Compactification of Perception Pairs and Spaces of Group Equivariant non-Expansive Operators

TL;DR

This paper introduces methods for compactifying perception pairs and spaces of group equivariant non-expansive operators, enabling their isometric embedding into compact spaces under certain conditions, which enhances their mathematical structure and analysis.

Contribution

It defines new notions of compact perception pairs and operators, and proves their embeddings into compact spaces under conditions like total boundedness and coverage.

Findings

Perception pairs with totally bounded measurements can be embedded into compact perception pairs.

Spaces of group equivariant non-expansive operators can be embedded into compact operator spaces.

The embeddings preserve isometric properties and satisfy compatibility conditions.

Abstract

We define the notions of a compact perception pair, compactification of a perception pair, and compactification of a space of group equivariant non-expansive operators. We prove that every perception pair with totally bounded space of measurements, which is also rich enough to endow the common domain with a metric structure, can be isometrically embedded in a compact perception pair. Likewise, we prove that if the images of group equivariant non-expansive operators in a given space form a cover for their common codomain, then the space of such operators can be isometrically embedded in a compact space of group equivariant non-expansive operators, such that the new reference perception pairs are compactifications of the original ones having totally bounded data sets. Meanwhile, we state some compatibility conditions for these embeddings and show that they too are satisfied by our constructions.