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

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

<p>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.  </p>

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

<p>TRACE OPTIMIZATION AND EIGENPROBLEMS IN DIMENSION REDUCTION METHODS, KOKIOPOULOU et al.</p>

<p>or the summary table in Section 2.7 in the thesis:</p>

<p><a href="http://webdocs.cs.ualberta.ca/~whitem/publications/thesis.pdf" rel="nofollow">Regularized Factor Models, Martha White</a></p>

CC BY-SA 3.0
2014-10-14T19:09:59.747Z
2014-10-14T19:09:59.747Z
Martha White
<p>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?</p>

<high-dimensional-geometry>
Is Locality preserving projections (LPP) method the same as Laplacian eigenmap method?
Mina
# Is Locality preserving projections (LPP) method the same as Laplacian eigenmap method?

(Asked by: username_0 on 2014-10-09)

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?

**Tags:** high-dimensional-geometry


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## 1 answers

### Answer by username_1 on 2014-10-14. Score: 1

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, username_1](http://webdocs.cs.ualberta.ca/~whitem/publications/thesis.pdf)


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