2022/10/08 by Faraz Ahmad, Ahmad, Faraz · 1 citation
Computer Science · Mathematics · #68T09 #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Mathematical Analysis and Transform Methods #Primary: 55N31 #Secondary: 46E10 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2210.04043
openalex publication_date 2022/10/08 · openalex created_date 2022/10/12 · openalex updated_date 2026/07/28
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.