cs.CGOct 20, 2026

A Bochner Weitzenböck Decomposition for Cellular Sheaf Laplacians, Curvature and Neural Sheaf Diffusion

Authors: Francesco Maria Guadagnuolo, Alessio Borgi, Dorina Thanou, Pietro Liò

Organizations: Sapienza University of Rome, Rome, Italy · Italian Institute of Artificial Intelligence (AI4I), Turin, Italy · University of Cambridge, Cambridge, United Kingdom · ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland

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Abstract

We extend the Bochner-Weitzenböck decomposition to cellular sheaf Laplacians, splitting it into a positive semidefinite term, the Bochner sheaf Laplacian, and a block diagonal term, the Forman curvature. We show that the Bochner Laplacians form a class of cellular sheaves that extends both connection and matrix-weighted Laplacians, and generalizes signed graphs to dd dimensions. We give a holonomy characterization of their kernel, prove the related Cheeger inequalities, and attach to them a random walk that generalizes the Gremban expansion. For the Forman term, we prove a discrete Bochner theorem, which bounds the first cohomology of a cellular sheaf with nonnegative curvature, and we read the whole decomposition as an opinion-dynamics model. We then apply this machinery to Sheaf Neural Networks, whose learnt Laplacian is otherwise unobserved. The Bochner interpretation yields a node-level conviction score that acts as an operational importance measure under frozen-model interventions, while the Forman profile provides a local descriptor of the learnt operator.

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Figure 1: The Sd−1\mathbb{S}^{d-1} and O(d)O(d) synchronization problems on a single edge, d=2d=2 . (a) Each node carries a unit vector. The angular factor W10⊤W_{10}^{\top} , here a rotation by 90∘90^{\circ} , transports x0x_{0} to the stalk of node 11 , and the frustration of the edge is the discrepancy between x1x_{1} and W10⊤x0W_{10}^{\top}x_{0} (dashed), weighted by the radial factor M10M_{10} . The ellipse below the edge draws M10M_{10} , with principal axes labelled by its largest and smallest eigenvalues w10+w^{+}_{10} and w10−w^{-}_{10} : a discrepancy along the first costs more than one along the second, while the dashed circle is the isotropic case of a connection Laplacian. (b) Each node carries an orthogonal frame gv∈O(d)g_{v}\in O(d) , drawn as its two columns, and the frustration is ∥g1−W10⊤g0∥M10,F\left\lVert g_{1}-W_{10}^{\top}g_{0}\right\rVert_{M_{10},F} .
Figure 2: The random walk of a Bochner matrix as a generalized Gremban expansion. (a) For the signed triangle with w01=−3w_{01}=-3 (dashed) and w02=1w_{02}=1 , the Gremban expansion doubles each node into v+v^{+} and v−v^{-} . From node 00 the walker moves to node 11 with probability 3/43/4 and to node 22 with probability 1/41/4 , in proportion to ∣w0u∣|w_{0u}| ; crossing the negative edge flips its sign, 0±→1∓0^{\pm}\to 1^{\mp} , and crossing the positive one keeps it, 0±→2±0^{\pm}\to 2^{\pm} . (b) For d>1d>1 each node is lifted to a sphere Sd−1\mathbb{S}^{d-1} , and the jump probability p0u(x^)=x^⊤P0ux^p_{0u}(\hat{x})=\hat{x}^{\top}P_{0u}\hat{x} depends on the direction the walker carries: in this example, from yy it reaches nodes 11 and 22 with probabilities 3/43/4 and 1/41/4 , from xx with 1/21/2 each (Definition 4.3 ).
Figure 3: The three examples of this section, with d=1d=1 . (a) The empty triangle of Example 5.1 : each arrow carries the restriction map from a vertex to an edge, all equal to 11 except F0⊴[0,2]=F1⊴[1,2]=2\mathcal{F}_{0\trianglelefteq[0,2]}=\mathcal{F}_{1\trianglelefteq[1,2]}=2 . (b) The star S4S_{4} of Example 5.2 , with maps 11 from the center and AA from the leaves. (c) The heptagonal bipyramid of Example 5.3 seen from above: the vertices AA and BB , one on each side of the equator, project onto the same point at the center. Next to each vertex is its value of qq , which sets the restriction maps of the construction.
