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I am working on machine learning methods to do dimensional reduction. And I am wondering are there any ways to determine whether the reduced variables have already captured most information of the original higher-dimensional variables?

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I have one approach in my mind. Try to run t-SNE on your original dataset. Play a bit with the perplexity and you should be able to observe some kind of clusters if your data is descriptive. Then you can perform your dimensionality reduction technique. On the reduced dataset you can again run t-SNE. If you still observe the same clusters, or at least the same amount / density, you can assume you didn't loose relevant information.

t-SNE tries to find similarities within your data based on sampling from probability distributions. It works quite well usually. Check out this:

http://lvdmaaten.github.io/tsne/

And this:

https://www.youtube.com/watch?v=RJVL80Gg3lA

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