Mathematics – Functional Analysis
Scientific paper
2008-07-11
Mathematics
Functional Analysis
Scientific paper
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 $2^{\Omega(\alpha(n))}$, where $\alpha$ is the inverse Ackermann function.
Johnson William B.
Naor Assaf
No associations
LandOfFree
The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-576413