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Preprint . 2025
License: CC BY
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Preprint . 2026
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A Continuous Generalization of the Cohn–Umans Framework via Quotient Topology and Fiber Morphisms

Authors: Harby, John;

A Continuous Generalization of the Cohn–Umans Framework via Quotient Topology and Fiber Morphisms

Abstract

We study when a continuous bilinear map of which matrix multiplication and dot-product attention are instances can be computed on equivalence classes of its inputs rather than on the inputs themselves. The exact case is governed by the universal property of the quotient topology, but it applies only when the map is genuinely constant on classes, which for similarity-based relations holds only in the degenerate case of exact duplicates. We therefore give the approximate version, which is the operationally relevant one: for a Lipschitz bilinear map and a relation whose classes have bounded diameter , computing on class representatives incurs an error bounded linearly in while reducing cost from to , where is the number of classes. This is an explicit accuracy-for-compute trade; the realized speedup is a constant factor determined empirically by the redundancy present in the data. We position the construction as a topological analogy to the Cohn–Umans group-algebra embedding rather than a generalization of it: the two achieve different kinds of reduction (exact asymptotic versus approximate constant-factor) by different mechanisms, and we make the distinction precise.

Keywords

FOS: Mathematics, Topology, Mathematics, Cohn-Umans

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
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