# Curvature-weighted Finsler Mixup

- ID: 1032
- Canonical URL: https://synthcore.org/idea/1032/curvature-weighted-finsler-mixup
- API JSON: https://synthcore.org/api/idea/1032.json
- API Markdown: https://synthcore.org/api/idea/1032.md
- Verification status: unverified
- Source: [arXiv:2607.16817](https://arxiv.org/abs/2607.16817)
- Category: geometry
- Solves: generalization, sample-efficiency, stability
- ML areas: data-augmentation, embedding, regularization
- Math tags: geometry, metric-geometry, optimal-transport, differential-geometry
- Ratings: usefulness 5/10; difficulty 5/10; novelty 5/10

## Idea description

Replace Euclidean Mixup with interpolation in a learned anisotropic embedding metric. Use the paper's distortion coefficient to weight the endpoints and add a consistency term requiring the model's interpolated prediction to respect the geometry-dependent mass allocation.

## Mathematical statement

The model coefficient is defined for curvature K, effective dimension N>1, interpolation fraction t in (0,1), and distance parameter theta by $$\tau^{(t)}_{K,N}(\theta)=\begin{cases}t^{1/N}\left(\frac{\sin(t\theta\sqrt{K/(N-1)})}{\sin(\theta\sqrt{K/(N-1)})}\right)^{(N-1)/N},&(N-1)\pi^2>K\theta^2>0,\\t,&K=0,\\t^{1/N}\left(\frac{\sinh(t\theta\sqrt{-K/(N-1)})}{\sinh(\theta\sqrt{-K/(N-1)})}\right)^{(N-1)/N},&K<0.\end{cases}$$ Here K is a scalar curvature lower-bound proxy, N is an effective dimension, theta is the distance between two endpoints, and t is the fraction from the first endpoint. The paper's Brunn–Minkowski inequality is $$\mathfrak{m}(Z_t(A,B))^{1/N}\geq\tau^{(1-t)}_{K,N}(\Theta(A,B))\mathfrak{m}(A)^{1/N}+\tau^{(t)}_{K,N}(\Theta(A,B))\mathfrak{m}(B)^{1/N},$$ where \mathfrak{m} is measure, Z_t(A,B) is the set of t-geodesic interpolants between sets A and B, and \Theta(A,B) is their transport distance. In the neural adaptation, use a learned embedding distance theta and treat scalar K as a tunable or batch-estimated curvature proxy; define target mixing weights from the two tau coefficients and match the network's prediction at the interpolated embedding to that weighted endpoint target.

## Key formulas

- $$\tau^{(t)}_{K,N}(\theta)=\begin{cases}t^{1/N}\left(\frac{\sin(t\theta\sqrt{K/(N-1)})}{\sin(\theta\sqrt{K/(N-1)})}\right)^{(N-1)/N},&(N-1)\pi^2>K\theta^2>0,\\t,&K=0,\\t^{1/N}\left(\frac{\sinh(t\theta\sqrt{-K/(N-1)})}{\sinh(\theta\sqrt{-K/(N-1)})}\right)^{(N-1)/N},&K<0.\end{cases}$$
- $$\mathfrak{m}(Z_t(A,B))^{1/N}\geq\tau^{(1-t)}_{K,N}(\Theta(A,B))\mathfrak{m}(A)^{1/N}+\tau^{(t)}_{K,N}(\Theta(A,B))\mathfrak{m}(B)^{1/N}.$$
- $$w_0=\frac{\tau^{(1-t)}_{K,N}(\theta)}{\tau^{(1-t)}_{K,N}(\theta)+\tau^{(t)}_{K,N}(\theta)},\qquad w_1=\frac{\tau^{(t)}_{K,N}(\theta)}{\tau^{(1-t)}_{K,N}(\theta)+\tau^{(t)}_{K,N}(\theta)},$$
- $$\mathcal{L}=\mathcal{L}_{\mathrm{CE}}(f(x_t),w_0y_0+w_1y_1)+\lambda\left[\max\left(0,\tau^{(1-t)}_{K,N}(\theta)p_0^{1/N}+\tau^{(t)}_{K,N}(\theta)p_1^{1/N}-p_t^{1/N}\right)\right]^2.$$

## Implementation notes

Integrate this at the embedding output of a classifier, before the final prediction head. Add a small positive anisotropic metric network g_phi(h)=softplus(W h+b)+epsilon, producing per-coordinate inverse scales; define theta(h0,h1)=sqrt(sum_j g_phi((h0+h1)/2)_j (h0_j-h1_j)^2). Start with a straight-line geodesic proxy h_t=(1-t)h0+t h1, since exact Finsler geodesics are unnecessary for the MVP. For each minibatch, randomly pair examples, sample t from Beta(0.4,0.4), compute theta, K, and fixed N such as the embedding dimension, evaluate tau with numerically stable sinc and sinhc branches, and construct w0,w1 using the normalized tau values. Run the prediction head on h_t and add the weighted-label cross-entropy. Optionally estimate p_i as the model confidence on endpoint samples and add the hinge penalty in the final formula; stop gradients through p_i initially to avoid collapse. K should first be a fixed sweep over {-1,0,1}; later learn K as a clipped scalar or estimate it from local triangle distortion. Train the metric network jointly but regularize log g_phi toward zero. The first experiment is CIFAR-10 with a small ResNet-18, comparing ordinary Mixup, manifold Mixup, and this method at identical augmentation counts and FLOPs. Measure validation accuracy, calibration, robustness to anisotropic input corruptions, and training-loss variance. Success means improved accuracy or corruption robustness at equal compute without unstable confidence or metric-scale collapse.

## Disclaimer

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