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E(n) Equivariant Topological Neural Networks

Claudio Battiloro, Ege Karaismailoğlu, Mauricio Tec, George Dasoulas, Michelle Audirac, Francesca Dominici

arXiv:2405.15429v5cs.LGcs.NE

TL;DR

Graph networks are limited in representing higher-order interactions, and geometric features remain little explored in topological deep learning. ETNNs bring E(n)-equivariant message passing to combinatorial complexes; across the reported applications, they match or surpass EMPSNs with substantially lower memory use and runtime.

  • Problem

    Graph networks struggle with higher-order interactions, and little is known about leveraging geometric features in topological deep learning.

    Source: §1 · pp. 1–2

  • Method

    ETNNs use E(n)-equivariant message passing over combinatorial complexes, combining scalar geometric invariants with updates to geometric features.

    Source: §1 · p. 2

  • Results

    ETNNs match or surpass EMPSNs with less than half the memory usage and almost half the runtime per epoch in the reported applications.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Combinatorial complexes support equivariant models for arbitrary hierarchical higher-order interactions across a wide class of topological and interaction structures.

    Source: §6 · p. 10

    There exist pairs of geometric-feature combinatorial complexes that ETNNs cannot distinguish for some collection of neighborhoods.

    Source: §3 · p. 6

Abstract

from arXiv · show

Graph neural networks excel at modeling pairwise interactions, but they cannot flexibly accommodate higher-order interactions and features. Topological deep learning (TDL) has emerged recently as a promising tool for addressing this issue. TDL enables the principled modeling of arbitrary multi-way, hierarchical higher-order interactions by operating on combinatorial topological spaces, such as simplicial or cell complexes, instead of graphs. However, little is known about how to leverage geometric features such as positions and velocities for TDL. This paper introduces E(n)-Equivariant Topological Neural Networks (ETNNs), which are E(n)-equivariant message-passing networks operating on combinatorial complexes, formal objects unifying graphs, hypergraphs, simplicial, path, and cell complexes. ETNNs incorporate geometric node features while respecting rotation, reflection, and translation equivariance. Moreover, being TDL models, ETNNs are natively ready for settings with heterogeneous interactions. We provide a theoretical analysis to show the improved expressiveness of ETNNs over architectures for geometric graphs. We also show how E(n)-equivariant variants of TDL models can be directly derived from our framework. The broad applicability of ETNNs is demonstrated through two tasks of vastly different scales: i) molecular property prediction on the QM9 benchmark and ii) land-use regression for hyper-local estimation of air pollution with multi-resolution irregular geospatial data. The results indicate that ETNNs are an effective tool for learning from diverse types of richly structured data, as they match or surpass SotA equivariant TDL models with a significantly smaller computational burden, thus highlighting the benefits of a principled geometric inductive bias. Our implementation of ETNNs can be found at https://github.com/NSAPH-Projects/topological-equivariant-networks.

1 INTRODUCTION

The introduction motivates topological deep learning as a way to represent higher-order interactions that graphs miss, then identifies geometric features as an open challenge for TDL. It introduces ETNNs, which combine E(n)-equivariant message passing with combinatorial complexes, and reports theoretical and empirical contributions.

1 INTRODUCTION

  • Graph networks usually struggle with higher-order, multi-way interactions, motivating TDL models over combinatorial topological spaces.These spaces support more sophisticated cell-adjacency schemes than ordinary graph adjacency.Source: §1 · pp. 1–2
  • Geometric information such as positions and velocities has received little attention in TDL, while existing equivariant approaches cover graphs or simplicial complexes rather than a general framework over topological spaces.The introduction identifies both as too inflexible for arbitrary higher-order interactions.Source: §1 · p. 2
  • ETNNs extend equivariant message passing to combinatorial complexes, which represent arbitrary higher-order interactions with hierarchy.Each layer passes messages over the complex using scalar geometric invariants and updates geometric features in their current directions.Source: §1 · p. 2
  • ETNNs match or surpass EMPSNs with less than half their memory usage and almost half their runtime per epoch across the reported applications.The applications include QM9 molecular-property prediction and land-use regression for hyper-local air-pollution estimation.Source: §1 · p. 3
  • The paper shows ETNNs are at least as expressive as established scalarization-based equivariant graph networks and can distinguish most cited counterexamples to k-body scalarization.It also derives E(n)-equivariant variants of several TDL models from the framework.Source: §1 · p. 3

2 COMBINATORIAL COMPLEXES AND EQUIVARIANCE

Combinatorial complexes provide a flexible representation of cells and their hierarchical relationships, while neighborhood functions define how those cells interact. CC message-passing networks update cell features through these neighborhoods and remain equivariant to relabeling.

