{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace a globally shared latent transformation group by a source-dependent collection of valid transformation paths. A feature at latent point z is transported only along paths whose transformed coordinate never reaches the singular locus, while homotopic paths are identified and composable paths are concatenated. This should let an equivariant model represent branched or incomplete symmetries that ordinary group-equivariant layers must discard.",
 "formulas_latex": [
  "$$\\mathcal{G}_{n}^{\\prime}=X_{n}/\\sim^{\\prime}$$",
  "$$(u(t),z)\\sim^{\\prime}(v(t),w)\\quad\\Longleftrightarrow\\quad w=z\\ \\text{and there exists a smooth endpoint-fixed homotopy }u(t,s)\\text{ with }u(t,0)=u(t),\\ u(t,1)=v(t),\\ u(t,s)\\cdot z^{n}\\neq 0\\ \\forall t,s.$$",
  "$$\\sigma^{\\prime}(u(t),z)=z,\\qquad \\tau^{\\prime}(u(t),z)=z(1)$$",
  "$$[(u_{2},w)]\\circ[(u_{1},z)]=[(u_{2}\\circ u_{1},z)]\\quad\\text{when }\\tau^{\\prime}(u_{1},z)=w.$$"
 ],
 "id": 2863,
 "implementation": "(1) Integration point: insert the layer between an encoder and an MLP head for data with a two-dimensional geometric latent coordinate z=(z_1,z_2), such as hyperbolic-disk images, articulated shapes, or graph nodes embedded in a disk. Let h(z)\\in\\mathbb{R}^{d} be the feature at z. Sample a short sequence of transformation increments g_1,\\ldots,g_K from a low-dimensional Lie algebra, and use the resulting path u_0=I, u_j=g_j u_{j-1} to transport h.\n\n(2) Pseudocode: inputs are z, h, n, and increments g[1:K]. Set z_cur=z and h_cur=h. For j=1,...,K, compute z_next=Action(g[j], z_cur); reject or shorten the step if \\|z_next^n\\|\u003cepsilon, where z^n means the complex latent coordinate raised to the integer n. Compute h_next=MLP_j([h_cur, z_cur, g[j]]) and optionally apply a residual transport h_cur=h_cur+alpha_j h_next. Store (z_cur,z_next) and continue. For two sampled paths u and v with the same endpoints, add a homotopy consistency loss only if a discretized interpolation u(t,s) satisfies min_{t,s}\\|Action(u(t,s),z)^n\\|\\geq epsilon: L_hom=\\|T(u,z)h-T(v,z)h\\|_2^2. For composable paths, impose L_comp=\\|T(u_2\\circ u_1,z)h-T(u_2,\\tau(u_1,z))T(u_1,z)h\\|_2^2.\n\n(3) The paper supplies the source/target, admissibility, homotopy, and concatenation rules. The action Action, transport MLP, step size, and epsilon are engineering choices. Estimate path validity by checking all K discretization points; estimate homotopy validity by checking a small S-by-K grid. Use a curriculum that begins with small increments and increases path length.\n\n(4) First experiment: train a small CNN encoder plus this transport layer on 64x64 synthetic images of hyperbolic-disk points with labels invariant under sampled non-singular SL(2,R)-like transformations, comparing against a standard global transformation-augmentation baseline and a conventional equivariant MLP. Use n=2 and n=3, with 8-step paths. Measure validation accuracy under transformations near the origin, the fraction of rejected paths, the homotopy and composition residuals, and optimization stability. Success means higher accuracy specifically on near-singularity examples at equal parameter count, reduced degradation under composed transformations, and no increase in exploding-loss or NaN frequency.",
 "math_summary": "The paper constructs arrows as equivalence classes of pairs (u(t),z), where u(t) is a smooth path of transformations and z is the source point. The equivalence relation requires the same source and identifies paths connected by a smooth endpoint-fixed homotopy u(t,s) satisfying u(t,s)\\cdot z^{n}\\neq 0 for every t,s; here n\\geq 1 is the singularity order and z^{n} is the transformed coordinate used to detect the singular set. The source and target maps are \\sigma'(u(t),z)=z and \\tau'(u(t),z)=z(1), so an arrow records both its starting latent state and its endpoint. Two arrows compose only when the first target equals the second source, and composition is path concatenation: [(u_{2},w)]\\circ[(u_{1},z)]=[(u_{2}\\circ u_{1},z)]. The neural adaptation uses a learned transport operator T(u,z) on features, with the groupoid law enforced only for composable arrows rather than imposing a global group representation.",
 "math_tags": [
  "lie-groups",
  "geometry",
  "differential-geometry",
  "topology"
 ],
 "ml_areas": [
  "graph-nn",
  "embedding",
  "geometry"
 ],
 "paper": {
  "arxiv_id": "2608.30077",
  "arxiv_url": "https://arxiv.org/abs/2608.30077",
  "summary_what_math_gives_to_ml": "The paper gives a constructive way to integrate incomplete, singular infinitesimal symmetries into a Lie groupoid rather than forcing them into a globally defined group action. The transferable asset is the combination of source-dependent arrows, path-homotopy equivalence constrained to avoid a singular set, and composition by concatenation. In neural networks this suggests a local-equivariant layer whose transformations are valid only for a given latent source point and whose path class is retained, allowing singular or branched symmetries without introducing invalid global coordinate transformations.",
  "title": "Lie groupoid integration of singular isometries of the Poincaré disk",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 8,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "stability",
  "generalization"
 ],
 "title": "Singularity-Aware Groupoid Transport Layer",
 "url": "https://synthcore.org/idea/2863/singularity-aware-groupoid-transport-layer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}