In this work, we continue the line of research on the complexity of distributions (Viola, Journal of Computing 2012), and study samplers defined by low degree polynomials. An $n$-tuple $P = (P_1,\dots, P_n)$ of functions $P_i \colon \mathbb{F}_2^m \to \mathbb{F}_2$ defines a distribution over $\{0,1\}^n$ in the natural way: draw $X$ uniformly at random from $\mathbb{F}_2^m$ and output $(P_1(X),\dots, P_n(X)) \in \{0,1\}^n$. We show that when $P$ is defined by polynomials of degree $d$, the total variation distance of $P$ from the product distribution $\mathrm{Ber}(1/3)^{\otimes n}$ is $1-o_n(1)$, where $o_n(1)$ is a vanishing function of $n$ for any constant degree $d$. For small values of $d$, we show the following concrete bounds. (i) For $d=1$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-Ω(n))$. (ii) For $d=2$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-Ω(\log(n)/\log\log(n)))$. (iii) For $d=3$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-Ω(\sqrt{\log\log(n)}))$. Our results extend the recent lower bound results for sampling distributions, which have mostly focused on local samplers, small depth decision trees, and small depth circuits. As part of our proof, we establish the following result, that may be of independent interest: for any degree-$d$ polynomial $P\colon\mathbb{F}_2^m \to \mathbb{F}_2$ it holds that $\Pr_X[P(X) = 1]$ is bounded away from $1/3$ by some absolute constant $δ= δ_d>0$. Although the statement may seem obvious, we are not aware of an elementary proof of this. The proof techniques rely on the structural results for low degree polynomials, saying that any biased polynomial of degree $d$ can be written as a function of a small number of polynomials of degree $d-1$.

翻译：在本工作中，我们延续了关于分布复杂性的研究路线（Viola, Journal of Computing 2012），并研究了由低次多项式定义的采样器。函数 $P_i \colon \mathbb{F}_2^m \to \mathbb{F}_2$ 构成的 $n$ 元组 $P = (P_1,\dots, P_n)$ 以自然方式定义了 $\{0,1\}^n$ 上的一个分布：从 $\mathbb{F}_2^m$ 中均匀随机抽取 $X$，输出 $(P_1(X),\dots, P_n(X)) \in \{0,1\}^n$。我们证明，当 $P$ 由次数为 $d$ 的多项式定义时，$P$ 与乘积分布 $\mathrm{Ber}(1/3)^{\otimes n}$ 之间的全变差距离为 $1-o_n(1)$，其中对于任意常数 $d$，$o_n(1)$ 是 $n$ 的消失函数。对于较小的 $d$ 值，我们给出了以下具体界值。(i) 当 $d=1$ 时，有 $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-Ω(n))$。(ii) 当 $d=2$ 时，有 $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-Ω(\log(n)/\log\log(n)))$。(iii) 当 $d=3$ 时，有 $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-Ω(\sqrt{\log\log(n)}))$。我们的结果扩展了近期关于采样分布的下界结果，这些结果主要集中于局部采样器、小深度决策树和小深度电路。作为证明的一部分，我们建立了以下可能具有独立意义的结果：对于任意次数为 $d$ 的多项式 $P\colon\mathbb{F}_2^m \to \mathbb{F}_2$，有 $\Pr_X[P(X) = 1]$ 被某个绝对常数 $δ= δ_d>0$ 限制在 $1/3$ 之外。尽管这一陈述看似显然，但我们尚未找到其初等证明。证明技术依赖于低次多项式的结构结果，即任何次数为 $d$ 的有偏多项式都可以写成少量次数为 $d-1$ 的多项式的函数。