# Subduction-Based Tangent Augmentation

- ID: 608
- Canonical URL: https://synthcore.org/idea/608/subduction-based-tangent-augmentation
- API JSON: https://synthcore.org/api/idea/608.json
- API Markdown: https://synthcore.org/api/idea/608.md
- Verification status: unverified
- Source: [arXiv:2607.08939](https://arxiv.org/abs/2607.08939)
- Category: architecture
- Solves: sample-efficiency, accuracy, stability
- ML areas: embedding, cnn, data-augmentation, regularization
- Math tags: differential-geometry, geometry, category-theory, lie-groups
- Ratings: usefulness 5/10; difficulty 6/10; novelty 6/10

## Idea description

Train a predictor on quotient-consistent tangent jets rather than only on transformed samples. Generate several local representatives of the same orbit, compute first-order feature perturbations, and aggregate them through a shared tangent module before prediction. This gives a structured alternative to treating augmented views as independent examples and can improve robustness to composed transformations.

## Mathematical statement

For a quotient \(X=M/\mathcal{G}\), the paper uses the quotient diffeology: a map \(p:U\to X\) is a plot exactly when it locally lifts through the quotient map \(\pi:M\to X\). The paper constructs a map from groupoid orbit-normal tangent data to quotient tangent data. In the extracted notation, \(N_2\mathcal{F}\) is the second-order normal object associated with the orbit foliation \(\mathcal{F}\), \(T_2M\) is the second-order tangent object of \(M\), and \(\lambda:N_2\mathcal{F}\to T_2(M/\mathcal{G})\) is a subduction satisfying \(T_2\pi=\lambda\circ(\operatorname{quot}\times\operatorname{quot})\). A subduction means that quotient tangent objects can be locally lifted from representative-space tangent objects. The neural adaptation uses this local-lift property by computing compatible jets for several transformed representatives and aggregating them into one quotient-level prediction. In the adapted formula, \(T_g\) is the input transformation, \(J_f\) is the encoder Jacobian, \(S\) is a sampled set of transformations, and \(\operatorname{Agg}\) is mean or attention aggregation.

## Key formulas

- $$p\colon U\to X/\mathcal{R}\text{ is a plot of the quotient diffeology}\iff p\text{ locally lifts along }\pi\colon X\to X/\mathcal{R}.$$
- $$\lambda\colon N_{2}\mathcal{F}\to T_{2}(M/\mathcal{G}),\qquad T_{2}\pi=\lambda\circ(\operatorname{quot}\times\operatorname{quot}).$$
- $$\mathcal{L}_{\mathrm{jet}}=\left\|\operatorname{Agg}_{g\in S}\left[J_f(T_gx)T_g\right]-J_f(x)\right\|_F^2.$$

## Implementation notes

Use this in an encoder-plus-head model when data have a known symmetry group, such as rotations, translations, permutations, or rigid motions. For each minibatch input \(x\), sample a small set \(S\) of transformations \(T_g\), always including the identity. Compute \(h_g=f_\theta(T_gx)\) and Jacobian-vector products \(J_f(T_gx)T_gv_j\) for a fixed set of random tangent directions \(v_j\); use Hutchinson directions if a full tangent basis is unavailable. Feed each representative jet \((h_g, J_f(T_gx)T_gv_1,\ldots,J_f(T_gx)T_gv_k)\) through a shared tangent head \(q_\psi\), then aggregate the resulting vectors over \(g\in S\) with a mean or permutation-invariant attention pool. Use the aggregate for prediction and add \(\lambda\mathcal{L}_{\mathrm{jet}}\), where the norm is estimated by comparing the sampled Jacobian-vector products of transformed and identity representatives. The paper's map \(\lambda\) is not explicitly constructed numerically; its role is the local-lift rule that quotient jets are represented by local jets of orbit representatives. The empirical quantities are sampled transformations, Jacobian-vector products, and aggregation error. Start with a four-layer CNN on rotated-MNIST, using two to four transformations per example. Compare ordinary augmentation, invariant pooling, and this jet module under equal training budgets. Evaluate clean accuracy, accuracy under composed and unseen transformations, performance at 10%, 25%, and 100% data, consistency loss, and wall-clock overhead. A positive signal is better low-data and composed-transformation accuracy at less than twice baseline training compute, with decreasing jet inconsistency during optimization.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.