Uniform 𝒞^k Approximation of G-Invariant and Antisymmetric Functions, Embedding Dimensions, and Polynomial Representations

Soumya Ganguly,Khoa Tran,Rahul Sarkar

arxiv(2024)

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Abstract
For any subgroup G of the symmetric group 𝒮_n on n symbols, we present results for the uniform 𝒞^k approximation of G-invariant functions by G-invariant polynomials. For the case of totally symmetric functions (G = 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 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 n2. 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.
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