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

要約

n$記号上の対称群$mathcal{S}_n$の任意の部分群$G$について、$G$不変多項式による$G$不変関数の一様な$mathcal{C}^k$近似の結果を示す。全対称関数($G = \mathcal{S}_n$)の場合、これはZaheerら(2018)の和分解Deep Sets ansatzをもたらすことを示す。特に、必要な埋め込み次元は、対象関数の正則性、所望の近似精度、$k$に依存しないことを示す。次に、同様の手順で、K$項の和として、逆対称関数の一様な$mathcal{C}^k$近似が得られることを示す。ここで、各項は、滑らかな全対称関数と、最大$binom{n}{2}次数の滑らかな逆対称同次多項式の積である。また、$K$の上下界を与え、$K$が対象関数の正則性、求める近似精度、$k$に依存しないことを示す。

要約(オリジナル)

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 $k$.

arxiv情報

提供元, 利用サービス

arxiv.jp, DeepL