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Principled Simplicial Neural Networks for Trajectory Prediction

T. Mitchell Roddenberry, Nicholas Glaze, Santiago Segarra

arXiv:2102.10058v3cs.LGcs.SI

TL;DR

Higher-order interactions call for neural networks that use simplicial-complex structure rather than pairwise graph relationships alone. The paper defines three design properties and proposes SCoNe, a convolutional architecture for trajectory prediction. It reports improved generalizability to unseen trajectories and best prediction on the real-data comparison, while one synthetic-data comparison excludes the chain and RNN methods from unseen-data testing.

  • Problem

    Some data involves higher-order relationships that are not adequately expressed by pairwise graphs, motivating neural architectures for data on simplicial complexes.

    Source: §1 · p. 1

  • Method

    The paper defines permutation equivariance, orientation equivariance, and simplicial awareness, then proposes SCoNe for trajectory prediction on dimension-2 simplicial complexes.

    Source: §1.1 · p. 1, §5 · p. 5

  • Results

    SCoNe with tanh and triangles performs similarly to projection onto ker(∆1) on reversed synthetic trajectories, and SCoNe yields the best trajectory prediction among competing methods on real data.

    Source: §6.2 · p. 8, §6.3 · p. 9

  • Takeaways & Limitations

    The authors report that designing for admissibility and integrating simplicial structure improves generalizability to unseen trajectories.

    Source: §1.1 · p. 1, §6.3 · p. 9

    On the reversed synthetic test set, the chain and RNN methods cannot be tested on unseen data.

    Source: §6.2 · p. 8

Abstract

from arXiv · show

We consider the construction of neural network architectures for data on simplicial complexes. In studying maps on the chain complex of a simplicial complex, we define three desirable properties of a simplicial neural network architecture: namely, permutation equivariance, orientation equivariance, and simplicial awareness. The first two properties respectively account for the fact that the node indexing and the simplex orientations in a simplicial complex are arbitrary. The last property encodes the desirable feature that the output of the neural network depends on the entire simplicial complex and not on a subset of its dimensions. Based on these properties, we propose a simple convolutional architecture, rooted in tools from algebraic topology, for the problem of trajectory prediction, and show that it obeys all three of these properties when an odd, nonlinear activation function is used. We then demonstrate the effectiveness of this architecture in extrapolating trajectories on synthetic and real datasets, with particular emphasis on the gains in generalizability to unseen trajectories.

1. Introduction

Graph neural networks leverage pairwise graph structure, but some data involves higher-order interactions that require a simplicial-complex model. This paper defines principles for simplicial neural networks and studies an admissible architecture for trajectory prediction.

1. Introduction

  • Simplicial complexes model higher-order relationships, such as group interactions, that are not adequately expressed by pairwise relationships alone.They are closed under restriction, so a group relationship also includes its constituent pairwise relationships.Source: §1 · p. 1

1.1. Contribution

  • The paper defines three properties for simplicial neural networks and shows that its trajectory-prediction architecture is more generalizable to unseen trajectories when designed around them.The properties are permutation equivariance, orientation equivariance, and simplicial awareness; the architecture requires an odd, nonlinear activation to satisfy them.Source: §1.1 · p. 1

2. Related work

Graph neural networks adapt convolutional methods to graph structure, while work on simplicial complexes uses algebraic topology to analyze higher-order data. This paper builds on those lines to develop a principled framework for simplicial neural networks.

2.1. Graph Neural Networks

  • Graph neural networks interleave nonlinear activations with graph-structured diffusion, with later architectures simplifying the diffusion operators for scalability.The cited progression moves from spectral-domain methods to low-order Chebyshev polynomials and then first-order diffusion.Source: §2.1 · p. 2

2.2. Signal Processing on Simplicial Complexes

  • Simplicial signal-processing work uses discrete Hodge theory and Hodge Laplacians to analyze flows on graphs and higher-order structures.One cited application formulates flow interpolation as an optimization problem under observation constraints.Source: §2.2 · p. 2

2.3. Simplicial Neural Networks

  • Prior simplicial neural-network work proposed architectures for edge-flow tasks, while this paper develops a principled framework and highlights orientation as an additional design concern.The authors contrast their framework with earlier architectures that, they say, do not address important notions of orientation.Source: §2.3 · p. 2

3. Background

A simplicial complex organizes nodes, edges, triangles, and higher-order simplices through a closure-under-restriction structure. Boundary and coboundary operators support the chain-complex and Hodge-Laplacian descriptions used to analyze chains and flows.

