Information Width

TL;DR

This paper extends Kolmogorov's concept of information to define 'information width', a measure that relates description complexity and information conveyance, applicable to various input sources and with implications for binary function spaces.

Contribution

It introduces the novel concept of information width, linking Kolmogorov's information theory to approximation theory and providing a unified framework for evaluating information from diverse sources.

Findings

Defined information width as an extension of Kolmogorov's entropy

Unified evaluation of information from different sources

Applied the concept to binary function spaces

Abstract

Kolmogorov argued that the concept of information exists also in problems with no underlying stochastic model (as Shannon's information representation) for instance, the information contained in an algorithm or in the genome. He introduced a combinatorial notion of entropy and information $I(x:\sy)$ conveyed by a binary string $x$ about the unknown value of a variable $\sy$. The current paper poses the following questions: what is the relationship between the information conveyed by $x$ about $\sy$ to the description complexity of $x$ ? is there a notion of cost of information ? are there limits on how efficient $x$ conveys information ? To answer these questions Kolmogorov's definition is extended and a new concept termed {\em information width} which is similar to $n$-widths in approximation theory is introduced. Information of any input source, e.g., sample-based, general side-information or a hybrid of both can be evaluated by a single common formula. An application to the space of binary functions is considered.