Paper:
The relational model is a ubiquitous representation of big-data, in part due to its extensive use in databases. However, recent progress in deep learning with relational data has been focused on (knowledge) graphs. In this paper we propose Equivariant Entity-Relationship Networks, a general class of parameter-sharing neural networks derived from the entity-relationship model. We prove that our proposed feed-forward layer is the most expressive linear layer under the given equivariance constraints, and subsumes recently introduced equivariant models for sets, exchangeable tensors, and graphs. The proposed feed-forward layer has linear complexity in the data and can be used for both inductive and transductive reasoning about relational databases, including database embedding, and the prediction of missing records. This provides a principled theoretical foundation for the application of deep learning to one of the most abundant forms of data.

Review:
The authors extend recent work in equivariant set encoding to the setting of entity-relation data.  Similar to the cited previous work, they encode sets of objects (in this case the tuples of a relational database) with a permutation invariant function. They use a parameter tieing scheme to enforce this invariance.
I don’t find the paper to be particularly well motivated. Relational DBs are not necessarily a setting where you would always want equivariance. While the ordering of tuples do not matter, relations are often asymmetrical. Further, the idea of concatenating all tuples of a relational DB to be passed through a particular feed forward layer is infeasible for all but the smallest datasets. Real world databases such as knowledge bases contains millions to billions of entries. Scaling issues aside, the experiments do not actually show that this method outperforms any reasonable baselines such as a simple tensor factorization. 
I also found it particularly hard to follow the descriptions of the methods. A few specific points:
- The text and notation in the beginning of section 2 could be a lot clearer. I had to read these paragraphs multiple times to pick out precisely what you were trying to say. Maybe have the set of relations be [R] like your other sets rather than a different script R. 
- In the next paragraph you say “A particular relation R is a multiset if it contains multiple copies of the same entity” but you previously defined R to be a set of instances and not entities. I think you need to be more consistent with your terminology since the differences between type level entities and instances is important for your definitions and as you noted in your first footnote, these terms are often used very differently.
- This explanation could be helped a lot by improving the figure. The caption and images are both very dense but still requires a lot of coreference. For example, labeling the entities and relations with their ids in figure 1a directly would make it much easier to mentally map to what you are explaining in the caption. Also figure 1c is not at all clear.
Lastly, the methodology seems quite incremental over the previous work. A lot of the context and background that was sent to the appendix should be included in the main paper, particularly the relation to related work
edits:
double ‘the’ in abstract “linear complexity in the the data “