Zero maskingBudget (fraction of the training pool)
Selection1%5%10%20%30%
High conviction0.51±0.17\mathbf{0.51}_{\pm 0.17}1.34±0.39\mathbf{1.34}_{\pm 0.39}2.18±0.70\mathbf{2.18}_{\pm 0.70}3.53±1.26\mathbf{3.53}_{\pm 1.26}4.55±1.83\mathbf{4.55}_{\pm 1.83}
Low conviction0.02±0.01\mathbf{0.02}_{\pm 0.01}0.04±0.04\mathbf{0.04}_{\pm 0.04}0.07±0.07\mathbf{0.07}_{\pm 0.07}0.20±0.14\mathbf{0.20}_{\pm 0.14}0.39±0.32\mathbf{0.39}_{\pm 0.32}
High degree0.37±0.160.37_{\pm 0.16}0.75±0.280.75_{\pm 0.28}1.06±0.401.06_{\pm 0.40}1.76±0.681.76_{\pm 0.68}2.46±0.982.46_{\pm 0.98}
Low degree0.03±0.020.03_{\pm 0.02}0.19±0.100.19_{\pm 0.10}0.38±0.210.38_{\pm 0.21}0.85±0.440.85_{\pm 0.44}1.35±0.741.35_{\pm 0.74}
Degree-matched (high)0.06±0.070.06_{\pm 0.07}0.20±0.170.20_{\pm 0.17}0.38±0.320.38_{\pm 0.32}0.57±0.510.57_{\pm 0.51}0.93±0.620.93_{\pm 0.62}
Table 1: Relative drop in performance (%) under zero masking, by selection strategy and budget. Mean ±s.e.{}_{\pm\text{s.e.}} across 8 datasets, each averaged over its folds. In each column, the most damaging and the least damaging selection are highlighted.
Table 2: 11 -Wasserstein distance between the Forman profile of the learnt normalized GeneralSheaf and the topological reference, by training epoch (rows; 00 is the untrained sheaf) and relative depth of the layer (columns). Darker cells are farther from the reference: the distance shrinks with training and with depth. Mean ±s.e.{}_{\pm\text{s.e.}} across 10 datasets, each averaged over its folds; 8 for the middle layer, which networks with L=2L=2 lack.
Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

 Checkpoint
ObservableUntrainedBest epoch
Spearman ρ\rho with degree0.07±0.100.07_{\pm 0.10}0.33±0.070.33_{\pm 0.07}
Spearman ρ\rho with betweenness0.05±0.080.05_{\pm 0.08}0.28±0.060.28_{\pm 0.06}
Spearman ρ\rho with confidence0.17±0.070.17_{\pm 0.07}0.66±0.050.66_{\pm 0.05}
Spearman ρ\rho with saliency0.07±0.100.07_{\pm 0.10}−0.18±0.09-0.18_{\pm 0.09}
∣| Cohen’s d∣d| , train split0.21±0.050.21_{\pm 0.05}0.54±0.090.54_{\pm 0.09}
Appendix
Table 3: Spearman correlation between the conviction and four node properties, and absolute Cohen’s dd between the conviction of correctly and incorrectly classified nodes, at the last diffusion layer. Untrained is the epoch- 00 checkpoint, a control.
Mean maskingBudget (fraction of the training pool)
Selection1%5%10%20%30%
High conviction0.50±0.16\mathbf{0.50}_{\pm 0.16}1.34±0.39\mathbf{1.34}_{\pm 0.39}2.19±0.72\mathbf{2.19}_{\pm 0.72}3.61±1.30\mathbf{3.61}_{\pm 1.30}4.66±1.85\mathbf{4.66}_{\pm 1.85}
Low conviction0.01±0.02\mathbf{0.01}_{\pm 0.02}0.08±0.05\mathbf{0.08}_{\pm 0.05}0.14±0.10\mathbf{0.14}_{\pm 0.10}0.36±0.17\mathbf{0.36}_{\pm 0.17}0.57±0.38\mathbf{0.57}_{\pm 0.38}
High degree0.38±0.160.38_{\pm 0.16}0.94±0.380.94_{\pm 0.38}1.35±0.591.35_{\pm 0.59}2.13±0.942.13_{\pm 0.94}2.91±1.172.91_{\pm 1.17}
Low degree0.03±0.020.03_{\pm 0.02}0.18±0.110.18_{\pm 0.11}0.41±0.220.41_{\pm 0.22}0.88±0.480.88_{\pm 0.48}1.43±0.801.43_{\pm 0.80}
Degree-matched (high)0.09±0.070.09_{\pm 0.07}0.30±0.210.30_{\pm 0.21}0.52±0.370.52_{\pm 0.37}0.77±0.630.77_{\pm 0.63}1.21±0.711.21_{\pm 0.71}
Appendix
Table 4: Budget sweep under mean masking, laid out as Table 1 . High conviction is the most damaging selection at every budget, low conviction the least.
