Relative-Difference Homotopy Consistency
Implementation & benchmark of arXiv:2607.03484 — Homotopic distances and group-like spaces
Source paper: Homotopic distances and group-like spaces arXiv:2607.03484 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace all pairwise consistency comparisons between m augmented views by a single group-valued relative-difference vector with m−1 components. Add a learned contractible-chart penalty so that the relative-difference map remains locally simple rather than merely numerically small. The construction is invariant to simultaneous left multiplication of every view, providing a useful gauge-invariant consistency signal.
Formulas
Mathematical statement
For a path-connected topological group G and maps f_i:X o G, the paper defines F_m:G^m o G^{m-1} by F_m(x_1,\ldots,x_m)=(x_1x_2^{-1},x_2x_3^{-1},\ldots,x_{m-1}x_m^{-1}). Theorem 7.2 states that the homotopic distance of the maps equals the LS-category of this composition: \mathsf{D}(f_1,\ldots,f_m)=\operatorname{cat}(F_m\circ(f_1,\ldots,f_m)). Here \operatorname{cat}(h) is the least integer n such that X is covered by n+1 open sets on each of which h is null-homotopic. The same reduction applies to group-like spaces and to diagonalized or distributional variants through sectional category. For neural outputs z_i=E(a_i(x))\in G, define r_i=z_i z_{i+1}^{-1}. The relative elements are invariant under the common gauge transformation z_i\mapsto g z_i. We approximate the category objective using K learned local contractible charts, with center c_k\in G and local logarithmic coordinates \log(c_k^{-1}r_i).
Implementation notes
1. Integration point: modify the encoder head in a multi-view or augmentation-consistency model. Given m augmented versions a_1(x),...,a_m(x), let the encoder produce group-valued outputs z_i. Start with G=SO(2), represented by a normalized two-vector (cos(theta_i),sin(theta_i)); later test SO(3) or unit quaternions. Avoid all pairwise comparisons. Compute the m−1 adjacent relative elements r_i=z_i z_{i+1}^{-1}; for SO(2), this is angle subtraction modulo 2pi. Add lambda_rel L_rel and lambda_chart L_chart to the task loss. 2. Pseudocode: z=[normalize(E(a_i(x))) for i in range(m)]; r=[group_mul(z[i],group_inv(z[i+1])) for i in range(m-1)]; Lrel=mean(norm(log(r_i))2); d[i,k]=norm(log(group_mul(group_inv(c[k]),r[i])))2; q=softmax(-d/tau,dim=k); Lchart=sum(q*d)+tau*sum(q*log(q)); loss=Ltask+lambda_rel*Lrel+lambda_chart*Lchart. Project trainable centers c_k back onto G after every optimizer step. 3. The paper supplies F_m and its category interpretation; K, tau, and loss weights are empirical. Monitor the geodesic radius of assigned relative elements and split or reject a chart if samples approach the logarithm cut locus. 4. First experiment: CIFAR-10 or STL-10 with four to eight random augmentations, a small CNN or ViT, and a SimCLR baseline using all pairwise cosine losses. Compare at equal encoder FLOPs. Measure linear-probe accuracy, held-out augmentation consistency, gradient variance, wall-clock time, and memory. Success means lower consistency cost as m grows, equal or better accuracy, and fewer unstable runs for rotation-like augmentations.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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