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 $k$.

翻译：摘要：对于$n$个符号上的对称群$\mathcal{S}_n$的任意子群$G$，我们提出了用$G$-不变多项式对$G$-不变函数进行均匀$\mathcal{C}^k$逼近的结果。对于全对称函数（$G = \mathcal{S}_n$）的情形，我们证明这会导出Zaheer等人（2018）的求和分解Deep Sets框架，其中内函数和外函数均可选为光滑函数，且内函数可独立于被逼近的目标函数选取。特别地，我们证明所需嵌入维数与目标函数的正则性、期望逼近精度以及$k$均无关。其次，我们证明类似方法可将反对称函数的均匀$\mathcal{C}^k$逼近表示为$K$项之和，其中每一项均为光滑全对称函数与次数不超过$\binom{n}{2}$的光滑反对称齐次多项式的乘积。我们还给出了$K$的上下界，并证明$K$与目标函数的正则性、期望逼近精度及$k$均无关。