Source-linked AI summary
Simplicial Neural Networks
Stefania Ebli, Michaël Defferrard, Gard Spreemann
TL;DR
The paper addresses learning and missing-data imputation when observations have higher-order relationships that graphs cannot represent. It develops simplicial neural networks using Laplacian-based convolution on simplicial complexes and evaluates them on coauthorship data. SNNs outperform simple imputation baselines and transfer across coauthorship complexes, while stronger comparisons remain future work.
Problem
Graphs capture pairwise relationships but are limited for higher-order data, motivating neural methods for learning on simplicial complexes.
Method
SNNs encode data as cochains on simplicial complexes and define convolution using simplicial Laplacians and spectral filters.
Results
SNNs outperform mean, median, and neighboring-simplex imputation baselines, and a model trained on one coauthorship complex is almost as accurate on another.
Takeaways & Limitations
Simplicial neural networks provide a framework for learning on higher-dimensional relational data and show preliminary utility for missing-citation imputation.
Abstract
from arXiv · showhide
We present simplicial neural networks (SNNs), a generalization of graph neural networks to data that live on a class of topological spaces called simplicial complexes. These are natural multi-dimensional extensions of graphs that encode not only pairwise relationships but also higher-order interactions between vertices - allowing us to consider richer data, including vector fields and $n$-fold collaboration networks. We define an appropriate notion of convolution that we leverage to construct the desired convolutional neural networks. We test the SNNs on the task of imputing missing data on coauthorship complexes.
1 Introduction
Simplicial complexes extend graphs to represent higher-dimensional relationships, motivating neural networks that can use global topological structure and support data such as vector fields and n-fold collaborations.
- Convolutional neural networks generalize beyond Euclidean spaces through operations designed for structures such as spheres, homogeneous spaces, and graphs.
- Graphs model pairwise relationships, whereas simplicial complexes provide higher-dimensional representations of complex systems.The paper positions simplicial complexes as a topological extension of graphs for richer relational data.
- Existing higher-order relational methods do not clearly connect convolution with the global topology of the underlying space.This motivates a framework grounded in the topological structure of simplicial complexes.
- The proposed approach is motivated by Hodge–de Rham theory and targets data naturally represented as vector fields or sensitive to global structure.
2 Proposed method
The method represents data as cochains on simplicial complexes, builds Laplacians from coboundary maps, and defines convolution spectrally through Laplacian eigenvectors. Low-degree polynomial filters make the resulting layers efficient, parameter-efficient, and localized across simplices.
- Simplicial complexes: Simplicial complexes are collections of finite sets closed under subsets, with vertices, edges, triangles, and higher-dimensional simplices representing increasingly complex relations.
- Cochains and coboundaries: Data are encoded as p-cochains, and coboundary maps transfer information from p-simplices to their higher-dimensional cofaces.
- Simplicial Laplacians: Simplicial Laplacians combine coboundary maps and their adjoints; their kernels encode the complex’s homological information.In practical settings, coboundaries can be represented sparsely, enabling efficient Laplacian computation.
- Simplicial Fourier analysis: The simplicial Fourier transform decomposes p-cochains using eigencochains of the corresponding Laplacian, providing the basis for spectral convolution.
- Simplicial convolution: Simplicial convolution is defined through pointwise multiplication in the Fourier domain, and convolutional layers apply learned filters followed by nonlinearities and bias.
- Localized convolutional layers: Low-degree polynomial filters reduce computation from O(|Kp|^2) to O(ξ|Kp|), reduce learned weights from O(|Kp|) to O(1), and yield N-local interactions.An N-local layer does not connect simplices more than N hops apart, so the operation can be interpreted as message passing.
3 Experimental results
SNNs were evaluated for imputing missing citations in coauthorship complexes, using higher-order collaboration structure. They outperformed simple baselines, produced errors concentrated near zero, and transferred nearly as well across complexes.
- Data: Coauthorship complexes represent papers as simplices, with citations assigned to collaborations among author subsets.The evaluation used complexes sampled from the Semantic Scholar Open Research Corpus, including CC1 and CC2.
- Method: SNNs imputed missing citations on 0-, 1-, and 2-cochains across missingness rates from 10% to 50%.Missing values were randomly introduced and replaced with the median of known citations as an initial input.
- Results: Prediction-error distributions accumulated close to zero on CC1, alongside mean-accuracy measurements over five samples.Figure 2 reports mean accuracy ± standard deviation and an absolute-error distribution for 40% missing citations.
- Results: SNNs well outperformed mean, median, and neighboring-simplex baselines for missing-citation imputation on CC1.The neighboring-simplex baseline uses the mean of the (k −1) and (k + 1) neighboring simplices.
- Results: An SNN trained on CC2 was almost as accurate on CC1 as an SNN trained on CC1.This cross-complex transfer result is shown by comparing Figures 2 and 3.
4 Conclusion and future work
The paper introduces a mathematical framework for neural networks on simplicial complexes and reports preliminary missing-data-imputation results. It identifies comparisons with stronger algorithms, vector-field problems, cross-dimensional operations, pooling, and expressive power as future directions.
- Conclusion: The paper introduces a mathematical framework for designing neural networks on data that live on simplicial complexes.It also provides preliminary results on their ability to impute missing data.
- Future work: Future work includes comparison with state-of-the-art imputation algorithms and applications to vector-field problems.Other directions include simplicial coarsening and pooling, cross-dimensional boundary and coboundary operations, and expressive-power analysis.
- Conclusion: The spectral language was key to developing and formulating the method, with localized filters that exploit space structure and share learnable weights.The paper describes this spectral view as extending convolution beyond homogeneous spaces.
A Supplementary material
The supplementary material defines how simplicial distance supports localized convolution and describes constructing and sampling coauthorship complexes for evaluation. It also specifies the accuracy and error measures used to assess missing-value imputation.
- Simplicial distance and localization: Simplicial distance counts steps between p-simplices connected through shared faces or cofaces, and degree-N filters do not couple simplices farther than N steps apart.This gives the convolutional layer a locality property analogous to spatial locality in ordinary CNNs.
- Simplicial projection: A bipartite paper-author graph is projected into a simplicial complex, with cochains assigned from weights on shared neighboring papers.In the coauthorship application, paper weights are citation counts and the cochain operation is a sum.
- Sampling papers: Each sampled coauthorship complex contains 80 papers selected by a random walk over papers linked by shared authors.The source corpus excludes papers with fewer than 5 citations or more than 10 authors.
- Mean accuracy and absolute error: Imputation accuracy is the percentage of missing values within 10% of the true value, while absolute error is the magnitude of their difference.For each missingness rate, the study reports mean accuracy ± standard deviation over 5 randomly damaged samples.