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

TL;DR

This paper investigates the minimal embedding dimension needed for Janossy pooling and Deep Sets to be universal in permutation-invariant neural networks, providing new lower bounds and theoretical insights.

Contribution

It establishes the first non-trivial lower bounds on embedding dimensions for Janossy pooling and refines bounds for Deep Sets, advancing understanding of their universality.

Findings

Provides the correct minimal embedding dimension for Deep Sets up to a constant factor.

Derives the first non-trivial lower bound for k-ary Janossy pooling when k > 1.

Uses a novel technique to establish these lower bounds.

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 $k > 1$.