Loss of information while projecting multidimensional data

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I'm interested in the evaluation of the loss of information after projecting multidimensional data. Since the dimensional reduction is a common tool to analyse data,a question about the loss of information seems to be reasonable, but I failed to find something useful, except comparing entropia before and after projection.

Can you help me and advice some technique?

Thank you very much!!!

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Perhaps what you are looking for is the Johnson-Lindenstrauss lemma...