2 COMBINATORIAL COMPLEXES AND EQUIVARIANCE

  • A combinatorial complex consists of nodes, cells, and an order-preserving rank function that organizes cells hierarchically and supports both hierarchical and set-type relations.Its cells include singletons, and the complex dimension is the maximum cell rank.Source: §2 · p. 3
  • Neighborhood functions assign neighboring cells and can express same-rank adjacencies or incidences between cells of different ranks.The general choices include up/down incidences based on containment and adjacencies based on shared cells.Source: §2 · p. 3
  • A topological signal assigns a real-valued feature to each cell, and CC message-passing networks are permutation-equivariant to relabeling the cells.A cell's feature vector is a collection of such signals.Source: §2 · p. 4
  • CC message-passing networks update each cell embedding by aggregating learnable messages from neighboring cells across selected neighborhoods.Different overlapping neighborhoods can encode different relation types, and message functions can incorporate orientation or shared-cell embeddings when available.Source: §2 · p. 4

3 E(N) EQUIVARIANT TOPOLOGICAL NEURAL NETWORKS

ETNNs combine E(n)-equivariant message passing with combinatorial complexes to model geometric features and heterogeneous higher-order interactions. In the reported molecular and geospatial tasks, they outperform or match comparison models while using fewer cells or benefiting from multi-level modeling.

E(n)-equivariant topological neural networks

  • ETNN layers use invariant geometric scalars and learnable processing before combining messages along original directions, yielding E(n)-equivariant updates.The framework can also use convex-hull volumes as invariants, though computing them at every training step can be expensive.Source: §3 · pp. 4–6

Expressiveness of ETNN

  • ETNNs are at least as powerful as EGNNs at distinguishing k-hop distinct graphs and are strictly more powerful in most cases, depending on how graphs are lifted into a CC.The paper also gives a pair of geometric CCs that ETNN cannot distinguish.Source: §3 · p. 6

Computational complexity of ETNN

  • For sufficiently sparse CC connectivity, ETNN complexity is linear in the number of cells, while fully connected cells yield quadratic cost.The paper reports empirical runtime comparisons with EGNN and EMPSN.Source: §3 · p. 6

Molecular combinatorial complexes

  • Molecular CCs represent atoms as 0-cells, bonds as 1-cells, and rings and larger functional groups as 2-cells, with a virtual cell linking cells across ranks.The max adjacency can connect cells according to the highest-rank cells they share, while rank-specific features avoid ad-hoc aggregation at atom or bond level.Source: §3 · pp. 7–8

Geospatial combinatorial complexes

  • Geospatial CCs represent points, road polylines, and census-tract polygons with spatially determined neighborhoods, supporting irregular multi-resolution data without regular grids or nested partitions.The air-pollution task uses Mercator coordinates for 0-cells; Euclidean distances in that coordinate space approximate geodesic distances in meters.Source: §3 · pp. 7, 10

Molecular property prediction

  • ETNN generally improves on EGNN and matches or outperforms reproduced EMPSNs, using almost two orders of magnitude fewer cells than EMPSN’s VR complexes.The molecular CC averages 39.8 cells per molecule versus 555.25 for the VR complex; ETNN also uses half the memory and almost half the runtime per epoch relative to EMPSNs.Source: §3 · p. 8, Table 1 · p. 9

Hyperlocal air pollution downscaling

  • ETNN outperforms EGNN, GNN, and MLP on air-pollution downscaling; ablations show the largest performance decrease when the virtual cell is removed.The geospatial results are averaged over 30 seeds.Source: Table 2 · p. 9, §3 · p. 10

6 CONCLUSION

ETNNs provide scalarization-based equivariant message passing on combinatorial complexes, which represent hierarchical higher-order interactions. The conclusion reports their expressiveness relative to geometric-graph architectures and effectiveness across molecular and geospatial applications.

