论文标题
高尺寸立方体的有效量化和较弱的覆盖率
Efficient quantization and weak covering of high dimensional cubes
论文作者
论文摘要
令$ \ mathbb {z} _n = \ {z_1，\ ldots，z_n \} $为design;也就是说，在[-1,1]^d $中的$ n $点$ z_j \的集合。我们研究了$ [-1,1]^d $的量化质量，该点的点点数的点数是$ {\ cal b} _d（\ cal b} _d（\ mathbb {z} _n，r）$ z_ $ z_的$ [\ cal b} _d（\ cal b} _d（\ mathbb {z} _n，r），我们专注于尺寸$ d $不小的情况（$ d \ geq 5 $），而$ n $不太大，$ n \ leq 2^d $。我们将设计$ {\ mathbb {d} _ {n，δ}} $定义为$ 2^{d-1} $设计在Cube $ [ - δ，δ]^d $，$ 0 \leqΔ\ leq leq 1 $的顶点上定义的。对于此设计，我们为{覆盖面积} vol $（[ - 1,1]^d \ cap {\ cal b} _d（\ mathbb {z} _n，r））$（{ - 1,1]^d \ cap {\ cal b} _d（\ cal cap {\ cal b} _d（\ cal cap {_n，r））的封闭式表达式和非常准确的近似值。我们提供了大规模数字研究的结果，确认了开发近似值的准确性以及设计的效率$ {\ Mathbb {d} _ {n，δ}} $。
Let $\mathbb{Z}_n = \{Z_1, \ldots, Z_n\}$ be a design; that is, a collection of $n$ points $Z_j \in [-1,1]^d$. We study the quality of quantization of $[-1,1]^d$ by the points of $\mathbb{Z}_n$ and the problem of quality of coverage of $[-1,1]^d$ by ${\cal B}_d(\mathbb{Z}_n,r)$, the union of balls centred at $Z_j \in \mathbb{Z}_n$. We concentrate on the cases where the dimension $d$ is not small ($d\geq 5$) and $n$ is not too large, $n \leq 2^d$. We define the design ${\mathbb{D}_{n,δ}}$ as a $2^{d-1}$ design defined on vertices of the cube $[-δ,δ]^d$, $0\leq δ\leq 1$. For this design, we derive a closed-form expression for the quantization error and very accurate approximations for {the coverage area} vol$([-1,1]^d \cap {\cal B}_d(\mathbb{Z}_n,r))$. We provide results of a large-scale numerical investigation confirming the accuracy of the developed approximations and the efficiency of the designs ${\mathbb{D}_{n,δ}}$.
