# Uniform Ck Approximation of G-Invariant and Antisymmetric Functions, Embedding Dimensions, and Polynomial Representations

arXiv: https://arxiv.org/abs/2403.01339 · HTML: https://mlsanity.com/paper/2403.01339/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 69 references that are in the map.
- First posted on arXiv 2 Mar 2024.

## Builds on (most cited first, 19 of 19)

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## Abstract (arXiv)

For any subgroup G of the symmetric group Sn on n symbols, we present results for the uniform Ck approximation of G-invariant functions by G-invariant polynomials. For the case of totally symmetric functions (G = Sn), we show that this gives rise to the sum-decomposition Deep Sets ansatz of Zaheer et al. (2018), where both the inner and outer functions can be chosen to be smooth, and moreover, the inner function can be chosen to be independent of the target function being approximated. In particular, we show that the embedding dimension required is independent of the regularity of the target function, the accuracy of the desired approximation, as well as k. Next, we show that a similar procedure allows us to obtain a uniform Ck approximation of antisymmetric functions as a sum of K terms, where each term is a product of a smooth totally symmetric function and a smooth antisymmetric homogeneous polynomial of degree at most binomn2. We also provide upper and lower bounds on K and show that K is independent of the regularity of the target function, the desired approximation accuracy, and k.