6 CONCLUSION

  • ETNNs provide scalarization-based equivariant message passing on combinatorial complexes, a flexible representation for hierarchical higher-order interactions.The framework serves as a proxy for equivariant models on a wide range of combinatorial topological spaces and interaction objects.Source: §6 · p. 10
  • ETNNs are at least as expressive as existing scalarization-based equivariant architectures for geometric graphs.Source: §6 · p. 10
  • Experiments demonstrate ETNN effectiveness on QM9 molecular data and a novel geographic regression task using irregular multi-resolution geospatial data.The geographic task illustrates using combinatorial complexes to model irregular multi-resolution data.Source: §6 · p. 10
  • The paper presents ETNNs as a unifying framework combining combinatorial and geometric equivariances beyond graph domains.Source: §6 · p. 10

A LIMITATIONS & FUTURE DIRECTIONS

ETNNs are presented as a unifying framework for scalarization-based E(n)-equivariant networks over combinatorial topological spaces, while the paper identifies methodological and scaling directions for future work.

A LIMITATIONS & FUTURE DIRECTIONS

  • ETNNs are presented as a first unifying framework for scalarization-based E(n)-equivariant networks over combinatorial topological spaces.The paper contrasts this scope with prior geometric-graph and simplicial-complex models.Source: A LIMITATIONS & FUTURE DIRECTIONS · p. 19
  • The methodological agenda includes richer or learned geometric invariants, scale invariance for geospatial tasks, temporal variants, and rank-specific symmetries.The paper notes that ETNNs cannot directly handle time-varying scenarios at this stage.Source: A LIMITATIONS & FUTURE DIRECTIONS · p. 19
  • Scaling to very large datasets remains a target, with message-passing operations and geometric-invariant computation identified as bottlenecks.Suggested directions include neighbor samplers and equivariant message-passing-free ETNNs.Source: A LIMITATIONS & FUTURE DIRECTIONS · pp. 19–20

B RELATED WORKS

  • The related-work review also covers expressivity results for simplicial and cell-complex message passing, heterophilic cellular-sheaf models, and architectures on path complexes and hypergraphs.It also notes models without message passing and a framework for inferring latent regular cell complexes.Source: B RELATED WORKS · p. 20
  • Prior equivariant TDL work included simplicial networks, persistent-homology augmentation, and Clifford-equivariant layers, but lacked a general framework spanning combinatorial topological spaces.The cited examples are EMPSNs (Eijkelboom et al., 2023), persistent-homology augmentation (Verma et al., 2024), and Clifford layers with simplicial message passing (Liu et al., 2024).Source: B RELATED WORKS · p. 20

C.1 GRAPHS

  • For graphs, the combinatorial-complex formulation uses rank-0 singleton cells and the usual node adjacency, recovering standard graph message passing and EGNNs.The text says edges need not be included as cells for this graph formulation.Source: §C.1 · p. 21

C.2 SIMPLICIAL COMPLEXES

  • A simplicial complex of order K contains simplices through rank K and requires every simplex's subsets to be included; its neighborhood choices recover simplicial message passing and EMPSNs.The listed neighborhoods are coboundary, boundary, up adjacency, and down adjacency.Source: §C.2 · pp. 21–22

C.3 CELL COMPLEXES

Cell complexes encode hierarchical cells without requiring every cell to be a simplex, while hypergraphs allow flexible higher-order groupings without general hierarchical or topological structure. The appendix also describes velocity inputs for ETNNs.

C.3 CELL COMPLEXES

  • A regular cell complex encodes its topological information in a face-poset hierarchy, and higher-rank cells can be attached to a graph's cycles.Its 0-skeleton is the node set and its 1-skeleton is the graph; order-2 cells can be attached to selected simple or induced cycles.Source: §C.3 · p. 22
  • Simplicial complexes are special cases of cell complexes in which each k-cell contains exactly k + 1 nodes.The order-2 example equates cycles of length up to three with triangles when node and edge sets match.Source: §C.3 · p. 22
  • The paper cautions that its cell-complex treatment may not be comprehensive for readers familiar with algebraic topology, prioritizing a consistent framework with generality, rigor, and applicability.The focus is on cell definitions and induced neighborhood functions for a unifying equivariant TDL framework.Source: §C.3 · p. 23
  • Cell complexes have a hierarchy grounded in topology, but homeomorphism requirements constrain which higher-order interactions they can represent.The paper contrasts this with the greater combinatorial flexibility of hypergraphs.Source: §C.3 · p. 22, §C.4 · p. 23

C.4 HYPERGRAPHS

  • Hypergraphs offer flexible higher-order groupings but generally lack the hierarchy and topology of cell complexes; ETNN neighborhood choices yield equivariant hypergraph message-passing networks.The rank function assigns rank 0 to nodes and rank 1 to hyperedges.Source: §C.4 · pp. 23–24

D ETNNS WITH VELOCITY TYPE INPUTS

  • ETNNs can accept node velocities alongside positions and non-geometric features by modifying the position update to include a velocity update.The passage identifies velocities as geometric node features.Source: D ETNNS WITH VELOCITY TYPE INPUTS · p. 24

E PROOF OF THEOREM 1

The proof establishes that ETNN feature updates are E(n)-invariant and position updates are E(n)-equivariant, then extends the argument to velocity inputs.

