Geometry-Aware SMP in Deep Learning

• The paper introduces geometry-aware SMP that integrates vertex coordinate data into message passing for enhanced geometric discrimination.
• It employs the GSWL color refinement and dual channels (boundary and coboundary) to capture geometric deformations and metric-dependent properties.
• Empirical results demonstrate superior performance in deformation classification, ECT regression, and topology discrimination compared to traditional methods.

Geometry-aware Simplicial Message Passing (SMP) refers to the class of equivariant neural architectures for learning and inference on simplicial complexes in which the geometric embedding (typically vertex coordinate data) is explicitly leveraged throughout the message-passing process. In contrast to classical message-passing on graphs or combinatorial simplicial complexes—which models only the abstract connectivity—geometry-aware SMP can distinguish and exploit geometric deformations, curvatures, and metric-dependent properties of the domain, enabling substantially richer expressivity in classification, regression, and geometric invariant tasks (Wang et al., 7 May 2026).

1. Underlying Formalism: GSWL Color Refinement

The geometric expressivity of geometry-aware SMP is grounded in the Geometric Simplicial Weisfeiler–Lehman (GSWL) color refinement. This test generalizes the classic Weisfeiler–Lehman (WL) algorithm for graphs—and its simplicial extension (SWL)—by incorporating initial vertex coordinate data. Given an embedded complex $(K, x)$, where $K$ is an abstract simplicial complex and $x: \mathsf{vert}(K) \to \mathbb{R}^d$ assigns Euclidean positions, the GSWL process proceeds via:

• Initialization:
  • Assign to each vertex $\sigma = \{v\}$ a color $c_\sigma^{(0)} = (0, x_v)$.
  • Assign to each higher simplex $\sigma = \{v_0,...,v_k\}$, $k \ge 1$, a color $c_\sigma^{(0)} = (k,\,\Phi_k(x_{v_0},...,x_{v_k}))$, where $\Phi_k$ is any permutation-invariant embedding, e.g., centroid or sorted tuple.
• At each iteration $\ell$, refine $K$0 via an injective hash of current color, the multiset of lower-dimensional boundary colors, and the multiset of higher-dimensional coboundary colors.

This process ensures that, after $K$1 rounds (where $K$2 exceeds the maximal simplex dimension), each color uniquely identifies the unordered set of constituent vertex coordinates for any given simplex. GSWL equivalence between complexes implies they are indistinguishable by any geometry-aware SMP up to depth $K$3 (Wang et al., 7 May 2026).

2. Geometry-Aware Simplicial Message Passing Architecture

Geometry-aware SMP operates on the full Hasse diagram of a simplicial complex, updating hidden states $K$4 for each simplex $K$5 and layer $K$6 using geometry-adapted messages. The pipeline is as follows:

• Initialization: $K$7, with $K$8 encoding each simplex's initialized GSWL color.
• Layer-wise update for each $K$9:
  • Message from faces (boundary): $x: \mathsf{vert}(K) \to \mathbb{R}^d$0
  • Message from cofaces (coboundary): $x: \mathsf{vert}(K) \to \mathbb{R}^d$1
  • Update: $x: \mathsf{vert}(K) \to \mathbb{R}^d$2, with all maps ($x: \mathsf{vert}(K) \to \mathbb{R}^d$3) typically MLPs.
• Readout: Final global representation or task-specific statistic $x: \mathsf{vert}(K) \to \mathbb{R}^d$4.

Both boundary and coboundary channels are essential for recovering higher-order geometric features (as shown by ablation experiments; see Section VI in (Wang et al., 7 May 2026)).

3. Geometric Expressivity Hierarchy and Theoretical Results

The expressivity of geometry-aware SMP is tightly characterized:

• For any depth $x: \mathsf{vert}(K) \to \mathbb{R}^d$5, geometry-aware SMPs cannot distinguish two complexes with identical GSWL-$x: \mathsf{vert}(K) \to \mathbb{R}^d$6 color multisets. This forms a sharp upper bound: no message-passing network of this form can discriminate more finely than GSWL-$x: \mathsf{vert}(K) \to \mathbb{R}^d$7 equivalence.
• On any finite family of embedded complexes, there exist parameters for geometry-aware SMPs such that their discrimination exactly matches GSWL up to round $x: \mathsf{vert}(K) \to \mathbb{R}^d$8.
• By supplementing the hidden state pipeline with skip-connections of the original vertex coordinates, one obtains universal approximation property for the class of geometric invariants realized by the Euler Characteristic Transform (ECT), meaning all geometric information up to finite sampling density can be extracted through SMP architectures (Wang et al., 7 May 2026).

