Elastic map

Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured k-means clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules). With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by A.N. Gorban, A.Y. Zinovyev and A.A. Pitenko in 1996–1998. Let be a data set in a finite-dimensional Euclidean space. Elastic map is represented by a set of nodes in the same space. Each datapoint has a host node, namely the closest node (if there are several closest nodes then one takes the node with the smallest number). The data set is divided into classes . The approximation energy D is the distortion which is the energy of the springs with unit elasticity which connect each data point with its host node. It is possible to apply weighting factors to the terms of this sum, for example to reflect the standard deviation of the probability density function of any subset of data points . On the set of nodes an additional structure is defined. Some pairs of nodes, , are connected by elastic edges. Call this set of pairs . Some triplets of nodes, , form bending ribs. Call this set of triplets . The stretching energy is , The bending energy is , where and are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the membrane, while the bending energy is referred to as the thin plate term. For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices.