论文标题
部分可观测时空混沌系统的无模型预测
Covering of high-dimensional sets
论文作者
论文摘要
令$（\ Mathcal {x}，ρ）$为公制空间，$λ$为在此空间上定义的borel量度，该空间是由$ \ Mathcal {x} $的开放子集生成的$σ$ -Algebra;此度量$λ$定义了$ \ Mathcal {x} $的Borel子集的量。主要情况是$ \ Mathcal {x} = \ Mathbb {r}^d $，$ρ$是Euclidean Metric，而$λ$是Lebesgue度量。在本文中，我们不会对小尺寸$ d $的情况非常关注，因为蛮力优化算法的构建良好覆盖方案的构建问题可能会攻击。相反，对于中等或大的维度（例如，$ d \ geq 10 $），如果不了解与高效覆盖设计有关的主要问题，那么几乎没有机会获得任何明智的东西。
Let $(\mathcal{X},ρ)$ be a metric space and $λ$ be a Borel measure on this space defined on the $σ$-algebra generated by open subsets of $\mathcal{X}$; this measure $λ$ defines volumes of Borel subsets of $\mathcal{X}$. The principal case is where $\mathcal{X} = \mathbb{R}^d$, $ρ$ is the Euclidean metric, and $λ$ is the Lebesgue measure. In this article, we are not going to pay much attention to the case of small dimensions $d$ as the problem of construction of good covering schemes for small $d$ can be attacked by the brute-force optimization algorithms. On the contrary, for medium or large dimensions (say, $d\geq 10$), there is little chance of getting anything sensible without understanding the main issues related to construction of efficient covering designs.
