Functorial Signal Representation: Foundations and Redundancy

In this paper we propose and lay the foundations of a functorial framework for representing signals. By incorporating an additional category-theoretic relative and generative perspective alongside the set-theoretic measure theory, the fundamental concept of redundancy is formulated in an arrow-theoretic way. The existing classic framework representing a signal as a vector in an appropriate linear space becomes a special case of the proposed framework. We also propose new definition of intra-signal redundancy using an isomorphism in a category, covering the translation case. Using category theory we provide a mathematical explanation for better signal compression performance of lossless differential encoding standards than classic representation techniques in certain cases (e.g. iconic images).