I recently stumbled across dimensionality reduction in machine learning. I know in that particular case, one may not even want to retain all information. However I wondered if it is possible with a non linear function to map a high dimensional space to a low dimensional space. I know this is not possible for linear functions, since a lower dimensional space can only be mapped to a subspace of a higher dimension. But as far as my understanding goes, the infinity between 0 and 1 in real numbers is as great as the infinity of all real numbers(I'm sorry for lacking the formally correct terms).
- Could that mean it is possible to find a mathematical function, that maps different dimensions to different intervals or am I making some crucial mistake?
- Are there alternative more sophisticated methods? (a reference would be enough)
- What are the minimum requirements for a vector space for this to be possible? After all I've assumed it to be a real numbered vector space. Is e.g. countable infinity and continuity enough or does it have to full fill more attributes? or maybe even less (only infinite)?
