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# Universal Representation of Permutation-Invariant Functions on Vectors and Tensors

*Proceedings of The 35th International Conference on Algorithmic Learning Theory*, PMLR 237:1134-1187, 2024.

#### Abstract

A main object of our study is multiset functions — that is, permutation-invariant functions over inputs of varying sizes. Deep Sets, proposed by Zaheer et al. (2017), provides a universal representation for continuous multiset functions on scalars via a sum-decomposable model. Restricting the domain of the functions to finite multisets of $D$-dimensional vectors, Deep Sets also provides a universal approximation that requires a latent space dimension of $O(N^D)$ — where $N$ is an upper bound on the size of input multisets. In this paper, we strengthen this result by proving that universal representation is guaranteed for continuous and discontinuous multiset functions through a latent space dimension of $O(N^D)$ (which we will further improve upon). We then introduce identifiable multisets for which we can uniquely label their elements using an identifier function, namely, finite-precision vectors are identifiable. Based on our analysis of identifiable multisets, we prove that a sum-decomposable model, for general continuous multiset functions requires only a latent dimension of $2DN$, as opposed to $O(N^D)$. We further show that both encoder and decoder functions of the model are continuous — our main contribution to the existing work which lacks such a guarantee. Additionally, this provides a significant improvement over the aforementioned $O(N^D)$ bound, derived for the universal representation of both continuous and discontinuous multiset functions. We then extend our results and provide special sum-decomposition structures to universally represent permutation-invariant tensor functions on identifiable tensors. These families of sum-decomposition models enable us to design deep network architectures and deploy them on a variety of learning tasks on sequences, images, and graphs.