manifold
High level manifold learning operators.
Isometric feature mapping.
Isometric feature mapping. Isomap is a widely used low-dimensional embedding methods, where geodesic distances on a weighted graph are incorporated with the classical multidimensional scaling. Isomap is used for computing a quasi-isometric, low-dimensional embedding of a set of high-dimensional data points. Isomap is highly efficient and generally applicable to a broad range of data sources and dimensionalities.
To be specific, the classical MDS performs low-dimensional embedding based on the pairwise distance between data points, which is generally measured using straight-line Euclidean distance. Isomap is distinguished by its use of the geodesic distance induced by a 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 between two nodes. The top n eigenvectors of the geodesic distance matrix, represent the coordinates in the new n-dimensional Euclidean space.
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.
This class implements C-Isomap that 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.
the data set.
the dimension of the manifold.
k-nearest neighbor.
C-Isomap algorithm if true, otherwise standard algorithm.
Laplacian Eigenmap.
Laplacian Eigenmap. Using the notion of the Laplacian of the nearest neighbor adjacency graph, Laplacian Eigenmap compute a low dimensional representation of the dataset that optimally preserves local neighborhood information in a certain sense. The representation map generated by the algorithm may be viewed as a discrete approximation to a continuous map that naturally arises from the geometry of the manifold.
The locality preserving character of the Laplacian Eigenmap algorithm makes it relatively insensitive to outliers and noise. It is also not prone to "short circuiting" as only the local distances are used.
the data set.
the dimension of the manifold.
k-nearest neighbor.
the smooth/width parameter of heat kernel e-||x-y||2 / t. Non-positive value means discrete weights.
Locally Linear Embedding.
Locally Linear Embedding. It has several advantages over Isomap, including faster optimization when implemented to take advantage of sparse matrix 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 describe the point as a linear combination of its neighbors. Finally, it uses an eigenvector-based optimization 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.
the data set.
the dimension of the manifold.
k-nearest neighbor.
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.