Local linear transformation embedding

Abstract

Dimensionality reduction is vital in many fields and locally linear embedding (LLE) is one of the most important approaches. However, LLE is unavoidable to derive the nonuniform wraps and folds when the data are of low sample density or unevenly sampled. LLE would also fail when the data are contaminated by even small noises. We have analyzed the performance of LLE and pointed out the reason why LLE fails. An improved algorithm, local linear transformation embedding (LLTE), is then proposed. Local linear transformation is performed on nearby points. The ‘Three-stage LLTE’ is also provided when the data has outliers. Comparing with LLE and Local tangent space alignment (LTSA), LLTE could derive more practical embedding than LLE and has wider application prospect than LTSA. Meanwhile, it exploits the tight relations between LLE/LLTE and LTSA. Several experiments and numerical results demonstrate the potential of our algorithm.