# Deep Models of Interactions Across Sets

arXiv: https://arxiv.org/abs/1803.02879 · HTML: https://mlsanity.com/paper/1803.02879/

## Key facts

- Cited by 69 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2018.

## Built on by (most cited first, 30 of 69)

- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
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- [Neural Functional Transformers](https://mlsanity.com/paper/2305.13546/) (2023)
- [Transformers Generalize DeepSets and Can be Extended to Graphs and Hypergraphs](https://mlsanity.com/paper/2110.14416/) (2021)
- [Equivariant Deep Weight Space Alignment](https://mlsanity.com/paper/2310.13397/) (2023)
- [Optimal-er Auctions through Attention](https://mlsanity.com/paper/2202.13110/) (2022)
- [Universal Neural Functionals](https://mlsanity.com/paper/2402.05232/) (2024)
- [Machine Learning Methods in Solving the Boolean Satisfiability Problem](https://mlsanity.com/paper/2203.04755/) (2022)
- [Equivariance Discovery by Learned Parameter-Sharing](https://mlsanity.com/paper/2204.03640/) (2022)
- [Learnable Polyphase Sampling for Shift Invariant and Equivariant Convolutional Networks](https://mlsanity.com/paper/2210.08001/) (2022)
- [Brauer's Group Equivariant Neural Networks](https://mlsanity.com/paper/2212.08630/) (2022)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [Making Vision Transformers Truly Shift-Equivariant](https://mlsanity.com/paper/2305.16316/) (2023)
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- [Inductive Cognitive Diagnosis for Fast Student Learning in Web-Based Online Intelligent Education Systems](https://mlsanity.com/paper/2404.11290/) (2024)
- [RetaGNN: Relational Temporal Attentive Graph Neural Networks for Holistic Sequential Recommendation](https://mlsanity.com/paper/2101.12457/) (2021)
- [Deep Permutation Equivariant Structure from Motion](https://mlsanity.com/paper/2104.06703/) (2021)
- [Deep Learning for Two-Sided Matching](https://mlsanity.com/paper/2107.03427/) (2021)
- [Learning Dynamic Attribute-factored World Models for Efficient Multi-object Reinforcement Learning](https://mlsanity.com/paper/2307.09205/) (2023)
- [Group invariant machine learning by fundamental domain projections](https://mlsanity.com/paper/2202.02164/) (2022)
- [Inductive Matrix Completion Using Graph Autoencoder](https://mlsanity.com/paper/2108.11124/) (2021)
- [Connecting Permutation Equivariant Neural Networks and Partition Diagrams](https://mlsanity.com/paper/2212.08648/) (2022)
- [Fast computation of permutation equivariant layers with the partition algebra](https://mlsanity.com/paper/2303.06208/) (2023)
- [A Unified Framework for Discovering Discrete Symmetries](https://mlsanity.com/paper/2309.02898/) (2023)
- [GradMetaNet: An Equivariant Architecture for Learning on Gradients](https://mlsanity.com/paper/2507.01649/) (2025)
- [Variational Inference Failures Under Model Symmetries: Permutation Invariant Posteriors for Bayesian Neural Networks](https://mlsanity.com/paper/2408.05496/) (2024)

## Abstract (arXiv)

We use deep learning to model interactions across two or more sets of objects, such as user-movie ratings, protein-drug bindings, or ternary user-item-tag interactions. The canonical representation of such interactions is a matrix (or a higher-dimensional tensor) with an exchangeability property: the encoding's meaning is not changed by permuting rows or columns. We argue that models should hence be Permutation Equivariant (PE): constrained to make the same predictions across such permutations. We present a parameter-sharing scheme and prove that it could not be made any more expressive without violating PE. This scheme yields three benefits. First, we demonstrate state-of-the-art performance on multiple matrix completion benchmarks. Second, our models require a number of parameters independent of the numbers of objects, and thus scale well to large datasets. Third, models can be queried about new objects that were not available at training time, but for which interactions have since been observed. In experiments, our models achieved surprisingly good generalization performance on this matrix extrapolation task, both within domains (e.g., new users and new movies drawn from the same distribution used for training) and even across domains (e.g., predicting music ratings after training on movies).