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

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.