The LLE and a linear mapping

• Published in: Pattern recognition Vol. 39; no. 9; pp. 1799 - 1804
• Main Authors: Wu, F.C., Hu, Z.Y.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.09.2006
• Subjects: Linear mapping; Locally linear embedding (LLE); Principal component analysis (PCA); Locally linear embedding (LLE); Principal component analysis (PCA); Linear mapping
• ISSN: 0031-3203; 1873-5142
• DOI: 10.1016/j.patcog.2006.03.019

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.