Hard gatingBudget (fraction of the training pool)
Selection1%5%10%20%30%
High conviction0.81±0.29\mathbf{0.81}_{\pm 0.29}1.80±0.41\mathbf{1.80}_{\pm 0.41}2.77±0.66\mathbf{2.77}_{\pm 0.66}4.32±1.27\mathbf{4.32}_{\pm 1.27}5.67±1.89\mathbf{5.67}_{\pm 1.89}
Low conviction0.05±0.030.05_{\pm 0.03}0.11±0.04\mathbf{0.11}_{\pm 0.04}0.18±0.09\mathbf{0.18}_{\pm 0.09}0.43±0.22\mathbf{0.43}_{\pm 0.22}0.71±0.45\mathbf{0.71}_{\pm 0.45}
High degree0.65±0.200.65_{\pm 0.20}1.37±0.371.37_{\pm 0.37}1.91±0.451.91_{\pm 0.45}2.83±0.732.83_{\pm 0.73}3.67±1.043.67_{\pm 1.04}
Low degree0.05±0.02\mathbf{0.05}_{\pm 0.02}0.26±0.120.26_{\pm 0.12}0.58±0.270.58_{\pm 0.27}1.20±0.591.20_{\pm 0.59}1.91±0.981.91_{\pm 0.98}
Degree-matched (high)0.12±0.050.12_{\pm 0.05}0.37±0.190.37_{\pm 0.19}0.68±0.350.68_{\pm 0.35}1.02±0.601.02_{\pm 0.60}1.44±0.861.44_{\pm 0.86}
Appendix
Table 5: As Table 4 , under hard gating. At the 1%1\% budget low degree is marginally the least damaging.
PruningBudget (fraction of the training pool)
Selection1%5%10%20%30%
High conviction1.65±0.96\mathbf{1.65}_{\pm 0.96}2.95±1.10\mathbf{2.95}_{\pm 1.10}4.02±1.16\mathbf{4.02}_{\pm 1.16}5.64±1.49\mathbf{5.64}_{\pm 1.49}7.08±2.04\mathbf{7.08}_{\pm 2.04}
Low conviction0.01±0.01\mathbf{0.01}_{\pm 0.01}0.07±0.05\mathbf{0.07}_{\pm 0.05}0.16±0.11\mathbf{0.16}_{\pm 0.11}0.42±0.28\mathbf{0.42}_{\pm 0.28}0.81±0.57\mathbf{0.81}_{\pm 0.57}
High degree1.36±0.771.36_{\pm 0.77}2.68±1.162.68_{\pm 1.16}3.48±1.243.48_{\pm 1.24}4.82±1.384.82_{\pm 1.38}5.77±1.575.77_{\pm 1.57}
Low degree0.04±0.030.04_{\pm 0.03}0.24±0.140.24_{\pm 0.14}0.54±0.300.54_{\pm 0.30}1.13±0.671.13_{\pm 0.67}1.88±1.101.88_{\pm 1.10}
Degree-matched (high)0.20±0.050.20_{\pm 0.05}0.52±0.210.52_{\pm 0.21}0.89±0.410.89_{\pm 0.41}1.29±0.761.29_{\pm 0.76}1.73±1.091.73_{\pm 1.09}
Appendix
Table 6: As Table 4 , under pruning.
Zero maskingRanking criterion
BinConvictionConfidenceDegree
10.07±0.070.07_{\pm 0.07}0.35±0.210.35_{\pm 0.21}0.38±0.210.38_{\pm 0.21}
20.09±0.080.09_{\pm 0.08}0.36±0.210.36_{\pm 0.21}0.42±0.220.42_{\pm 0.22}
30.06±0.200.06_{\pm 0.20}0.39±0.260.39_{\pm 0.26}0.41±0.240.41_{\pm 0.24}
40.17±0.250.17_{\pm 0.25}0.42±0.250.42_{\pm 0.25}0.37±0.270.37_{\pm 0.27}
50.37±0.260.37_{\pm 0.26}0.48±0.380.48_{\pm 0.38}0.45±0.290.45_{\pm 0.29}
Appendix
Table 7: Decile experiment under zero masking. The training pool is split into ten equal bins by each ranking criterion (columns), from the lowest-scoring bin 11 to the highest-scoring bin 1010 (rows), and one bin is perturbed at a time. The conviction has the largest drop at bin 1010 , and most of its growth occurs in the last two bins.
