Rapid and brief communication The LLE and a linearmapping

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