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Geometric Structures on Graphs: a Holonomy-Based Discretization of Curvature
Hao Li, Yuhan Peng, Junwen Dong
TL;DR
The paper addresses how to discretize curvature on graphs without reducing it initially to scalar descriptors. It builds curvature observations from normalized logarithmic loop holonomy, rectifies them into SPD metric responses, and calibrates the construction on known sphere geometry and synthetic transport recovery. The experiments support the holonomy–curvature relation and metric-compatible representation, while the recovery study remains a fixed-mesh consistency check.
Problem
Existing graph-learning uses often compress curvature into scalar descriptors, motivating a construction that obtains curvature from local metrics, edge transports, and their loop composition.
Method
The framework uses normalized logarithmic holonomy as finite-loop curvature, commutator and divergence aggregations, and gauge-equivariant SPD-preserving metric updates.
Results
The unit-sphere calibrations verify the holonomy–curvature relation and metric representation, while recovered edge transports yield declining curvature error as correspondences increase.
Takeaways & Limitations
The construction provides graph-level geometric objects whose metric evolution and semantic propagation can be separated while remaining coupled through curvature-derived information.
Takeaways & Limitations
The aggregated loop-curvature signal does not recover the canonical Ricci contraction on an arbitrary graph, and the synthetic recovery study does not establish a mesh-refinement limit or arbitrary-application training signal.
Abstract
from arXiv · showhide
We propose a holonomy-based framework for discretizing curvature on graphs equipped with local symmetric positive-definite metrics. Each vertex carries a fibre metric \(g_i\), and each directed edge carries a reversible metric-compatible transport \(F_{ij}\). The ordered product around an oriented triangular loop \(\mathcal C\) gives a holonomy \(H_{\mathcal C}\), whose normalized logarithm \(Ω_{\mathcal C}=-s_{\mathcal C}^{-1}\operatorname{Log}(H_{\mathcal C})\) is used as a finite-loop curvature observation. Thus the construction discretizes the geometric principle that infinitesimal holonomy is controlled by curvature, rather than treating holonomy as a heuristic feature. Since \(Ω_{\mathcal C}\) lies in the \(g_i\)-orthogonal Lie algebra, it is not itself a velocity of an SPD metric. We therefore introduce two aggregation mechanisms: a commutator with a symmetric response matrix, producing symmetric Ricci-type metric responses, and an incidence-aware covariant divergence of curvature-induced edge fluxes, reflecting the relation between trace and covariant divergence. The resulting responses are locally orthogonal-gauge equivariant and can drive exponential updates that preserve positive definiteness. We also give a reversible metric-compatible parametrization of edge transports, allowing orthogonal edge factors, loop scales, weights, and response matrices to be learned while respecting the graph geometry. Known-geometry calibrations on the unit sphere test the holonomy--curvature relation, curvature preservation under nontrivial local metric representations, and the empirical recovery of edge transports from local observations.
1. Introduction
The paper organizes graph interactions as local metrics and metric-compatible edge transports, obtaining curvature from nontrivial loop holonomy rather than prescribing scalar curvature. It then separates curvature observation, metric-response construction, and geometry-conditioned feature propagation.
- Motivation: Graph curvature is obtained from the failure of local edge interactions to compose trivially around closed loops, rather than from a prescribed scalar descriptor.The framework treats graph interactions as discrete geometric data with local metrics and edgewise parallel transports.
- Graph geometric data: Each vertex carries an SPD fibre metric, while each directed edge carries a reversible metric-compatible discrete parallel transport.The transports preserve the local fibre geometry and provide the data from which loop holonomy is formed.
- Holonomy and curvature: For an oriented triangular loop, ordered edge transports form holonomy, whose normalized logarithm is a finite-loop curvature observation at an explicitly chosen scale.The scale may use embedding or area information when available, but remains a specified normalization on an abstract graph.
- Metric response: Because curvature observations are gi-skew-adjoint rather than symmetric, a commutator with a symmetric response matrix produces a direction suitable for SPD metric updates.The commutator represents noncommutative coupling between curvature and metric-response channels without redefining curvature.
- Curvature aggregation: An alternative aggregation preserves face–edge incidence by converting loop curvature into antisymmetric edge fluxes and applying a covariant graph divergence.Direct loop aggregation and divergence aggregation are kept separate because they encode different geometric information.
- Graph-learning realization: Metric evolution uses SPD-preserving exponential updates, while semantic features are propagated separately through curvature-conditioned message passing.The framework couples geometry construction and task-dependent information exchange without conflating their roles.
2. Graph geometric structures
The framework equips graph vertices with metric fibres and edges with reversible metric-compatible transports, interpreted as discrete connection data. Selected oriented triangular loops yield finite-scale curvature observations through holonomy and normalized logarithms, with explicit gauge and scale choices.
