Learning Neural Representations and Local Embedding for Nonlinear Dimensionality Reduction Mapping

• Published in: Mathematics (Basel) Vol. 9; no. 9; p. 1017
• Main Authors: Wu, Sheng-Shiung, Jong, Sing-Jie, Hu, Kai, Wu, Jiann-Ming
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.05.2021
• Subjects: Approximation; Centroids; data support approximation; data visualization; distance preserving mapping; Embedding; Graphical representations; Learning; Mapping; Mathematical models; Mathematics; Neighborhoods; nonlinear dimensionality reduction mapping; Principal components analysis; Reduction; topology preservation; Training; unsupervised learning
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math9091017

This work explores neural approximation for nonlinear dimensionality reduction mapping based on internal representations of graph-organized regular data supports. Given training observations are assumed as a sample from a high-dimensional space with an embedding low-dimensional manifold. An approximating function consisting of adaptable built-in parameters is optimized subject to given training observations by the proposed learning process, and verified for transformation of novel testing observations to images in the low-dimensional output space. Optimized internal representations sketch graph-organized supports of distributed data clusters and their representative images in the output space. On the basis, the approximating function is able to operate for testing without reserving original massive training observations. The neural approximating model contains multiple modules. Each activates a non-zero output for mapping in response to an input inside its correspondent local support. Graph-organized data supports have lateral interconnections for representing neighboring relations, inferring the minimal path between centroids of any two data supports, and proposing distance constraints for mapping all centroids to images in the output space. Following the distance-preserving principle, this work proposes Levenberg-Marquardt learning for optimizing images of centroids in the output space subject to given distance constraints, and further develops local embedding constraints for mapping during execution phase. Numerical simulations show the proposed neural approximation effective and reliable for nonlinear dimensionality reduction mapping.