3.1. Simplicial Complexes

  • An abstract simplicial complex is a set X of finite subsets of V closed under restriction; its elements are simplices, with nodes corresponding to 0-simplices, edges to 1-simplices, and triangles to 2-simplices.For a k-simplex, faces are the (k−1)-simplices that are subsets of it, and cofaces are the (k+1)-simplices having it as a face.Source: §3.1 · p. 2
  • Each simplex is assigned an orientation, and the vector space Ck has oriented k-simplices as a basis; its elements are k-chains.Reversing a basis simplex's orientation multiplies it by −1.Source: §3.1 · p. 2, §3.2 · p. 2

3.2. Boundary Operators and Hodge Laplacians

  • Boundary operators map k-chains to their faces through an alternating sum, and successive boundary maps compose to zero.Together, the chain spaces and boundary maps form a chain complex; the adjoint maps are coboundary operators.Source: §3.2 · p. 3
  • The Hodge Laplacian acts on k-chains, and the Hodge decomposition separates chains into boundary-related components and a harmonic component.The dimension of the Laplacian's kernel equals the kth Betti number, which counts k-dimensional holes.Source: §3.2 · p. 3
  • For 1-chains, the decomposition distinguishes triangle-boundary flows, node-gradient flows, and harmonic flows that are neither.Harmonic chains have zero coefficient sum around every triangle and at every node, representing a smooth, conservative flow in the paper's account.Source: §3.2 · p. 3

4. Admissible Neural Architectures

The paper proposes three properties for neural networks acting on simplicial-complex chains: independence from arbitrary node labels and simplex orientations, and dependence on the complex's higher-order structure. It calls architectures satisfying all three admissible.

4. Admissible Neural Architectures

  • The paper frames admissible neural architectures as maps between chain spaces, parameterized by weights and boundary operators.The input and output chains may have different orders.Source: §4 · p. 3

4.1. Permutation Equivariance

  • Permutation equivariance requires relabeling the simplicial complex and input to produce the correspondingly relabeled output.This guards against dependence on arbitrary node indexing.Source: §4.1 · pp. 3–4

4.2. Orientation Equivariance

  • Orientation equivariance requires the network's output to transform consistently when simplex orientations are changed.The orientation choice is arbitrary, while nodes are left unoriented.Source: §4.2 · p. 4

4.3. Simplicial Awareness

  • Simplicial awareness requires the output to depend on higher-order simplices, rather than ignoring dimensions of the underlying complex.For example, a node-to-node network that ignores triangles would fail awareness of order 2.Source: §4.3 · p. 4

4.4. Admissibility

  • An architecture is admissible when it satisfies permutation equivariance, orientation equivariance, and simplicial awareness.The authors present admissibility as a design suggestion and motivate it as a way to reflect domain symmetries and promote generalization to unseen structures.Source: §4.4 · pp. 4–5

5. Trajectory Prediction with SCoNe

SCoNe predicts a trajectory’s next node on a 2-dimensional simplicial complex by processing the trajectory as a 1-chain. Its admissibility depends on activation-function properties that support orientation equivariance and use of higher-order simplicial information.