- Local metric fibres and edge transport: A metric graph structure assigns each vertex a fibre Ei with an SPD metric gi and each directed edge a linear transport between neighbouring fibres.The construction can represent local metric data, learned latent metrics, and finite identifications between locally flat patches.
- Geometric interpretation: In the piecewise-flat interpretation, loop curvature arises from the failure of finite patch identifications to compose trivially around interfaces or hinges.The edge maps are connection data rather than ordinary coordinate-transition Jacobians.
- Local metric fibres and edge transport: Metric compatibility is a modelling constraint requiring edge transports to preserve fibre inner products, and such transports admit orthogonal-factor parametrizations.The orthogonal factors generate metric-compatible transports, while reversibility supplies the paired directed-edge structure.
- Loops and holonomy: Selected triangular faces are oriented once, re-rooted at each base vertex, and represented by ordered closed walks whose holonomy is an automorphism of the base fibre.For a rooted loop, holonomy is formed by composing the three directed transports in order.
- Loops and holonomy: The continuous small-loop formula motivates using normalized logarithmic holonomy as a finite graph curvature observation at an effective positive scale.The scale may be induced by an unsigned area when geometric information exists, or specified as a graph normalization otherwise.
- Loops and holonomy: Metric compatibility places loop holonomy in the gi-orthogonal group, so its logarithm and normalized curvature observation are gi-skew-adjoint.The principal logarithm is used under an admissibility condition excluding eigenvalues on the closed negative real axis.
- Gauge changes: The construction has local orthogonal gauge symmetry, but does not claim general linear coordinate covariance for the commutator formulation.Broader covariance would require a more carefully typed tensor formulation.
3. Holonomy-based curvature observations
The paper uses normalized logarithmic holonomy around oriented triangular loops as finite-scale curvature observations, while distinguishing direct loop aggregation from incidence-aware covariant divergence.
- Direct loop aggregation: Direct aggregation combines normalized loop-curvature observations at a vertex but does not recover the canonical Ricci contraction on an arbitrary graph.It retains local loop curvature while compressing directional and face-incidence information.
- Ricci contraction calibration: A frame-resolution condition allows local loop aggregation to approximate Ricci contraction in the smooth calibration regime.The approximation uses direction-resolved observations and trace-quadrature weights.
- Holonomy-based curvature observations: Normalized logarithmic holonomy provides a finite-loop curvature observation derived from the holonomy–curvature relation.The loop scale normalizes the observation without asserting a unique intrinsic curvature for an abstract combinatorial graph.
- Incidence-based covariant divergence: Covariant divergence first assigns face curvature to oriented boundary edges and then sums signed outgoing fluxes at vertices.This construction preserves face–edge incidence information and is related to divergence as the negative adjoint of a covariant edge increment.
- Geometric weights and gauge conditions: Loop scales, edge conductances, and vertex masses play distinct roles in scaling observations and aggregating curvature fluxes.Without a shared reference frame, conductances should come from gauge-invariant edge data or an equivariant module.
4. Ricci-type metric responses and evolution
The paper converts skew-adjoint loop-curvature observations into symmetric metric responses using commutators, and applies them through gauge-equivariant exponential updates that preserve positive definiteness.
- Symmetry mismatch: Loop-curvature observations are skew-adjoint, whereas SPD metric velocities must be symmetric, creating the central response-construction mismatch.The obstruction is explicit because g_iK_i is antisymmetric and cannot directly update a symmetric metric.
- Commutator response: A symmetric response matrix and commutator convert loop curvature into a symmetric Ricci-type metric response.The response matrix controls how curvature couples to metric degrees of freedom rather than creating curvature itself.
- Divergence response: The divergence-type response uses the covariant divergence of the curvature edge cochain instead of direct loop aggregation.This preserves the alternative incidence-based mechanism in the metric-evolution pipeline.
- Symmetry and equivariance: The resulting responses are symmetric and locally orthogonal-gauge equivariant.The equivariance follows from conjugation under local gauge transformations when response matrices transform compatibly.
- Metric evolution: Exponential retraction keeps updated metrics SPD for every step size because its exponent is symmetric.This replaces a Euclidean update that may leave the SPD cone.
5. A geometry-aware learning realization
The learning realization separates geometry updates from semantic message passing and parameterizes transports, responses, and conductances while preserving compatibility and gauge constraints.
- Layer design: The proposed layer is a family of geometric layers rather than a fully specified neural architecture, leaving conservative learnable choices.The construction separates structural constraints from task-specific architectural decisions.
- Transport parametrization: Orthogonal factors parameterize edge transports while preserving metric compatibility at every layer.They can be generated from skew-symmetric matrix exponentials or orthogonalization parameterizations.
