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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 5815429, 9 pages
Research Article

Parallelization of Eigenvalue-Based Dimensional Reductions via Homotopy Continuation

Institute of Electronics, Chinese Academy of Sciences, North Fourth Ring Road West 98, Beijing 100190, China

Received 12 December 2015; Accepted 22 February 2016

Academic Editor: Jinyun Yuan

Copyright © 2016 Size Bi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper investigates a homotopy-based method for embedding with hundreds of thousands of data items that yields a parallel algorithm suitable for running on a distributed system. Current eigenvalue-based embedding algorithms attempt to use a sparsification of the distance matrix to approximate a low-dimensional representation when handling large-scale data sets. The main reason of taking approximation is that it is still hindered by the eigendecomposition bottleneck for high-dimensional matrices in the embedding process. In this study, a homotopy continuation algorithm is applied for improving this embedding model by parallelizing the corresponding eigendecomposition. The eigenvalue solution is converted to the operation of ordinary differential equations with initialized values, and all isolated positive eigenvalues and corresponding eigenvectors can be obtained in parallel according to predicting eigenpaths. Experiments on the real data sets show that the homotopy-based approach is potential to be implemented for millions of data sets.