Dimensionality reduction for a mapping

I have a dataset that I measured in 100 dimensions (i.e. each sample contains 100 values). I want to dimensionality reduce the data to 10 dimensions (i.e find a mapping where 10 values per sample can reconstruct the 100 values per sample). With PCA this is very simple; you remove the 90 smallest eigenvalues.

Now, lets say that for each sample, there is a linear transformation to a simpler set of 30 dimensions (lets call it QWERTY measuments) if you set values for some external parameters (the coefficients of the linear transformation changes when you change the external parameters - the equations to calculate the coefficents of the linear transformation from the external parameters are non-linear, however they are analytic and known exactly). If you apply regular linear PCA to these 30 dimensions, you can easily get dimensionality reduction to 5 dimensions with 98% of the variance explained.

What I want to do is dimensionality reduce the 100 dimension data set to 10 dimensions, only I want those 10 new dimensions to be able to reconstruct the 30 QWERTY measuments, whatever the external parameters are. In other words, I want 10 values per sample and the known coefficents of the linear transformation from the external parameters to be able to reconstruct the 30 dimension data with 98% of the variance explained.

What method can I use? All numbers were for illustration only, I just want to know the name of the method that solves this problem.

• Who's to say that's possible? Whether 98% of the information in your original p variables can be captured in only m<p dimensions is completely contingent on the data themselves. – gung Jan 19 '16 at 2:30
• If I choose the 5 external parameters arbitrarily, I will always be able to reduce the 30 dimensions to 5 dimensions with at least 98% variance explained. Assume this to be known. The question is how to do this without doing PCA for the infinite number of combinations of the 5 external parameters. – Sam Baker Jan 19 '16 at 2:47