Description
The notion of a factor is arguably one of the most powerful concepts in algebra. Factoring something is essentially forgetting information in a "smart" way. Here "smart" means that it is consistent with certain prescribed operations. A similar notion (also called factor) arises in the theory of dynamical systems and it is a key component in Furstenberg's program. Quite surprisingly Szemeredi's famousregularity lemma can also be viewed as finding an approximate factor of a graph and this idea can be given a precise meaning using non-standard analysis. Similar approximate factors appear in higher order Fourier analysis, a topic started by W.T. Gowers. We show that there is a rich connection between all the above subjects. Motivated by this, we turn to deep learning (the most dominant branch of artificial intelligence) where "smart forgetting" also called "abstraction" is a key ingredient. It usually comes in the form of a dimension reduction. We discuss how abstraction in deep learning is related to the above subjects.