Zero masking, avg_zBudget (fraction of the training pool)
Selection1%5%10%20%30%
High conviction0.56±0.17\mathbf{0.56}_{\pm 0.17}1.68±0.57\mathbf{1.68}_{\pm 0.57}2.65±0.98\mathbf{2.65}_{\pm 0.98}4.10±1.61\mathbf{4.10}_{\pm 1.61}5.18±2.21\mathbf{5.18}_{\pm 2.21}
Low conviction0.01±0.01\mathbf{0.01}_{\pm 0.01}0.02±0.03\mathbf{0.02}_{\pm 0.03}0.02±0.06\mathbf{0.02}_{\pm 0.06}0.13±0.18\mathbf{0.13}_{\pm 0.18}0.34±0.29\mathbf{0.34}_{\pm 0.29}
High degree0.37±0.160.37_{\pm 0.16}0.75±0.280.75_{\pm 0.28}1.06±0.401.06_{\pm 0.40}1.76±0.691.76_{\pm 0.69}2.46±0.982.46_{\pm 0.98}
Low degree0.03±0.020.03_{\pm 0.02}0.19±0.100.19_{\pm 0.10}0.38±0.210.38_{\pm 0.21}0.85±0.440.85_{\pm 0.44}1.35±0.741.35_{\pm 0.74}
Degree-matched (high)0.07±0.040.07_{\pm 0.04}0.19±0.150.19_{\pm 0.15}0.27±0.260.27_{\pm 0.26}0.52±0.490.52_{\pm 0.49}0.77±0.550.77_{\pm 0.55}
Appendix
Table 8: As Table 1 , with nodes ranked by the layer-averaged score avg_z . The effect is unchanged, and slightly sharper.
Soft gating 
Ranking scoreRelative drop (%)
Confidence0.67±0.370.67_{\pm 0.37}
Conviction3.09±0.953.09_{\pm 0.95}
Degree4.19±1.954.19_{\pm 1.95}
Appendix
Table 9: Relative drop under soft gating, which attenuates the outgoing message of every training node in proportion to its score (gate floor gmin⁡=0.1g_{\min}=0.1 ); with no budget, each ranking score gives one number.
Zero masking, highBudget (fraction of the training pool)
DatasetSelection1%5%10%20%30%
Amazon RatingsDegree1.25±0.171.25_{\pm 0.17}1.75±0.281.75_{\pm 0.28}2.14±0.322.14_{\pm 0.32}3.14±0.323.14_{\pm 0.32}3.90±0.273.90_{\pm 0.27}
 Conviction1.16±0.171.16_{\pm 0.17}2.31±0.262.31_{\pm 0.26}2.93±0.292.93_{\pm 0.29}3.81±0.283.81_{\pm 0.28}4.48±0.284.48_{\pm 0.28}
CiteseerDegree−0.04±0.16-0.04_{\pm 0.16}−0.39±0.21-0.39_{\pm 0.21}−0.65±0.35-0.65_{\pm 0.35}−1.25±0.82-1.25_{\pm 0.82}−1.74±1.05-1.74_{\pm 1.05}
 Conviction0.06±0.170.06_{\pm 0.17}0.41±0.210.41_{\pm 0.21}0.62±0.220.62_{\pm 0.22}1.20±0.311.20_{\pm 0.31}0.67±0.460.67_{\pm 0.46}
CoraDegree0.46±0.170.46_{\pm 0.17}0.87±0.200.87_{\pm 0.20}0.99±0.370.99_{\pm 0.37}1.53±0.441.53_{\pm 0.44}2.21±0.452.21_{\pm 0.45}
Appendix
Table 10: Budget sweep under zero masking, dataset by dataset, for the high-degree and high-conviction selections. Unlike the other tables, the spread is the standard error across the ten folds of each dataset.
