On Sampling Lower Bounds for Polynomials

2026-05-01

AI Summary

This work investigates whether samplers defined by constant-degree polynomials over $\mathbb{F}_2$ can effectively approximate the product distribution $\mathrm{Ber}(1/3)^{\otimes n}$. By analyzing the total variation distance between the target distribution and the output of low-degree polynomial maps applied to uniformly random inputs from $\mathbb{F}_2^m$, the paper establishes strong lower bounds for any constant degree $d$: specifically, the distance approaches 1 with bounds exponential in $n$ for $d=1$, $\log n / \log\log n$ for $d=2$, and $\sqrt{\log\log n}$ for $d=3$. The proof hinges on structural theorems for low-degree polynomials, leveraging the fact that biased degree-$d$ polynomials can be expressed as functions of few degree-$(d-1)$ polynomials, thereby revealing an inherent limitation preventing their output distributions from closely approximating bias $1/3$.

Abstract

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 $d-1$.

Problem

• sampling lower bounds
• low-degree polynomials
• total variation distance
• distribution complexity
• Bernoulli product distribution

Innovation

• polynomial sampling
• total variation distance
• low-degree polynomials
• distribution complexity
• Boolean functions