# Johnson and Lindenstrauss lemma for hamming space

A result of Johnson and Lindenstrauss shows that a set of $n$ points in high dimensional Euclidean space can be mapped into an $O(\frac{\log n}{\epsilon^2})$- dimensional Euclidean space such that the distance between any two points changes by only a factor of $(1\pm \epsilon).$

I am looking for reference for a similar result for Hamming space- i.e. a set of $n$ binary vectors can be mapped into a very low dimensional space such that the hamming distance between any pair of vectors is approximately preserved.

## 1 Answer

Consider the $n$ vectors $e_1,\ldots,e_n$ of weight $1$, and the zero vector $e_0$. The Hamming distance between $e_0$ to any $e_i$ is $1$. Let $\varphi$ be a map into a Hamming cube of dimension $m$, and suppose that it preserves distances up to a factor of $C$. In particular, the $n$ points $\varphi(e_1),\ldots,\varphi(e_n)$ are within distance $C$ of $\varphi(e_0)$. On the other hand, the number of points at distance at most $C$ from $\varphi(e_0)$ is $O(m^C)$. We conclude that $m = \Omega(n^{1/C})$ is polynomial in $n$ rather than logarithmic in $n$.

• At this point, I think you should be able to read the paper yourself and compare your perception of it to the problem highlighted in my answer. If you need someone to talk to about your research, I suggest finding someone local. This site is not the correct forum for such discussions. – Yuval Filmus Apr 5 '16 at 8:40