E PROOF OF THEOREM 1

  • Feature updates are E(n)-invariant because positions enter only through geometric invariants and the features contain no geometric information.The position-update proof then establishes the complementary equivariance property.Source: E PROOF OF THEOREM 1 · pp. 24–25
  • The position update is E(n)-equivariant, completing the proof that the feature and position updates are invariant and equivariant, respectively.The argument generalizes the EGNN proof of Satorras et al. (2021).Source: E PROOF OF THEOREM 1 · p. 25
  • With velocity inputs, the proof separately establishes E(n)-equivariance of the velocity and position updates.The text states that the position update is trivially equivariant and concludes that the updates are invariant and equivariant, respectively.Source: E PROOF OF THEOREM 1 · p. 25

F EXPRESSIVENESS OF ETNNS

ETNN expressivity depends on how a graph is lifted into a combinatorial complex: cells can expand receptive fields and distinguish geometric graphs with fewer layers than graph-based EGNNs. The section analyzes this theoretically and reports experiments on chain graphs and counterexample structures.

Geometric Augmented Hasse Graph of a Combinatorial Complex

  • Under shared message functions, shared aggregators, and sum-of-distance linear invariants, ETNNs correspond to EGNNs on geometric augmented Hasse graphs with a modified position update.The Hasse graph represents cells as nodes, connects neighboring cells, and assigns higher-order cells positions as functions of their constituent nodes.Source: F EXPRESSIVENESS OF ETNNS · p. 26

Expressivity of ETNN

  • An ETNN can distinguish k-hop distinct geometric graphs in M layers when a cell's receptive field covers the whole graph after M layers, under the stated assumptions.The required layers depend on the chosen skeleton-preserving lift and its neighborhood functions.Source: F EXPRESSIVENESS OF ETNNS · p. 27
  • The layer bound assumes skeleton-preserving lifts producing undirected Hasse graphs, and the authors call it potentially pessimistic because the modified position update may distinguish graphs sooner.They report that the homogeneous-function, common-aggregation, and invariant assumptions simplify the analysis but are not important in practice.Source: F EXPRESSIVENESS OF ETNNS · pp. 27–28
  • A skeleton-preserving lift makes ETNN strictly more powerful than an EGNN when a cell reaches the whole-graph receptive field in fewer layers than any EGNN node.The comparison concerns distinguishing k-hop distinct graphs.Source: F EXPRESSIVENESS OF ETNNS · p. 27

Discussion

  • A virtual node connected to every node can be viewed as a cell containing the whole graph, giving a VN-EGNN whole-graph receptive field in one layer.The discussion presents Proposition 2 as a generalization of the VN-EGNN expressivity result.Source: F EXPRESSIVENESS OF ETNNS · p. 28

Experimental Validation; Counterexamples Structures

  • ETNN experiments confirm a distinguishability advantage over standard EGNNs on lifted chain graphs and distinguish every tested counterexample except the 4-body chiral case.The authors also point to higher-order interactions as a practical approach to oversquashing.Source: F EXPRESSIVENESS OF ETNNS · pp. 29–30

Counterexamples Structures

  • In the tested counterexample task, ETNN distinguishes all structures except the 4-body chiral example, using up/down incidences and adjacencies for the tested lifts.Both invariant and equivariant ETNN versions were tested.Source: F EXPRESSIVENESS OF ETNNS · p. 30

G COMPLEXITY ANALYSIS

ETNN layer cost depends on the complex's connectivity and selected neighborhood functions. With sufficiently sparse connectivity, the cost is linear in the number of cells, while dense connectivity can make it quadratic.

