Any-Dimensional Invariant Universality

2026-05-21

AI Summary

While existing machine learning models can process inputs of arbitrary dimensions—such as graphs or point clouds of varying scales—they lack rigorous theoretical guarantees of universality, as classical universal approximation theorems apply only to fixed-dimensional settings. This work introduces the first systematic definition and verification framework for universality across arbitrary input dimensions. By constructing an infinite-dimensional topological space encompassing all finite-dimensional inputs and their limits, and leveraging symmetry analysis together with the theory of compact families, we reveal fundamental limitations in the cross-dimensional universality of mainstream architectures and propose a concise, effective correction. Building upon this framework, we design a novel model that achieves provable universality over arbitrary dimensions, with formal guarantees rooted in function approximation theory.

Abstract

Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain poorly understood, as universality is traditionally studied for models accepting inputs of a fixed size, defined on a compact subset of their domain. In sharp contrast, any-dimensional models can be viewed as sequences of functions defined on growing-sized inputs, and it is not clear in which sense they can be universal. We develop a systematic approach to establish any-dimensional universality, by identifying any-dimensional functions with a unique function taking inputs in a suitable infinite-dimensional limit space containing inputs of all finite sizes as well as their limits. Using the symmetries of these inputs and relations between inputs of different sizes, we show that this limit space admits a natural topology with rich families of compact sets on which any-dimensional universality can be established. We illustrate our approach by showing that several existing architectures fail to be universal, and we propose simple modifications that restore universality.

Problem

• any-dimensional universality
• machine learning models
• input size variability
• universal approximation
• infinite-dimensional limit space

Innovation

• any-dimensional universality
• infinite-dimensional limit space
• symmetry
• compact sets
• universal approximation