Rapid and brief communication: The LLE and a linear mapping

Authors:
F. C. Wu;Z. Y. Hu
Affiliations:
National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, P.O. Box 2728, Beijing 100080, People's Republic of China;National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, P.O. Box 2728, Beijing 100080, People's Republic of China
Venue:
Pattern Recognition
Year:
2006

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Abstract

The locally linear embedding (LLE) is considered an effective algorithm for dimensionality reduction. In this short note, some of its key properties are studied. In particular, we show that: (1) there always exists a linear mapping from the high-dimensional space to the low-dimensional space such that all the constraint conditions in the LLE can be satisfied. The implication of the existence of such a linear mapping is that the LLE cannot guarantee a one-to-one mapping from the high-dimensional space to the low-dimensional space for a given data set; (2) if the LLE is required to globally preserve distance, it must be a PCA mapping; (3) for a given high-dimensional data set, there always exists a local distance-preserving LLE. The above results can bring some new insights into a better understanding of the LLE.