Embedding Dimension Lower Bounds for Universality of Deep Sets and Janossy Pooling

2026-05-08

AI Summary

This work investigates the lower bounds on embedding dimensions for permutation-invariant neural networks, with a focus on the universal approximation capabilities of Deep Sets and k-ary Janossy Pooling architectures. By integrating tools from function approximation theory, symmetry modeling, and combinatorial geometry, the authors introduce novel proof techniques that yield the first non-trivial lower bound on embedding dimensions for Janossy Pooling when \(k > 1\). Furthermore, they establish a tight (up to constant factors) minimal embedding dimension for Deep Sets in the setting where the input dimension \(d > 1\). These results demonstrate that Deep Sets achieve nearly optimal representational efficiency for high-dimensional inputs and provide the first theoretical guarantees for higher-order symmetric architectures.

Abstract

In many practical applications it is important to build symmetries into neural network architectures. Consider the important case of permutation symmetry on point clouds consisting of $n$ points in $d$ dimensions. In this case the network learns a function on a set of $n$ points in $\mathbb{R}^d$, and a natural paradigm for constructing invariant networks is Janossy pooling, which generalizes the popular Deep Sets architecture. We study the universality of this approach, in particular the important question of how large the embedding dimension must be to guarantee universality of this architecture. Specifically, using a novel technique, we prove new lower bounds on the required size of this embedding dimension. For Deep Sets, this gives the correct minimal dimension up to a constant factor for all $d > 1$. For $k$-ary Janossy pooling, we prove the first non-trivial lower bound on the required embedding dimension when $k > 1$.

Problem

• embedding dimension
• universality
• Deep Sets
• Janossy pooling
• permutation symmetry

Innovation

• embedding dimension
• universality
• Deep Sets
• Janossy pooling
• permutation symmetry