Locally-Linear Embedding Algorithm

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A Locally-Linear Embedding Algorithm is a non-linear dimensionality reduction algorithm that uses an eigenvector-based optimization algorithm to find the low-dimensional embedding of points.



References

2014

  • (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Nonlinear_dimensionality_reduction#Locally-linear_embedding Retrieved:2014-3-20.
    • Locally-Linear Embedding (LLE) [1] was presented at approximately the same time as Isomap. 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. LLE has no internal model.

      LLE computes the barycentric coordinates of a point Xi based on its neighbors Xj. The original point is reconstructed by a linear combination, given by the weight matrix Wij, of its neighbors. The reconstruction error is given by the cost function E(W).

       : [math] E(W) = \sum_i |{\mathbf{X}_i - \sum_j {\mathbf{W}_{ij}\mathbf{X}_j}|}^\mathsf{2} [/math]

      The weights Wij refer to the amount of contribution the point Xj has while reconstructing the point Xi. The cost function is minimized under two constraints:

      (a) Each data point Xi is reconstructed only from its neighbors, thus enforcing Wij to be zero if point Xj is not a neighbor of the point Xi and

      (b) The sum of every row of the weight matrix equals 1.

       : [math] \sum_j {\mathbf{W}_{ij}} = 1 [/math]

      The original data points are collected in a D dimensional space and the goal of the algorithm is to reduce the dimensionality to d such that D >> d. The same weights Wij that reconstructs the ith data point in the D dimensional space will be used to reconstruct the same point in the lower d dimensional space. A neighborhood preserving map is created based on this idea. Each point Xi in the D dimensional space is mapped onto a point Yi in the d dimensional space by minimizing the cost function

       : [math] C(Y) = \sum_i |{\mathbf{Y}_i - \sum_j {\mathbf{W}_{ij}\mathbf{Y}_j}|}^\mathsf{2} [/math]

      In this cost function, unlike the previous one, the weights Wij are kept fixed and the minimization is done on the points Yi to optimize the coordinates. This minimization problem can be solved by solving a sparse N X N eigen value problem (N being the number of data points), whose bottom d nonzero eigen vectors provide an orthogonal set of coordinates. Generally the data points are reconstructed from K nearest neighbors, as measured by Euclidean distance. For such an implementation the algorithm has only one free parameter K, which can be chosen by cross validation.

  1. S. T. Roweis and L. K. Saul, Nonlinear Dimensionality Reduction by Locally Linear Embedding, Science Vol 290, 22 December 2000, 2323–2326.

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