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Uniform $\mathcal{C}^k$ Approximation of $G$-Invariant and Antisymmetric Functions, Embedding Dimensions, and Polynomial Representations

Soumya Ganguly, Khoa Tran, Rahul Sarkar

TL;DR

This work establishes that $G$-invariant and $n$-antisymmetric functions on $\Omega^n$ can be uniformly approximated in the $\mathcal{C}^k$ norm by, respectively, $G$-invariant and $n$-antisymmetric polynomials. In the totally symmetric case, the approximation admits a Deep Sets–style decomposition with smooth inner and outer maps, where the embedding dimension is $M=\binom{n+d}{n}$ and crucially independent of the target function, the approximation accuracy, and the order $k$. For $n$-antisymmetric functions, the paper shows the approximation can be written as a finite sum of $K$ terms, each a product of a symmetric polynomial and an antisymmetric homogeneous polynomial of degree at most $\binom{n}{2}$, with $K$ independent of those factors; it also provides detailed bounds on the minimal number of module generators and exact counts in special cases (e.g., Catalan numbers when $d=2$). Collectively, these results yield symmetry-preserving, low-embedding-dimensional representations that mitigate the curse of dimensionality and facilitate efficient polynomial-based approximations for invariant and antisymmetric function classes.

Abstract

For any subgroup $G$ of the symmetric group $\mathcal{S}_n$ on $n$ symbols, we present results for the uniform $\mathcal{C}^k$ approximation of $G$-invariant functions by $G$-invariant polynomials. For the case of totally symmetric functions ($G = \mathcal{S}_n$), 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 $\mathcal{C}^k$ 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 $\binom{n}{2}$. 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 $k$.

Uniform $\mathcal{C}^k$ Approximation of $G$-Invariant and Antisymmetric Functions, Embedding Dimensions, and Polynomial Representations

TL;DR

This work establishes that -invariant and -antisymmetric functions on can be uniformly approximated in the norm by, respectively, -invariant and -antisymmetric polynomials. In the totally symmetric case, the approximation admits a Deep Sets–style decomposition with smooth inner and outer maps, where the embedding dimension is and crucially independent of the target function, the approximation accuracy, and the order . For -antisymmetric functions, the paper shows the approximation can be written as a finite sum of terms, each a product of a symmetric polynomial and an antisymmetric homogeneous polynomial of degree at most , with independent of those factors; it also provides detailed bounds on the minimal number of module generators and exact counts in special cases (e.g., Catalan numbers when ). Collectively, these results yield symmetry-preserving, low-embedding-dimensional representations that mitigate the curse of dimensionality and facilitate efficient polynomial-based approximations for invariant and antisymmetric function classes.

Abstract

For any subgroup of the symmetric group on symbols, we present results for the uniform approximation of -invariant functions by -invariant polynomials. For the case of totally symmetric functions (), 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 . Next, we show that a similar procedure allows us to obtain a uniform approximation of antisymmetric functions as a sum of terms, where each term is a product of a smooth totally symmetric function and a smooth antisymmetric homogeneous polynomial of degree at most . We also provide upper and lower bounds on and show that is independent of the regularity of the target function, the desired approximation accuracy, and .
Paper Structure (17 sections, 24 theorems, 64 equations, 2 tables)

This paper contains 17 sections, 24 theorems, 64 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Our contributions
  4. Structure of the paper
  5. Notation
  6. Ck approximation via G-invariant and totally symmetric polynomials
  7. Ck approximation via n-antisymmetric polynomials
  8. Representations of the totally symmetric polynomials
  9. Representations of the n-antisymmetric polynomials
  10. Finite generation of n-antisymmetric polynomials and the case n2
  11. Bounds for minimum module generators of n-antisymmetric polynomials for ng2
  12. Acknowledgments and disclosure of funding
  13. Necessary background for commutative algebra
  14. Necessary results from representation theory
  15. Polynomial algebras and modules induced by a group action
  16. ...and 2 more sections

Key Result

Theorem 5

Given $d,n \geq 1$ and $\Omega \in \mathbb{R}^d$ compact set, if $\bm{\eta}: \Omega^n \to \mathbb{R}^m$ satisfy assumption A then for any $n$-antisymmetric, continuous function $f: \Omega^n \to \mathbb{R}$, there exists a unique $g:\bm{\eta}(\Omega^n) \to \mathbb{R}$ that is continuous and odd, sati where $\bm{\eta}(\Omega^n)$ is equipped with the topology induced from $\mathbb{R}^m$.

Theorems & Definitions (38)

  • Definition 1: $G$-invariant function
  • Definition 2: Totally symmetric function
  • Definition 3: $n$-antisymmetric function
  • Definition 4: Inner, Outer Functions, Embedding dimension
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Definition 8: $\mathcal{C}^k$ norm
  • Theorem 9: $\mathcal{C}^k$ approximation: $G$-invariant function
  • Corollary 10: $\mathcal{C}^k$ approximation: totally symmetric function
  • ...and 28 more