AI 中文总结
该研究通过构造实例证明多项式时间低次猜想不成立，指出低次不可区分性、均匀分布、置换不变性和独立重采样并不必然导致多项式时间难度。
AI 中文摘要
低次方法及其相关的下界被广泛用于指导算法设计，并为平均情况推断、高维统计、随机优化等相关问题提供计算难度的证据。这导致了低次猜想的提出，该猜想预测当植根分布与均匀空分布之间的低次优势保持有界时，在独立噪声存在的情况下，没有高效的区分器能够成功。几项工作已经产生了对这一猜想变体或对更高时间复杂度算法版本的反例，但该猜想在标准的二进制、多项式时间形式中仍然未被解决。我们通过给出该设定下的一个例子族来驳斥多项式时间低次猜想。对于每个固定的整数r≥3，我们构造了一个在简单图上的置换不变分布P_n，其中Q_n=G(n,1/2)，使得P_n在至多D_n=Θ((log n)^{r-1})条边上的每一个边际都是均匀的。因此，通过度D_n，低次优势为零。然而，在每条边独立重新采样以固定正率的情况下，一个确定性的秩测试可以在多项式时间内区分由此产生的分布与Q_n。该构造选择了一个Reed-Muller码的子空间，其中非零多项式具有小的绝对偏置，选择评估向量没有短线性依赖的点，并在这些向量对上评估一个随机交替双线性形式。我们的结果表明，低次不可区分性、均匀空分布、置换不变性和独立重采样本身并不意味着多项式时间难度，并表明一个有效的普遍猜想必须附加一个额外的条件。
英文摘要
The low-degree method and its associated lower bounds are widely used to guide algorithm design and to provide evidence of computational hardness in average-case inference, high-dimensional statistics, random optimization, and related problems. This led to the low-degree conjecture, which predicts that when the low-degree advantage between a planted distribution and a uniform null distribution remains bounded, no efficient distinguisher can succeed after independent noise, provided that the planted distribution has permutation symmetry. Several works have produced counterexamples to variants of this conjecture or to versions for algorithms with higher time complexity, but the conjecture remained open in its standard binary, polynomial-time formulation. We disprove the polynomial-time low-degree conjecture by giving a family of examples in this setting. For every fixed integer $r\geq3$, we construct a permutation-invariant distribution $\mathbb{P}_n$ on simple graphs, with $\mathbb{Q}_n=G(n,1/2)$, such that every marginal of $\mathbb{P}_n$ on at most $D_n=Θ((\log n)^{r-1})$ edges is uniform. Therefore, the low-degree advantage is zero through degree $D_n$. Nevertheless, after every edge is independently resampled at a fixed positive rate, a deterministic rank test strongly distinguishes the resulting distribution from $\mathbb{Q}_n$ in polynomial time. The construction chooses a subspace of a Reed--Muller code whose nonzero polynomials have small absolute bias, selects points whose evaluation vectors have no short linear dependencies, and evaluates a random alternating bilinear form on pairs of these vectors. Our result shows that low-degree indistinguishability, a uniform null distribution, permutation invariance, and independent resampling do not by themselves imply polynomial-time hardness, and suggests that a valid general conjecture must impose an additional condition.
Comments17 pages