G COMPLEXITY ANALYSIS

  • With sufficiently sparse complex connectivity, ETNN's overall complexity is linear in the number of cells; in the worst case of all-to-all connectivity, it is quadratic.The complexity discussion considers up/down incidences and adjacencies as neighborhood functions.Source: G COMPLEXITY ANALYSIS · pp. 31–32
  • ETNN processes neighborhood functions in a loop, so its implementation's forward and backward passes scale linearly with their number.A different implementation could parallelize this processing and make runtime constant with respect to neighborhood count.Source: G COMPLEXITY ANALYSIS · pp. 32–33
  • The compared ETNN configuration is one of the most computationally intensive, while lighter configurations can achieve better learning performance.The runtime and memory comparisons use that intensive configuration.Source: G COMPLEXITY ANALYSIS · pp. 32–33
  • ETNN uses three times as many neighborhoods as EGNN-graph-W in the compared configuration, yet its memory use is less than half that of EMPSN.The authors also report reduced computational burden relative to EMPSN, without giving the numerical runtime values in these passages.Source: G COMPLEXITY ANALYSIS · pp. 32–33

H.1 MOLECULAR DATA

The molecular-data discussion contrasts graphs, cell complexes, and hypergraphs as representations of molecular interactions. Each accommodates some higher-order structure, but each has a stated representational limitation.

Molecular graphs

  • Molecular graphs represent atoms as nodes and bonds as edges, but do not explicitly model higher-order multiway interactions such as information from rings.The passage notes multi-view and relational graphs as efforts to represent different atomic relations.Source: §H.1 · p. 33

Molecular cell complexes

  • Molecular cell complexes represent atoms, bonds, and rings at ranks 0, 1, and 2, but cannot explicitly represent motifs such as functional groups that are not cycles.Message-passing networks on molecular CW complexes have shown superior performance to models on molecular graphs.Source: §H.1 · p. 33

Molecular hypergraphs

  • Molecular hypergraphs can represent bonds, rings, and functional groups as hyperedges, but do not explicitly represent hierarchy among those interactions.Atoms are represented as nodes.Source: §H.1 · p. 33

H.2 GEOSPATIAL DATA

The geospatial discussion describes challenges in preserving multiresolution information and contrasts existing spatial and hierarchical graph approaches with geospatial combinatorial complexes. The following molecular-property-prediction subsection states that it provides dataset, featurization, and configuration details.

H.2 GEOSPATIAL DATA

  • Aggregating multiresolution spatial data to a common resolution can lose information, while cited hierarchical GNN approaches can admit multilevel features.The discussion contrasts these approaches with proximity graphs used in single-resolution settings.Source: §H.2 · p. 33
  • Prior geography-based hierarchical GNN work cited here does not use coordinate information directly and therefore does not explore equivariant architectures.The passage also says those hierarchical graph constructions rely heavily on application specifics, whereas geospatial CCs offer a more generally applicable formalism.Source: §H.2 · pp. 33–34
  • The discussion distinguishes TDL, which models relations and data distributions over topological spaces, from TDA, which describes their mathematical properties and shape.Source: §H.2 · p. 34

I.1 MOLECULAR PROPERTY PREDICTION

  • The molecular property prediction subsection covers dataset information, data featurization, and model parameter configuration.Source: §I.1 · p. 34

I.1.1 DATASET STATISTICS

The QM9 appendix describes its dataset, rank-specific molecular features, and parameter choices for evaluating ETNNs. The model uses features for atoms, bonds, rings, and functional groups, with configurations calibrated for a fair comparison.

I.1.1 DATASET STATISTICS

  • QM9 contains over 130,000 molecules and 12 property-prediction targets.Table 7 provides dataset statistics for molecules, atoms, bonds, rings, and functional groups.Source: §I.1.1 · p. 34, Table 7 · p. 34

I.1.2 CELL FEATURES

  • ETNN uses rank-specific features for atoms, bonds, rings, and functional groups, including categorical encodings and structural or chemical descriptors.Atom features include one-hot atom types and atomic-number functions; bond features encode bond type, conjugation, and ring membership; ring and functional-group features include structural and chemical attributes.Source: §I.1.2 · pp. 34–35

I.1.3 PARAMETER CONFIGURATION

  • Configurations use property-dependent learning rates and widths calibrated to keep the parameter space near 1.5M parameters.All configurations run for 1,000 epochs on EGNN’s train/validation/test splits, with batch normalization of geometric invariants and gradient clipping at 1.0.Source: §I.1.3 · p. 35

I.2.1 DATASET STATISTICS

The geospatial task predicts fine-resolution PM2.5 from a multi-rank complex of measurement points, roads, and census tracts. Features combine local geography, traffic, land use, and coarse pollution estimates, with spatially mindful splits and coordinate inputs.

