Track: Track2; Team name: Fernando Espinosa; Model: SheafTSP

[ferdavid1]

SheafTSP is a spectral convolutional network whose diffusion operator is
a learned, weighted sheaf Laplacian. The model builds a Hansen–Ghrist
graph sheaf [6] over the 1-cell adjacency structure of the lifted cell
complex and diffuses the signal with a polynomial filter on that
operator, following the Hilbert-bundle framework of Tandon et al. [1].

Each layer:

1. learns SO(d) restriction maps from an antisymmetrized skew generator
   through the Cayley transform, so R_vu = R_uv^{-1} holds by
   construction and the maps are orientation-equivariant (an exponential
   map projection is available as an option);
2. weights each edge with a Gaussian kernel on the transport-consistency
   distance ||s_i − R_e s_j|| (Eq. 8 of [1]);
3. assembles the degree-normalized sheaf Laplacian
   L̂ = D^{-1/2} δᵀKδ D^{-1/2}, whose spectrum lies in [0, 2];
4. applies a PPR-initialized scalar polynomial filter on the sheaf lazy
   walk P = I − L̂/2 with K = 10 hops (Eq. 10 of [1]);
5. adds a bounded kernel-alignment regularizer on the transports
   (Eq. 15 of [1], λ = 0.01, signal and bandwidth detached).

Alongside the diffusion machinery, the node embedding carries an
unnormalized count term t_v = |B1||B2|·1, derived from the incidence
matrices of the lifted complex and exact under the all-3-cliques
lifting (Σ t_v = 6 × #triangles). The embedding is assembled with no
post-hoc normalization, because LayerNorm on the sum erases the count
magnitudes. The two challenge tasks want opposite treatments of
normalization, and the architecture keeps an operator-normalized
diffusion signal and a strictly linear count term side by side in one
embedding, protected from each other.

At stalk dimension d = 2 the transports are SO(2) rotations, so the
learned operator specializes to a magnetic Laplacian [5] whose per-edge
phases are learned from data. Measured on trained checkpoints, the
first layer selects mild rotations under homophily (mean |θ| = 9.9°)
and near-antipodal maps under heterophily (mean |θ| = 165.8°), flipping
anti-correlated neighbors into consistency.

Results

Official 72-run grid (12 GraphUniverse settings × 3 seeds × 2 tasks),
run with the unmodified evaluation protocol:

Task	Score
Community detection (accuracy, higher is better)	0.4735
Triangle counting (MSE / triangle, lower is better)	0.0108

Per-run values: 2026_tdl_challenge/outputs/sheaf_tsp_full_grid/results.json.
run_evaluation.ipynb is changed on the single permitted line
(MODEL_CONFIG = "cell/sheaf_tsp").

Files

topobench/nn/backbones/cell/sheaf_tsp.py                      backbone
topobench/nn/wrappers/cell/sheaf_tsp_wrapper.py               wrapper
configs/model/cell/sheaf_tsp.yaml                             model config
configs/transforms/model_defaults/sheaf_tsp.yaml              lifting defaults
topobench/transforms/liftings/graph2cell/clique_cell_lifting.py  all-3-cliques lifting
configs/transforms/liftings/graph2cell/clique_cell.yaml       lifting config
topobench/transforms/data_manipulations/triangle_degree.py    verification transform
test/nn/backbones/cell/test_sheaf_tsp.py                      backbone tests
test/nn/wrappers/cell/test_sheaf_tsp_wrapper.py               wrapper tests
test/transforms/liftings/graph2cell/test_clique_cell_lifting.py  lifting tests
test/transforms/data_manipulations/test_TriangleDegree.py     transform tests
docs/sheaf_tsp/                                               reviewer docs

Documentation for reviewers is under docs/sheaf_tsp/: an overview of
the approach (hypotheses, each equation, design principles, grid
results), architecture flow diagrams, and an executable notebook that
verifies the claimed properties (orthogonality, orientation
equivariance, exact counting) on a toy graph.

The CellCliqueLifting transform adds every 3-clique as a 2-cell; the
result is canonical and permutation-invariant, which the cycle-basis
lifting is not (a cycle basis depends on node order and undercounts
triangles in dense neighborhoods).

Backbone and wrapper are auto-discovered through
ModelExportsManager/WrapperExportsManager; no __init__.py edits.

Tests

• 29 tests across four suites: backbone (orthogonality, orientation
  equivariance, spectrum bounds, dense/sparse agreement, the shipped
  configuration, sparse autoselect), wrapper (count injection,
  regularizer gating), lifting (K4 completeness, permutation
  invariance), and TriangleDegree
• per-file coverage: backbone 99%, wrapper 100%, lifting 100%,
  TriangleDegree 100%
• ruff check and ruff format clean
• includes a small idempotency fix for the hypergraph dataset
  download() (junk directories inside some hosted archives made
  re-downloads fail), which was surfacing as an order-dependent
  loader-test failure

Checklist

• The PR has a clear, explanatory title
• The code passes linting (ruff check and ruff format --check clean)
• Unit tests are added and passing (29 SheafTSP tests; full suite
  1004 passed, 0 failed)
• The code is PEP8 compliant
• Documentation follows numpydoc conventions with parameter and
  return annotations
• results.json from the unmodified official evaluation protocol
  is included, with heatmap and OOD visualizations

References

1. Tandon, Gould, Bhatia, Dominici, Ribeiro & Battiloro, "Consistent
   Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves"
   (2026), arXiv:2605.06395
2. Bodnar, Di Giovanni, Chamberlain, Liò & Bronstein, "Neural Sheaf
   Diffusion" (NeurIPS 2022), arXiv:2202.04579
3. Bamberger, Barbero, Dong & Bronstein, "Bundle Neural Networks for
   message diffusion on graphs" (2024), arXiv:2405.15540
4. Chen, Chen, Villar & Bruna, "Can Graph Neural Networks Count
   Substructures?" (NeurIPS 2020), arXiv:2002.04025
5. Zhang, He, Brugnone, Perlmutter & Hirn, "MagNet: A Neural Network
   for Directed Graphs" (NeurIPS 2021), arXiv:2102.11391
6. Hansen & Ghrist, "Toward a spectral theory of cellular sheaves",
   J. Appl. Comput. Topol. 3(4):315–358, 2019

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