| Many problems in information processing involve some form of dimensionality reduction. In this thesis, we introduce Locality Preserving Projections (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 Component Analysis (PCA)---a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional 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 Beltrami 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. Theoretical analysis shows that PCA, LPP, and Linear Discriminant Analysis (LDA) can be obtained from different graph models. Central to this is a graph structure that is inferred on the data points. LPP finds a projection that respects this graph structure. We have applied our algorithms to several real world applications, e.g. face analysis and document representation. |
