Learning Transfers: Kan Extensions of Topological Invariants of Learned Representations

The gist A representation transfers if it stays usable once the task has changed, and this paper makes that structure explicit and computable by defining transfer discrepancy as the supremum over target objects of the distance between that extension and the observed target invariant, rather than against the source.

The Categorical Transfer Framework

The framework models transfer learning by setting up a categorical structure where a source task is a small category A, a change of task is a functor J: A → B, and a representation is a functor F: A → V into a cocomplete category of invariants. The target-side invariant L(b) for an object b in B is defined as the left Kan extension LanJ F evaluated at b, which records the source invariants lying over b together with every identification imposed by the source morphisms visible over b. The transfer score is then computed as CompJ (F, G) = supb dV L(b), G(b), scoring transfer against the invariant that the change of task forces rather than against the source.

Key Properties of Kan Extensions

The paper proves several propositions detailing how Kan extensions behave in different settings. Proposition 1 states that for every J: A → B, every F: A → V and every b ∈ Ob(B) one has (LanJ F)(b) ∼= colimJ↓b F πb. Specifically, for forgetting structure, it shows (LanJ F)(•) ∼= colimF a∈Ob(A) F(a), the coproduct in V, for every functor F: A → V. For class refinement, it shows (LanJ F)(d) ∼= F(c) when c is unique. For layer collapse, it shows (LanJ F)(•) ∼= F(n), the value at the last layer of the chain and not the coproduct of all of them.

Invariants and Distances

The paper introduces an invariant target as a triple (V, S, dV) where V admits all small colimits, S ⊆ Ob(V) is a class of objects closed under isomorphism whose isomorphism classes form a set S/∼=, and dV is an extended pseudometric on S/∼=. The transfer discrepancy CompJ (F, G) is defined as supb dV [(LanJ F)(b)], [G(b)]. For chain complexes (ChK), the invariant target is (ChK, Sch, dch) where dch takes only the two values 0 and 1. For persistence modules (PersK), it is (PersK, Persq-tame K, dI) where dI is the interleaving distance.

Evaluation and Bounds

The paper establishes that for modules of finite type, the discrepancy is the bottleneck distance computed without approximation. For persistence modules, CompnatJ (F, G) is defined as the interleaving distance of the two diagrams. Theorem 13 shows that dB I (LanJ F, LanJ F') ≤ dA I (F, F') for all functors F, F': A → PersK. Corollary 14 provides a bound on the difference between weighted transfer scores S and S' in terms of the source and target distances.

Transfer Detection in Latent Space

The paper tests these concepts on sampled manifolds and learned latent clouds. Experiment 1 scores degrees n ≤ q = 2 with weights w0 = 0.25 and w1 = w2 = 1. The objectwise baseline ObjBaseT J,q compares the sources with the observed target one by one and drops the transport along J entirely. Experiment 2 tests whether a geometric partition is a refinement, and Table 1 confirms that the baseline is infinite on all ten rows and orders none of them. Experiment 3 samples a learned gluing, where the pushout of Proposition 20 differs from the coproduct.

Conclusion

The paper adds what topological work on transfer left out: a structural map between tasks, and the invariant it forces on the target. The discrepancy reads the morphisms of the source and not only its objects (Proposition 6), and its vanishing is strictly weaker than exact transferability (Theorem 7). The colimit is also computable, by a cokernel presentation (Proposition 10), a Lipschitz bound (Theorem 13) and exact evaluation by dB (Proposition 16). The paper demonstrates that the discrepancy is non-expansive in both of its arguments, as shown by Theorem 13. The final result shows that the score is computable without approximation for every n and b. The findings are validated across three experiments involving sampled manifolds and learned latent clouds.

How it works

1. Model the Tasks A small categories A and B, change of task J: A → B, 2. Read Invariants F: A → V from the source, G: B → V observed, 3. Force the Target L(b):= (LanJ F)(b) ∼= colimJ↓b F πb, 4. Transfer Score CompJ (F, G) = supb dV L(b), G(b).

Key Properties of Kan Extensions

The paper proves several propositions detailing how Kan extensions behave in different settings. Proposition 1 states that for every J: A → B, every F: A → V and every b ∈ Ob(B) one has (LanJ F)(b) ∼= colimJ↓b F πb. Specifically, for forgetting structure, it shows (LanJ F)(•) ∼= colimF a∈Ob(A) F(a), the coproduct in V. For class refinement, it shows (LanJ F)(d) ∼= F(c) when c is unique. For layer collapse, it shows (LanJ F)(•) ∼= F(n), the value at the last layer of the chain and not the coproduct of all of them.

How it works

1. Color Transfer Discrepancy Scoring: Implement a transfer score that score transfer against the invariant that the change of task forces rather than against the source, specifically using CompJ (F, G):= sup b∈Ob(B) dV [(LanJ F)(b)], [G(b)] to explicitly detect structural changes rather than just distributional shifts.

2. Topological Invariant Preservation: AI systems can be engineered to preserve global topological structures by comparing the observed target invariant against the left Kan extension of the source functor along the change of task, which ensures that the structure that the target has to exhibit is the left Kan extension of the source functor along the change of task.

3. Structural Control Rejection: The system can be designed to actively reject structural controls by scoring against every structural control that we pose, ensuring that the score recovers the intended change of task and rejects every structural control that we pose.

4. Bottleneck Distance Accuracy: For models on finite-type modules, the discrepancy score will compute the bottleneck distance, computed without approximation, ensuring high fidelity in comparing persistence diagrams for learned latent clouds.

5. Exact Transferability Detection: The system can determine if a representation is exactly transferable by checking if LanJ F ∼= G in V B, and this provides a strict criterion beyond the weaker condition where the vanishing discrepancy does not force an isomorphism of functors.

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 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.