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Are "Locality Preserving projections(LPP)" and "Laplacian eigenmap" one method for dimension reduction, only under different names? I have not seen any article refer to them as the same method, in fact in LPP paper, they cite Laplacian eigenmap as a non-linear dimension reduction and say that LPP is a linear dim reduction method. But both algorithms have exactly same description. Are they a same method, or am I missing something?

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There is a subtle difference, which can be difficult to recognize. They both minimize the same objective:

$\sum_{i,j} w_{ij} || y_i - y_j ||^2_2$

However, they parametrize the predictor on y differently. Locality preserving projections is linear because the parameterization is linear: $y_i = x_iW$. Laplacian eigenmaps is non-linear because the parametrization uses a kernel: $y_i = k_iA$. Kernel LPP is equivalent to Laplacian eigenmaps.

For a good description and a summary showing the connections between all these different dimensionality reduction algorithms, see the paper:

TRACE OPTIMIZATION AND EIGENPROBLEMS IN DIMENSION REDUCTION METHODS, KOKIOPOULOU et al.

or the summary table in Section 2.7 in the thesis:

Regularized Factor Models, Martha White

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