用范畴论定义可迁移表示，精准衡量模型在任务变化时的结构保持能力。
Learning Transfers: Kan Extensions for Neural Invariants
- 将任务变化建模为函子，用左肯扩展定义应保留的表示结构。
- 提出转移差异度量，基于目标对象上扩展与实际不变量的距离计算。
- 在有限型单参数模上可精确计算瓶颈距离，适合研究表示稳定性者。
表示的可迁移性指其在任务变更后仍保持可用性。传统评估仅报告目标准确率或数据分布距离，却未说明何种表示结构应持续存在。本文显式定义并可计算该结构：将任务视为小范畴，任务变化为函子，表示为到不变量范畴的函子。目标需具备的结构是源函子沿任务变化函子的左肯扩展。我们的转移差异度量为所有目标对象上该扩展与观测目标不变量之间距离的上确界，因此评分依据的是任务变化所强制的不变结构，而非源结构。我们证明了在伴随范畴、链复形及持久模中的余核表示。对有限型单参数模，证明差异度量即为瓶颈距离，且无需近似即可计算。还在采样流形和学习潜空间云上测试，结果表明该得分能正确识别预期的任务变化，并拒绝所有结构控制干扰。
原文摘要 · Abstract (English)
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.
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