Johnson-Lindenstrauss引理几乎刻画了希尔伯特空间，但并不完全是
The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite
摘要：
设$X$是满足Johnson-Lindenstrauss引理（简称J-L引理）的赋范空间，在这个意义上，对于任何整数$n$和任何$x_1,\ldots,x_n\in X$，存在线性映射$L:X\to F$，其中$F\subseteq X$是维数为$O(\log n)$的线性子空间，使得对于所有$i,j\in \{1,\ldots, n\}$，$\|x_i-x_j\|\le\|L(x_i)-L(x_j)\|\le O(1)\cdot\|x_i-x_j\|$。我们证明了这意味着$X$在以下意义上几乎是欧几里德的：$X$的每个$n$维子空间嵌入到希尔伯特空间中，失真为$2^{2^{O(\log^*n)}}$。另一方面，我们证明了存在一个满足J-L引理的赋范空间$Y$，但对于每个$n$，存在一个$n$维子空间$E_n\subseteq Y$，其欧氏失真至少为$2^{Ω(α(n))}$，其中$α$是逆Ackermann函数.
英文摘要：
Let $X$ be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer $n$ and any $x_1,\ldots,x_n\in X$ there exists a linear mapping $L:X\to F$, where $F\subseteq X$ is a linear subspace of dimension $O(\log n)$, such that $\|x_i-x_j\|\le\|L(x_i)-L(x_j)\|\le O(1)\cdot\|x_i-x_j\|$ for all $i,j\in \{1,\ldots, n\}$. We show that this implies that $X$ is almost Euclidean in the following sense: Every $n$-dimensional subspace of $X$ embeds into Hilbert space with distortion $2^{2^{O(\log^*n)}}$. On the other hand, we show that there exists a normed space $Y$ which satisfies the J-L lemma, but for every $n$ there exists an $n$-dimensional subspace $E_n\subseteq Y$ whose Euclidean distortion is at least $2^{Ω(α(n))}$, where $α$ is the inverse Ackermann function.
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