论文标题
高，更高甚至更高维度的矩阵的测量浓度效应
A measure concentration effect for matrices of high, higher, and even higher dimension
论文作者
论文摘要
令$ n> m $，让$ a $为$（M \ times n）$ - 全等级的矩阵。然后，显然，估计$ \ | ax \ | \ leq \ | a \ | \ | x \ | $持有$ x $和$ ax $的欧几里得规范，以及作为分配的矩阵标准的频谱规范。我们研究了所有$ x $的集合，对于固定$Δ<1 $，相反，$ \ | ax \ | \ | \geqΔ\，\ | a \ | \ | \ | x \ | $ holds。事实证明，在高维情况下，一旦$δ$降至界限以下，这些集合几乎是完整的空间，这取决于$ a $的极端单数值和尺寸比率。这种效果与随机投影定理有很大关系，该定理在数据科学中起着重要作用。作为副产品，我们计算该定理准确处理的概率。
Let $n>m$, and let $A$ be an $(m\times n)$-matrix of full rank. Then obviously the estimate $\|Ax\|\leq\|A\|\|x\|$ holds for the euclidean norm of $x$ and $Ax$ and the spectral norm as the assigned matrix norm. We study the sets of all $x$ for which, for fixed $δ<1$, conversely $\|Ax\|\geqδ\,\|A\|\|x\|$ holds. It turns out that these sets fill, in the high-dimensional case, almost the complete space once $δ$ falls below a bound that depends on the extremal singular values of $A$ and on the ratio of the dimensions. This effect has much to do with the random projection theorem, which plays an important role in the data sciences. As a byproduct, we calculate the probabilities this theorem deals with exactly.
