论文标题
部分可观测时空混沌系统的无模型预测
A degree 4 sum-of-squares lower bound for the clique number of the Paley graph
论文作者
论文摘要
我们证明，Paley图的集团数量的4度（SOS）放松在Prime Number $ p $的顶点上的数字至少具有$ω（p^{1/3}）$。这与普遍认为的猜想相反，即Paley图的实际集团数为$ O（\ Mathrm {polylog}（p））$。我们的结果可能被视为Deshpande和Montanari（2015）的降低，他们在$ p $ pertices上显示出相同的下限（最高$ \ mathrm {polylog}（p）$），对$ p $ pertices上的Erdős-rényi随机图的可能性很高，其clique number具有很高的概率$ o（log o o（pog o o（p））$（p）$（p）$（p）$（p）。我们还表明，我们的下界对假单胞菌的Feige-Krauthgamer结构是最佳的，这使Kelner的论点降低了。最后，我们提出数字实验，表明paley图的度量4 SOS松弛的值可能会缩放为$ O（p^{1/2-ε}）$对于某些$ε> 0 $，并且给出矩阵标准计算，并给出一个矩阵规范计算，表明对随机图的sos降低范围的较低限制策略不会立即传输到Paley图。综上所述，我们的结果表明，对于Paley图的集团数量，SOS可能会破坏“ $ \ sqrt {p} $屏障”，但证明它最多可以将指数从$ 1/2 $提高到$ 1/3 $。
We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number $p$ of vertices has value at least $Ω(p^{1/3})$. This is in contrast to the widely believed conjecture that the actual clique number of the Paley graph is $O(\mathrm{polylog}(p))$. Our result may be viewed as a derandomization of that of Deshpande and Montanari (2015), who showed the same lower bound (up to $\mathrm{polylog}(p)$ terms) with high probability for the Erdős-Rényi random graph on $p$ vertices, whose clique number is with high probability $O(\log(p))$. We also show that our lower bound is optimal for the Feige-Krauthgamer construction of pseudomoments, derandomizing an argument of Kelner. Finally, we present numerical experiments indicating that the value of the degree 4 SOS relaxation of the Paley graph may scale as $O(p^{1/2 - ε})$ for some $ε> 0$, and give a matrix norm calculation indicating that the pseudocalibration proof strategy for SOS lower bounds for random graphs will not immediately transfer to the Paley graph. Taken together, our results suggest that degree 4 SOS may break the "$\sqrt{p}$ barrier" for upper bounds on the clique number of Paley graphs, but prove that it can at best improve the exponent from $1/2$ to $1/3$.