- Semantic aggregation: Semantic features can be updated by an equivariant or invariant message-passing operator conditioned on scalar loop statistics.Examples include norms of normalized logarithmic holonomies and gauge-invariant contractions.
- Gauge assumptions: A conductance prior based on common-coordinate metric representations is not gauge-invariant for unregistered fibres.Unregistered fibres require gauge-invariant edge data or a separately designed equivariant module.
- Two-stage realization: The layer first updates geometry from transports and holonomies, then propagates semantic features using a task-specific operator.This division avoids conflating arbitrary edge transformations with metric-compatible parallel transport.
6. Structural checks and special cases
The structural checks establish flat-loop consistency and clarify where curvature and metric responses originate. They also identify cycle coverage and response-matrix choice as important scope and modeling conditions.
- Flat-loop consistency: Flat holonomy makes every loop-curvature term, aggregated response, and exponential update trivial, so both metric updates leave g_i unchanged.This provides a consistency check for the full construction rather than an arbitrary matrix-valued message-passing rule.
- Connection data: Metric-compatible transports exist for arbitrary SPD metrics, but choosing O_ij = I is globally flat because transports telescope around every loop.Nontrivial curvature therefore resides in the edgewise orthogonal factors rather than node metrics alone.
- Connection data: Learning only {g_i} with fixed O_ij = I cannot produce nonzero holonomy curvature; learned connection degrees of freedom or external geometric data are required.
- Response matrix: The response matrix B selects how curvature observations change the metric, but its choice is nonunique because graphs lack a canonical smooth Ricci-tensor contraction.A fixed choice B_C,i = g_i provides a baseline, while learned responses allow loop- and node-dependent coupling and should be evaluated against it.
- Cycle coverage: Graphs without selected cycles have zero direct and divergence-type curvature responses, while cell completions or longer cycles require explicit modeling choices not assumed here.
7. Geometric calibration and synthetic consistency checks
The checks calibrate holonomy against known sphere curvature, verify representation consistency under nontrivial local metrics, and test transport recovery from noisy correspondences. Results support numerical consistency while remaining limited to fixed-mesh and analytically controlled settings.
- Holonomy–curvature calibration: On the unit sphere, exact Levi–Civita transports with s_C = A_C reproduce θ_C = A_C and unit curvature to numerical precision.Across four mesh levels from 20 to 1280 faces, the largest observed |θ_C − A_C| is below 4×10−16.
- Holonomy–curvature calibration: The flat control F_ij = I produces zero loop holonomy and therefore cannot recover the sphere’s curvature.This control isolates the role of nontrivial edge transport in generating curvature observations.
- Metric representation calibration: A nontrivial spatially varying SPD field preserves curvature after comparison in a common whitened representation, with residuals at numerical precision.The unwhitened difference is at most 3.5 × 10−13, while compatibility, telescoping, and g_i-skew residuals remain numerically negligible.
- Metric representation calibration: The metric field is not claimed to define the Levi–Civita connection of the auxiliary SPD field, and the test isolates a representation identity rather than full-model irrelevance of g_i.Local metrics may still enter loop scales, weights, response matrices, and SPD evolution.
- Synthetic transport recovery: With noisy vector correspondences, both learned edge-transport error and induced whitened curvature error decrease as the correspondence count M increases.At M = 256, mean curvature error is 3.44 × 10−1 versus 1.41 for the flat control O_ij = I.
- Synthetic transport recovery: The synthetic results establish a fixed-mesh consistency trend, not a mesh-refinement theorem or evidence that arbitrary applications provide an analogous training signal.The checks verify one concrete recovery route while task-level usefulness remains for separate evaluation.
8. Discussion, scope, and next steps
The framework turns loop holonomy into curvature-driven, SPD-preserving graph updates while exposing the modelling choices involved. Its scope remains limited by loop availability, logarithm branch and stability issues, and the absence of general continuum, stability, or task-performance validation.
- Scope and interpretation: Nontrivial loop holonomy supplies local curvature observations, while commutator responses convert them into symmetric metric directions for SPD-preserving updates.The construction also makes the role of additional modelling choices explicit.
- Limitations: Triangle-based loops may be unavailable in sparse graphs, requiring longer cycles or a cell-complex completion whose scale normalization is not determined here.
- Limitations: The logarithm requires a branch choice and may become numerically unstable near the excluded spectrum.
- Limitations: The response matrix B is a modelling choice rather than a replacement for smooth Ricci contraction.
- Limitations: Sphere calibrations do not establish a general continuum limit, learned-dynamics stability theorem, or predictive advantage on a learning task.
- Next steps: Future work includes cell-complex area and orientation calculus, broader frame covariance, response-matrix and loop-scale comparisons, and learning-task evaluation.