation), a merchant or trade ...
kernel matrix (that is positive semidefinite) using a constant-shifting method, in order to relate it to kernel PCA such that the generalization property naturally emerges .H. Choi, S. Choi, Robust Kernel Isomap, Pattern Recognition, Vol. 40, No. 3, pp. 853-862, 2007


See also

*
Kernel PCA 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. Using a kernel, the originally linear operations of PCA are performed ...
* Spectral clustering * Nonlinear dimensionality reduction


References

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External links


Isomap webpage at Stanford university

Initial article by Tenenbaum et al.

Global versus local methods in nonlinear dimensionality reduction at MIT by Tenenbaum et al.
Computational statistics