5. Trajectory Prediction with SCoNe

  • SCoNe predicts the next node from a trajectory on a 2-dimensional simplicial complex, with candidate outputs restricted to neighbors of the terminal node.The task excludes trajectories that terminate at their current node.Source: §5 · p. 5, §5.2 · p. 5

5.1. Representation of Trajectories as 1-Chains

  • A trajectory is represented as a 1-chain by turning each adjacent-node step into an oriented edge and summing those edges.This representation retains much of the sequential information while making the input suitable for operators on chains.Source: §5.1 · p. 5

5.2. An Admissible Architecture for 1-Chains

  • SCoNe processes the edge-chain input through layers, maps the result to a 0-chain, and applies softmax over nodes neighboring the terminal node.The architecture maps from C1 to C1, then uses the boundary map to obtain a 0-chain for the neighborhood-based decision.Source: §5.2 · pp. 5–6
  • SCoNe uses information only from an L-hop simplicial neighborhood of the terminal node, allowing computation on a localized region of a large complex.The stated runtime sums per-layer costs involving the numbers of edges and triangles and the layer feature dimensions.Source: §5.2 · p. 6
  • SCoNe is admissible only when its continuous, elementwise activation is both odd and nonlinear.Oddness gives orientation equivariance; nonlinearity permits dependence on ∂2, whereas a linear activation fails simplicial awareness of order 2. The paper proposes tanh.Source: §5.2 · p. 6

6. Experiments

The experiments compare SCoNe with projection, simplicial-network, and sequence-based baselines on synthetic and real trajectory-prediction tasks. Results highlight the roles of admissibility, triangles, and local aggregation in prediction and generalization.

6.1. Methods

  • SCoNe uses three layers with 16 hidden features per layer and is compared with kernel projections, SCNN, S2CCNN, a Markov chain, and an RNN.The sequence-based baselines learn node-specific decisions and cannot generalize to unseen structures.Source: §6.1 · pp. 6–7

6.2. Synthetic Dataset

  • The synthetic dataset contains 1,000 trajectories over a triangulated 400-point complex, with 800 for training and 200 for testing.A separate evaluation orients edges to align with training direction; non-admissible activations perform similarly to tanh when the test set follows that rule, but fail on the reversed case.Source: §6.2 · pp. 7–8
  • On the standard synthetic split, the RNN performs best, while both kernel-projection methods outperform random guessing and ker(∆1) beats ker(∂1).The authors attribute the RNN result to the volume of training examples and the kernel methods’ performance to trajectories’ harmonic-subspace interpretation.Source: §6.2 · pp. 7–8
  • With tanh, including triangles improves SCoNe, and tanh outperforms sigmoid and ReLU; identity performs poorly across experiments.The authors relate tanh’s advantage to its oddness and SCoNe’s advantage over SCNN and S2CCNN possibly to admissibility’s added regularization.Source: §6.2 · p. 8
  • On reversed synthetic trajectories, SCoNe with tanh and triangles performs similarly to projection onto ker(∆1), and both outperform competing approaches.The reversed test set changes trajectory direction relative to training.Source: §6.2 · p. 8

6.3. Real Data

  • The real-data evaluations use ocean drifter trajectories around Madagascar and shortest paths on a Berlin map represented as a cubical complex.The Berlin trajectories are split 80/20 for training and testing; the cubical case illustrates the architecture’s flexibility to other chain complexes.Source: §6.3 · pp. 8–9
  • SCoNe yields the best trajectory prediction across the competing methods on the Ocean Drifters and Berlin datasets.It also outperforms projection onto ker(∆1), an admissible approach.Source: Table 1 · p. 7, §6.3 · p. 9

7. Conclusion

The conclusion presents a principled approach to neural networks on simplicial complexes. It combines permutation equivariance, orientation equivariance, and simplicial awareness to support generalizable architectures.

7. Conclusion

  • The paper constructs neural networks for simplicial-complex data that account for node permutations, simplex orientations, and higher-order structure.The authors frame these properties as supporting systems that transfer and scale well.Source: §7 · p. 9
  • The conclusion identifies permutation equivariance as a core graph-neural-network property and extends the design considerations to orientation equivariance and simplicial awareness.Simplicial awareness means accounting for higher-order structures.Source: §7 · p. 9
  • The authors characterize the resulting neural networks as principled and generalizable for data supported on simplicial complexes.Source: §7 · p. 9

Supplementary Material

The supplement explains practical representations and implementation details for SCoNe. It covers chain and boundary-map representations, the vector-and-matrix algorithm, and elementwise activations on 1-chains.