Zero masking, lowBudget (fraction of the training pool)
DatasetSelection1%5%10%20%30%
Amazon RatingsDegree0.02±0.050.02_{\pm 0.05}−0.02±0.08-0.02_{\pm 0.08}0.05±0.120.05_{\pm 0.12}0.48±0.180.48_{\pm 0.18}0.73±0.200.73_{\pm 0.20}
 Conviction−0.02±0.03-0.02_{\pm 0.03}−0.02±0.08-0.02_{\pm 0.08}0.07±0.090.07_{\pm 0.09}0.13±0.100.13_{\pm 0.10}0.56±0.220.56_{\pm 0.22}
CiteseerDegree−0.04±0.04-0.04_{\pm 0.04}−0.05±0.11-0.05_{\pm 0.11}−0.12±0.16-0.12_{\pm 0.16}−0.15±0.22-0.15_{\pm 0.22}−0.29±0.35-0.29_{\pm 0.35}
 Conviction0.08±0.090.08_{\pm 0.09}−0.12±0.18-0.12_{\pm 0.18}−0.30±0.20-0.30_{\pm 0.20}−0.62±0.27-0.62_{\pm 0.27}−1.60±0.67-1.60_{\pm 0.67}
CoraDegree0.00±0.000.00_{\pm 0.00}0.09±0.060.09_{\pm 0.06}0.09±0.140.09_{\pm 0.14}0.11±0.190.11_{\pm 0.19}0.14±0.180.14_{\pm 0.18}
Appendix
Table 11: As Table 10 , for the low-degree and low-conviction selections.
PruningBudget (fraction of the training pool)
Selection1%5%10%20%30%
High conviction7.7±3.17.7_{\pm 3.1}15.9±3.515.9_{\pm 3.5}22.0±3.422.0_{\pm 3.4}31.3±3.131.3_{\pm 3.1}39.4±2.839.4_{\pm 2.8}
Low conviction0.5±0.10.5_{\pm 0.1}2.7±0.52.7_{\pm 0.5}5.6±1.05.6_{\pm 1.0}11.6±1.811.6_{\pm 1.8}18.0±2.518.0_{\pm 2.5}
High degree10.2±3.410.2_{\pm 3.4}23.5±5.123.5_{\pm 5.1}32.7±5.532.7_{\pm 5.5}44.2±5.144.2_{\pm 5.1}51.6±4.351.6_{\pm 4.3}
Low degree0.4±0.10.4_{\pm 0.1}1.9±0.61.9_{\pm 0.6}4.2±1.24.2_{\pm 1.2}8.7±2.58.7_{\pm 2.5}13.3±3.713.3_{\pm 3.7}
Degree-matched (high)5.0±1.35.0_{\pm 1.3}11.5±1.411.5_{\pm 1.4}16.6±1.316.6_{\pm 1.3}23.6±1.123.6_{\pm 1.1}28.3±1.728.3_{\pm 1.7}
Appendix
Table 12: Percentage of the graph’s edges deleted under pruning, by selection strategy and budget; the performance cost of the same selections is in Table 6 . Selecting by degree deletes the most edges.
Table 13: W1W_{1} distance of the unnormalized GeneralSheaf from the topological reference (top) and the zero reference (bottom). The profile aligns with neither.
Table 14: W1W_{1} distance of the other Laplacian, with maps trained unnormalized and evaluated normalized (Appendix D.2 ), from the topological reference. Once normalized, the maps align with it as training proceeds.
Table 15: W1W_{1} distance of the other Laplacian, with maps trained normalized and evaluated unnormalized (Appendix D.2 ), from the topological reference (top) and the zero reference (bottom). The maps lose their alignment and stay far from the zero reference.
Table 16: W1W_{1} distance between the two Laplacians evaluated normalized, built from the maps trained normalized and unnormalized (cross-comparison, Appendix D.2 ). They stay apart and neither converge nor diverge.
Table 17: As Table 16 , with both Laplacians evaluated unnormalized. The distances are of the same size.
Table 18: W1W_{1} distance of the unnormalized JointSheafParams, started at the identity sheaf, from the topological reference (top) and the zero reference (bottom), the latter measuring the deviation from the identity. The maps deviate through the layers and are drawn back with training.
Table 19: W1W_{1} distance of the normalized JointSheafParams, started at the identity sheaf, from the topological reference, which here measures the deviation from the identity. The deviation grows with training. Tolokers is excluded since its training is unstable ( 99 datasets, 77 for the middle layer). On Cora, PubMed and Questions training was interrupted, and these datasets enter with the folds that completed (between two and six of ten).
Table 20: Normalized GeneralSheaf on Roman Empire, d=5d=5 , five layers, against the topological reference.
Table 21: Unnormalized GeneralSheaf on Citeseer, d=5d=5 , four layers, against the topological and the zero reference.
Table 22: Unnormalized JointSheafParams on Questions, d=5d=5 , four layers, started at the identity sheaf, against the topological and the zero reference. In the first layer the profile is constant, so its correlation is undefined.
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