论文标题
部分可观测时空混沌系统的无模型预测
The first eigenvector of a distance matrix is nearly constant
论文作者
论文摘要
令$ x_1，\ dots，x_n $是公制空间中的点，并定义距离矩阵$ d \ in \ mathbb {r}^{n \ times n} $ by $ {d} _ {ij {ij} = d（x_i，x_jj）$。 Perron-Frobenius定理意味着\ Mathbb {r}^n _ {} $具有与最大特征值相关的非阴性条目中的特征向量$ v \。我们证明，从\ Mathbb {r}^n $中的内部产品$ \ mathbb {1} \ in \ mathbb {r}^n $是大的$$ \ left \ left \ langle v，\ mathbb {1} \ rangle \ rangle \ geq \ geq \ geq \ frac} c c c c c c c c c c cd， \ | v \ | _ {\ ell^2} \ cdot \ | \ m马理{1} \ | _ {\ ell^2} $$，每个条目都满足$ v_i \ geq \ geq \ | | _ | _ {\ ell^ell^ell^2}/\ sqrt {4n} $。两种不平等都很尖锐。
Let $x_1, \dots, x_n$ be points in a metric space and define the distance matrix $D \in \mathbb{R}^{n \times n}$ by ${D}_{ij} = d(x_i, x_j)$. The Perron-Frobenius Theorem implies that there is an eigenvector $v \in \mathbb{R}^n_{}$ with non-negative entries associated to the largest eigenvalue. We prove that this eigenvector is nearly constant in the sense that the inner product with the constant vector $\mathbb{1} \in \mathbb{R}^n$ is large $$ \left\langle v, \mathbb{1} \right\rangle \geq \frac{1}{\sqrt{2}} \cdot \| v\|_{\ell^2} \cdot \|\mathbb{1} \|_{\ell^2}$$ and that each entry satisfies $v_i \geq \|v\|_{\ell^2}/\sqrt{4n}$. Both inequalities are sharp.