I.2.1 DATASET STATISTICS

  • The geospatial complex contains 3,946 measurement points, 550 roads, and 151 census tracts at a target resolution of approximately 22 m.The measurement targets are aggregated at 0.0002° resolution after removing roads and tracts without measurements.Source: §I.2.1 · p. 36

I.2.2 CELL FEATURES

  • Point, road, and tract features combine local amenities and densities, traffic, land use, and coarse census-tract PM2.5 estimates.Point features include OSM-derived measures; road features use 2021 traffic data; tract features use PLUTO land-use variables and coarse PM2.5 estimates.Source: §I.2.2 · pp. 36–37, Table 9 · p. 37

I.2.3 PARAMETER CONFIGURATION

  • Training uses 70% of census tracts, with the remaining 30% split equally between validation and test; models are selected by validation RMSE and reported by test RMSE.The tract-based split partitions points by their corresponding tract, and the loss is computed only on training nodes.Source: §I.2.3 · p. 37

I.3 GEOMETRIC FEATURES

  • Geometric inputs are 0-cell coordinates inherited by higher ranks: 3D atom coordinates for QM9 and latitude and longitude for the geospatial task.Source: §I.3 · p. 38

J ABLATION STUDY AND REPRODUCIBILITY FOR MOLECULAR PROPERTY PREDICTION

The molecular study compares ETNN configurations across higher-order cells, adjacency and incidence choices, features, and virtual cells, while documenting reproducibility constraints. Several higher-order configurations reach near-SotA results, improve on EGNN baselines, and outperform TopNet.

Reproducibility

  • QM9 reproducibility is limited by differing data splits across prior models, and the authors could not reproduce EMPSN results.They use EGNN’s splits, reproduce EGNN with its original codebase, and report EMPSN results from the original paper and reproduced implementations for comparison.Source: J ABLATION STUDY AND REPRODUCIBILITY… · p. 38

Ablation dimensions

  • The ablation varies which atoms, bonds, rings, and functional groups form cells, and whether max, up/down, or combined adjacencies are used.Combining max with up/down adjacencies yields a heterogeneous setting.Source: J ABLATION STUDY AND REPRODUCIBILITY… · p. 39
  • Incidence ablations range from no inter-rank message exchange to adjacent-rank exchange or communication across all ranks.The k-up/down incidences allow cells to communicate across different rank gaps.Source: J ABLATION STUDY AND REPRODUCIBILITY… · p. 39
  • Higher-order feature ablations add bond, ring, or functional-group features; virtual cells are excluded from messaging and readout in these experiments.With max adjacency, a virtual cell is equivalent to using the max adjacency without including that cell.Source: J ABLATION STUDY AND REPRODUCIBILITY… · p. 39

Framework configurations

Results

  • Several higher-order ETNN configurations reach near-SotA results and significantly improve over EGNN baselines; ETNN also consistently outperforms TopNet.The tables report mean absolute error for molecular property predictions; the supplied passages do not provide the specific cell values.Source: J ABLATION STUDY AND REPRODUCIBILITY… · pp. 39–40, Table 11 · p. 40

K ARCHITECTURE DESCRIPTION

ETNN processes rank-specific cell features and geometric invariants through equivariant message-passing layers, then pools cell representations for graph-level prediction. Its EMP layers aggregate neighbor messages by adjacency and sender rank, with an optional update to node positions.

Architecture

  • ETNN embeds rank-specific input features, computes geometric invariants, applies EMP layers, and pools cell representations for graph-level prediction.The pipeline can use cells’ own non-geometric features, features averaged from constituent nodes, or membership vectors before embedding.Source: K ARCHITECTURE DESCRIPTION · p. 42

Equivariant Message Passing

  • EMP layers form adjacency-specific messages from sender and receiver representations and geometric invariants, aggregate messages by sender rank, then update each cell representation.A rank-specific update MLP processes concatenated adjacency-wise aggregates and adds the update to the cell’s hidden representation.Source: K ARCHITECTURE DESCRIPTION · p. 42
  • EMP layers can optionally update node positions using a weighted sum of differences between each node and its neighboring nodes.The weights are learnable functions of the messages passed from neighboring nodes.Source: K ARCHITECTURE DESCRIPTION · p. 43
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