Categorical Framework for Understanding Transfer Learning in Neural Networks
A new preprint introduces a mathematical framework using Kan extensions from category theory to formally define and measure structural invariants in transfer learning. The work addresses a gap in standard transfer learning evaluation, which typically relies on target accuracy or distributional discrepancy without specifying what structural property is supposed to transfer. The framework could help detect representation collapses that preserve classification accuracy while silently destroying topology relevant to transfer.
Researchers have submitted a preprint to arXiv proposing a category-theoretic formalization of transfer learning, centered on the concept of Kan extensions. The core idea is to model source and target task domains as categories connected by a 'task-change functor,' and to define the universally transferred invariant as the left Kan extension of the source representation along that functor. A 'transfer discrepancy' metric is then derived by comparing the target invariant against this universal transferred invariant, rather than making direct source-to-target comparisons. The paper proves finite computational formulas for this construction in chain complexes and persistence modules, and shows that for persistence-valued one-parameter invariants, the discrepancy reduces to bottleneck distances between barcodes from topological data analysis. Controlled experiments on neural latent point clouds are reported to validate that the score correctly identifies the task functor and flags representation collapses that standard accuracy metrics would miss.
What's missing
As a preprint, this work has not yet undergone peer review. The experimental validation is described as 'controlled' but the scale, datasets, and baselines used are not detailed in the abstract, leaving open questions about empirical generalizability. The computational cost of evaluating Kan extensions and bottleneck distances at scale in practical deep learning pipelines is not addressed. It is also unclear how the framework handles non-discrete or continuous task-change functors beyond the finite-type one-parameter case.
What different sources said
- arXiv cs.LGCenter
Learning Transfers: Kan Extensions for Neural Invariants
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