Performance comparison between reduction techniques If I want to compare the performance of PCA and PLS with nonlinear data, what do I have to measure ?
I know that I can use the MSE (but if I do not use a linear profile, I can't estimate the parameters of the linear regression, so how do I calculate it?
For the PLS, the data should follow a normal ditribution or it is an unsupervised method.
Which other methods would you propose to me for dimension reduction techniques?
 A: As you did not provide specific details on the problem I will give a short but comprehensive review of the dimension reduction techniques.
Generaly two problem statements considered:
Unsupervised dimension reduction.
We have a set of observations $S =\{X_i\}_{i=1}^{N}$ and need to find a reduced representation for it.


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*Linear models. Here PCA is a the most popular technique. MDS is another approach you can try.

*Nonlinear models. Here we have a lot of different techiques: LTSA (I would suggest this in general), Isomap, LLE, and lots of their modifications.


All this methods (and a litlle bit more) are provided by special (yet free) MatLab toolbox.
Supervised dimension reduction (aka Feature Extraction).
We have a set of observations (both regressors and responses) 
$$S =\{X_i, Y_i\}_{i=1}^{N}, X_i\in\mathbb{R}^p, Y_i\in\mathbb{R}^q$$ 
and need to find a reduced representation for $X$s which preserves information about $Y$s. And we assume that the foolowing model holds: 
$$Y=f(X)+\varepsilon=g(\tilde{X}) +\varepsilon, \tilde{X}=XB, B\in\mathbb{R}^{p\times d}, d <p.$$
Verbally it means that we expect the regression $Y=f(X)$ to lye on some hyperplane of lower dimensionality. This problem statement considered in the case when we finally need a regression to be constructed.


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*Linear models. PLS is the most popular (and probably the only linear) technique.

*Nonlinear models. MAVE, OPG, SAMM, SIR and some others are employed in this problem. Yocan find some MatLab implementations here.


In both problem statement dimensionalty selection is another individual problem.
I can give more exhaustive comments on any of the subjects provided if you give more details on your problem and possibly choose the approach.
