Xiaofei He, Partha Niyogi
Many problems in information processing involve some form of dimen- sionality reduction. In this paper, we introduce Locality Preserving Pro- jections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Com- ponent Analysis (PCA) – a classical linear technique that projects the data along the directions of maximal variance. When the high dimen- sional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by ﬁnding the optimal linear approximations to the eigenfunctions of the Laplace Bel- trami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is deﬁned everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets.