Unified Functorial Signal Representation III: Foundations, Redundancy, $L^0$ and $L^2$ functors

TL;DR

This paper develops a category-theoretic framework for signal representation, unifying classical and modern approaches, and introduces novel concepts of redundancy and compression based on functorial and arrow-theoretic ideas.

Contribution

It introduces a functorial, category-theoretic foundation for signal representation, generalizing classical vector space models and defining new notions of intra-signal redundancy.

Findings

Unified framework encompassing classical and modern signal representations

Novel definition of intra-signal redundancy using category theory

Explanation of differential encoding advantages through categorical concepts

Abstract

In this paper we propose and lay the foundations of a functorial framework for representing signals. By incorporating additional category-theoretic relative and generative perspective alongside the classic set-theoretic measure theory the fundamental concepts of redundancy, compression are formulated in a novel authentic arrow-theoretic way. The existing classic framework representing a signal as a vector of appropriate linear space is shown as a special case of the proposed framework. Next in the context of signal-spaces as a categories we study the various covariant and contravariant forms of $L^0$ and $L^2$ functors using categories of measurable or measure spaces and their opposites involving Boolean and measure algebras along with partial extension. Finally we contribute a novel definition of intra-signal redundancy using general concept of isomorphism arrow in a category covering the translation case and others as special cases. Through category-theory we provide a simple yet precise explanation for the well-known heuristic of lossless differential encoding standards yielding better compressions in image types such as line drawings, iconic image, text etc; as compared to classic representation techniques such as JPEG which choose bases or frames in a global Hilbert space.