Supplementary Material

  • The supplement outlines vector and matrix representations, the real-vector form of SCoNe, experimental hyperparameters, projection-method implementation, complexity analysis, and the cubical-complex implementation.It also includes the proof of the need for odd, nonlinear activations.Source: Supplementary Material · p. 10

A.1. Chains as Real Vectors

A.2. Boundary Maps as Matrices

  • The boundary maps ∂1 and ∂2 are represented by matrices B1 and B2, with dimensions set by the sizes of their chain spaces.B1 is the signed incidence matrix, and the first Hodge Laplacian is represented using the boundary matrices.Source: §A.2 · p. 10
  • Algorithm S-2 gives the SCoNe trajectory-prediction procedure using real vectors and boundary matrices.The supplement describes it as a vector-and-matrix version of Algorithm 1.Source: A. Representing Chains and Boundary… · p. 10, §A.2 · p. 11

A.3. Application of Elementwise Activation Functions

  • An activation on a 1-chain is defined by applying φ to its real coefficients, equivalently by applying φ elementwise to its vector representation.The supplement states that SCoNe uses an odd, elementwise activation on C1.Source: §A.3 · p. 11

B. Implementation of SCoNe

The implementation section reports SCoNe’s training configuration and points to supplementary tables for experiment hyperparameters and layer-computation complexities.

B. Implementation of SCoNe

  • SCoNe uses three layers with 16 hidden dimensions, Adam, cross-entropy loss, and weight decay of 5e−5.The Adam parameters are β1 = 0.9 and β2 = 0.99; other hyperparameters vary by experiment.Source: B. Implementation of SCoNe · p. 11
  • Hyperparameters were selected by hand; synthetic standard and reversed experiments share them, while transfer-learning settings differ slightly.Table S-2 covers synthetic and real-data experiments.Source: Table S-2 · p. 12
  • Table S-3 compares computational complexities for alternative operation orderings when computing a single SCoNe layer’s intermediate representations.Its best-use cases depend on the relative sizes of the simplicial-complex levels.Source: Table S-3 · p. 12

C. Implementation of Nullspace Projection Methods

The nullspace-projection comparison adjusts the boundary readout so it can predict a next node rather than always returning zero.

C. Implementation of Nullspace Projection Methods

D. Computational Complexity Analysis

The analysis establishes a per-layer runtime for SCoNe and explains how sparsity and operation order affect its computation.

D. Computational Complexity Analysis

E. Proof of Proposition 1

The proof establishes permutation equivariance for SCoNe and shows that, under the stated continuity and elementwise-activation assumptions, admissibility requires an odd, nonlinear activation.

E. Proof of Proposition 1

F. Admissibility of Previous Simplicial Neural Networks

The appendix compares admissibility conditions for earlier simplicial networks, explains the flow components of the Hodge decomposition, and extends the boundary and Laplacian construction to cubical complexes.

F. Admissibility of Previous Simplicial Neural Networks

  • Ebli et al.’s architecture is orientation equivariant for elementwise φ iff φ is odd, and it satisfies awareness of orders k−1 and k+1; unlike SCoNe, it does not require nonlinear φ for that awareness.Bunch et al.’s all-level architecture is admissible iff φ is odd; nonlinear φ is required when only one level is output.Source: F. Admissibility of Previous… · p. 15

G. Chains, Flows, and the Hodge Decomposition

  • A 1-chain models an oriented edge flow, with reversing the measurement direction reversing its sign.Source: G. Chains, Flows, and the Hodge… · p. 16
  • The Hodge decomposition distinguishes curly triangle-boundary flows, node-gradient flows, and harmonic flows that are neither curly nor gradient.Gradient-flow integrals depend only on path endpoints; harmonic flows have zero circulation around triangles and zero incident-flow sum at nodes.Source: G. Chains, Flows, and the Hodge… · p. 16

H. Hodge Laplacians for Cubical 2-Complexes

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