arXiv：cs.LG· Luciano Melodia·· 13 小时前
学习迁移的范畴论刻画：用 Kan 扩展分析学习表示拓扑不变量的迁移
Learning Transfers: Kan Extensions of Topological Invariants of Learned Representations
AI 导读
研究者把任务视为小范畴、任务变换视为函子 J，用左 Kan 扩展 Lan_JF 刻画源表示与任务变换在目标上强制保留的不变量。在持续同调上，该扩展是逗号范畴上的余核，其函子化差异在 interleaving 下为 1-Lipschitz；有限样本下差异即瓶颈距离，可无近似计算。在采样流形、训练编码器和预训练图像模型上的实验里，全部精确预测均成立，但该差异并不追踪线性探针的准确率。
正文
Abstract:When a learned representation is reused for a new task, which of its structure should survive? A task is a small category, a change of task is a functor $J$, and the invariants of a trained representation form a functor $F$. The left Kan extension $\operatorname{Lan}_JF$ is the invariant that the source and the change of task force on the target: by its universal property, every comparison of the source with an observed target invariant factors through it. The transfer discrepancy is the largest distance, over the target objects, between this extension and the observed invariant. For persistent homology the extension is a cokernel over comma categories, and a functorial version of the discrepancy, which also compares the maps between target objects, is $1$-Lipschitz under interleavings. On finite samples the discrepancy is a bottleneck distance, computed without approximation. For data glued from two pieces whose exclusive parts lie farther apart than the filtration scale, a Mayer-Vietoris sequence identifies what extending homology loses, while extending chains loses nothing. For a merge of labelled point clouds, the degree-zero discrepancy vanishes if and only if some $1$-Lipschitz classifier has a margin above the filtration scale at every sample point. In experiments on sampled manifolds, trained encoders and pretrained image models, every exact prediction holds on every sample. The discrepancy of truncated invariants recovers a hidden merge of separated parts far above chance and tells the intended target of a merge from a permuted control, but across pretrained models it does not track the accuracy of a linear probe.
| Comments: | Repository this https URL |
| Subjects: | Machine Learning (cs.LG); Algebraic Topology (math.AT); Category Theory (math.CT) |
| MSC classes: | 18A40, 55N31, 68T05, 68T07 |
| Cite as: | arXiv:2606.07627 [cs.LG] |
| (or arXiv:2606.07627v4 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2606.07627 arXiv-issued DOI via DataCite |
Submission history
From: Luciano Melodia [view email]
[v1]
Sat, 30 May 2026 15:34:08 UTC (30 KB)
[v2]
Thu, 23 Jul 2026 14:38:43 UTC (48 KB)
[v3]
Fri, 14 Aug 2026 16:20:56 UTC (875 KB)
[v4]
Fri, 9 Oct 2026 07:24:47 UTC (159 KB)
来源：arXiv：cs.LG · arxiv.org