dimensionality reduction, LLE, fuzzy partition

We assume that a given high-dimensional data is embedded into a differentiable manifold. At the beginning, this manifold is unknown. The purpose is to characterize this manifold using its substantial parameters extracted from the sampled data. We would like to chacterize the closeness between data points. In this research, we focus on the locally linear embedding (LLE) algorithm. If we assume that two points are close enough, the LLE determines their non-zero closeness. The sum of the closeness values between a fixed point and all points that are close enough, is normalized to 1. The values of closeness are stored in an adjacency matrix W of the corresponding weighted graph representation of the data. If, for example, we assume that the point x has two neighbours, y and z producing non-singular correlation matrix, we showed that the closeness between the points x and y is equal to ... Another way to determine the closeness values is by their extraction from a fuzzy partition. We found cases in which we can redesign the basic function so that the weights given by the LLE and by the fuzzy partition coincide. The corresponding F1-transform component (determined by the weighted inner product given by the same fuzzy partition unit) is compared with lower-dimensional embedding given by the LLE.