mlsanityData to 7 Oct 2026

On the Limitations of Representing Functions on Sets

arXiv 1901.09006 ↗First posted 25 Jan 2019cs.LG cs.AI cs.NEcited by 89

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Abstract (arXiv)

Recent work on the representation of functions on sets has considered the use of summation in a latent space to enforce permutation invariance. In particular, it has been conjectured that the dimension of this latent space may remain fixed as the cardinality of the sets under consideration increases. However, we demonstrate that the analysis leading to this conjecture requires mappings which are highly discontinuous and argue that this is only of limited practical use. Motivated by this observation, we prove that an implementation of this model via continuous mappings (as provided by e.g. neural networks or Gaussian processes) actually imposes a constraint on the dimensionality of the latent space. Practical universal function representation for set inputs can only be achieved with a latent dimension at least the size of the maximum number of input elements.

Key facts. Cited by 89 papers in the map (data through 7 Oct 2026). First posted on arXiv 25 Jan 2019.

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Citing papers per year: 2021: 22; 2022: 22; 2023: 16; 2024: 12; 2025: 6; 2026: 11.

89 papers.

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  1. 2021You are AllSet: A Multiset Function Framework for Hypergraph Neural Networkscited by 124
  2. 2021Accurate Learning of Graph Representations with Graph Multiset Poolingcited by 120
  3. 2021GENESIS-V2: Inferring Unordered Object Representations without Iterative Refinementcited by 109
  4. 2023Everything is Connected: Graph Neural Networkscited by 85
  5. 2021Reconstruction for Powerful Graph Representationscited by 65
  6. 2021Universal Approximation of Functions on Setscited by 62
  7. 2022Theory of Graph Neural Networks: Representation and Learningcited by 60
  8. 2022Message passing all the way upcited by 47
  9. 2022Equivariant Hypergraph Diffusion Neural Operatorscited by 45
  10. 2021You Never Cluster Alonecited by 38
  11. 2022Low Dimensional Invariant Embeddings for Universal Geometric Learningcited by 38
  12. 2023Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theoremcited by 33
38 more
  1. 2022Likelihood-Free Parameter Estimation with Neural Bayes Estimatorscited by 31
  2. 2023Sumformer: Universal Approximation for Efficient Transformerscited by 30
  3. 2022Graph Neural Networks with Adaptive Readoutscited by 29
  4. 2022PELICAN: Permutation Equivariant and Lorentz Invariant or Covariant Aggregator Network for Particle Physicscited by 26
  5. 2022Variable-Input Deep Operator Networkscited by 24
  6. 2022The Neural Process Family: Survey, Applications and Perspectivescited by 23
  7. 2021Learning Prototype-oriented Set Representations for Meta-Learningcited by 19
  8. 2021Multiset-Equivariant Set Prediction with Approximate Implicit Differentiationcited by 17
  9. 2022Set Norm and Equivariant Skip Connections: Putting the Deep in Deep Setscited by 17
  10. 2021ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Cloudscited by 15
  11. 2022Learnable Commutative Monoids for Graph Neural Networkscited by 13
  12. 2023Practical Equivariances via Relational Conditional Neural Processescited by 13
  13. 2024An end-to-end attention-based approach for learning on graphscited by 11
  14. 2022Exponential Separations in Symmetric Neural Networkscited by 10
  15. 2024On permutation-invariant neural networkscited by 10
  16. 2021SLOSH: Set LOcality Sensitive Hashing via Sliced-Wasserstein Embeddingscited by 9
  17. 2021Task-Adaptive Neural Network Search with Meta-Contrastive Learningcited by 8
  18. 2022Graph Ordering Attention Networkscited by 8
  19. 2025Fourier Sliced-Wasserstein Embedding for Multisets and Measurescited by 8
  20. 2021Reversible Action Design for Combinatorial Optimization with Reinforcement Learningcited by 7
  21. 2023Postprocessing of Ensemble Weather Forecasts Using Permutation-invariant Neural Networkscited by 7
  22. 2024DE-HNN: An effective neural model for Circuit Netlist representationcited by 7
  23. 2021Set-to-Sequence Methods in Machine Learning: a Reviewcited by 6
  24. 2022Breaking the Curse of Dimensionality in Multiagent State Space: A Unified Agent Permutation Frameworkcited by 6
  25. 2022Neural Estimation of Submodular Functions with Applications to Differentiable Subset Selectioncited by 6
  26. 2023Polynomial Width is Sufficient for Set Representation with High-dimensional Featurescited by 6
  27. 2024On the H"older Stability of Multiset and Graph Neural Networkscited by 6
  28. 2022Equilibrium Aggregation: Encoding Sets via Optimizationcited by 5
  29. 2024Optimal Equivariant Architectures from the Symmetries of Matrix-Element Likelihoodscited by 5
  30. 2021Deep Synoptic Monte Carlo Planning in Reconnaissance Blind Chesscited by 4
  31. 2022DCSF: Deep Convolutional Set Functions for Classification of Asynchronous Time Seriescited by 4
  32. 2023Neural approximation of Wasserstein distance via a universal architecture for symmetric and factorwise group invariant functionscited by 4
  33. 2023Transformer Choice Net: A Transformer Neural Network for Choice Predictioncited by 4
  34. 2024Spectral Convolutional Conditional Neural Processescited by 4
  35. 2021Optimal sequential decision making with probabilistic digital twinscited by 3
  36. 2021GP-ConvCNP: Better Generalization for Convolutional Conditional Neural Processes on Time Series Datacited by 3
  37. 2021PICASO: Permutation-Invariant Cascaded Attentional Set Operatorcited by 3
  38. 2024Quantification using Permutation-Invariant Networks based on Histogramscited by 3

50 most cited of 89 All 89 (JSON)