Learning Transfers: Kan Extensions for Neural Invariants

Abstract

A representation transfers if it stays usable once the task has changed. Standard evaluations report target accuracy or a distance between data distributions, but neither says which structure of the representation is meant to survive. Here we make that structure explicit and computable. A task is a small category, a change of task is a functor, and a representation is a functor into a category of invariants. The structure the target has to exhibit is the left Kan extension of the source functor along the change of task. Our transfer discrepancy is the supremum over target objects of the distance between that extension and the observed target invariant, so we score transfer against the invariant the change of task forces rather than against the source. We prove cokernel presentations of the extension over comma categories, in chain complexes and in persistence modules. On one-parameter modules of finite type we show the discrepancy is the bottleneck distance, computed without approximation. We also test on sampled manifolds and on learned latent clouds whether the score recovers the intended change of task and rejects every structural control that we pose.