Generalized Low-Computational Cost Laplacian Eigenmaps

Abstract

Dimensionality reduction (DR) is a methodology used in many fields linked to data processing, and may represent a preprocessing stage or be an essential element for the representation and classification of data. The main objective of DR is to obtain a new representation of the original data in a space of smaller dimension, such that more refined information is produced, as well as the time of the subsequent processing is decreased and/or visual representations more intelligible for human beings are generated. The spectral DR methods involve the calculation of an eigenvalue and eigenvector decomposition, which is usually high-computational-cost demanding, and, therefore, the task of obtaining a more dynamic and interactive user-machine integration is difficult. Therefore, for the design of an interactive IV system based on DR spectral methods, it is necessary to propose a strategy to reduce the computational cost required in the calculation of eigenvectors and eigenvalues. For this purpose, it is proposed to use locally linear submatrices and spectral embedding. This allows integrating natural intelligence with computational intelligence for the representation of data interactively, dynamically and at low computational cost. Additionally, an interactive model is proposed that allows the user to dynamically visualize the data through a weighted mixture.