A strictly weaker expressivity results for combinatorial SMPs and GIN/GCN classes, establishing the following practical model inclusion:

Model	Distinguishes Embeddings?	Maximal Expressivity
Combinatorial SMP	No	SWL
GIN/GCN	Partially	Standard WL
Geometry-Aware SMP	Yes	GSWL (geometry-injective)

4. Algorithmic Implementation and Computational Complexity

Geometry-aware SMP admits efficient implementation with computational complexity linear in the number of simplices and layers. The high-level pseudocode is:

$\sigma = \{v\}$3

Per-layer cost is $x: \mathsf{vert}(K) \to \mathbb{R}^d$9, where $\sigma = \{v\}$0 is the number of simplices. Geometric computations (e.g., centroids, lengths) are $\sigma = \{v\}$1 per simplex (Wang et al., 7 May 2026).

5. Integration with Geometric Invariants

A central application of geometry-aware SMP is the computation and approximation of the Euler Characteristic Transform (ECT). Given an embedded complex $\sigma = \{v\}$2, the ECT associates to each direction and threshold the Euler characteristic of the sublevel set. Geometry-aware SMP can be parameterized so that its readout matches the ECT vector at a discretized set of directions and thresholds, thereby capturing all geometric and topological information that is ECT-injective. Completeness and stability of ECT for Euclidean embeddings ensure the suitability of geometry-aware SMPs for tasks requiring geometric fidelity, including mesh classification, deformation recognition, and geometric statistic regression (Wang et al., 7 May 2026).

6. Empirical Performance and Applications

Empirical studies demonstrate the following:

• On synthetic deformation classification, geometry-aware SMP achieves 1.000 accuracy, outperforming both MLPs and combinatorial SMP.
• In regression of sampled ECT, geometry-aware SMP delivers MSE 0.094, surpassing combinatorial SMP (MSE 0.281).
• On the FAUST 10-way human body pose dataset, geometry-aware SMP reaches 0.838 ± 0.075 accuracy, outperforming DeepSets and GCN/GIN with coordinates.
• On the MANTRA triangulation dataset (various topologies), geometry-aware SMP attains near-perfect accuracy for all topologies sampled (Wang et al., 7 May 2026).

Coboundary ablation experiments confirm that omitting coboundary messages sharply degrades performance, especially for higher-order geometric tasks (e.g., Gaussian curvature regression).

A plausible implication is that geometry-aware SMP architectures are optimal—in the sense of GSWL expressivity—for any learning task where discrimination between different geometric realizations, not just combinatorial types, is essential.

7. Broader Connections: Geometry-Aware SMP in Other Domains

While the term "geometry-aware SMP" appears primarily in deep learning contexts related to geometric combinatorics, other domains leverage "geometry-aware" paradigms with problem-specific adaptations. For example, in symbolic mathematical problem solving, "geometry-aware symbolic math problem solving" (as instantiated by GAPS) refers to architectures that incorporate symbolic/geometric primitives into the decoding process to exploit explicitly the structure of geometric mathematics (Zhang et al., 2024). However, these do not perform message passing on complexes but rather ingest diagrammatic and symbolic elements in a programmatic or sequence-based architecture.

In spectral theory, the geometric uniformization of the spectrum of certain pentadiagonal SMP matrices (arising from the Strong Moment Problem) uses conformal mapping from the spectral surface to a hyperbolic comb domain, where geometric parameters (slit heights, capacities) encode spectral intervals and perturbations, providing a geometry-aware parametrization of operator classes in the sense of embedding Hilbert–Schmidt perturbations into geometric data (Ensgraber et al., 2013). This is distinct from the message-passing formalism, but illustrates that geometry-aware methodologies pervade a variety of mathematical and computational frameworks.