Optimal compression of approximate inner products and dimension reduction

Abstract: Let $X$ be a set of $n$ points of norm at most $1$ in the Euclidean space $R^k$, and suppose $\varepsilon>0$. An $\varepsilon$-distance sketch for $X$ is a data structure that, given any two points of $X$ enables one to recover the square of the (Euclidean) distance between them up to an {\em additive} error of $\varepsilon$. Let $f(n,k,\varepsilon)$ denote the minimum possible number of bits of such a sketch. Here we determine $f(n,k,\varepsilon)$ up to a constant factor for all $n \geq k \geq 1$ and all $\varepsilon \geq \frac{1}{n^{0.49}}$. Our proof is algorithmic, and provides an efficient algorithm for computing a sketch of size $O(f(n,k,\varepsilon)/n)$ for each point, so that the square of the distance between any two points can be computed from their sketches up to an additive error of $\varepsilon$ in time linear in the length of the sketches. We also discuss the case of smaller $\varepsilon>2/\sqrt n$ and obtain some new results about dimension reduction in this range. In particular, we show that for any such $\varepsilon$ and any $k \leq t=\frac{\log (2+\varepsilon^2 n)}{\varepsilon^2}$ there are configurations of $n$ points in $R^k$ that cannot be embedded in $R^{\ell}$ for $\ell < ck$ with $c$ a small absolute positive constant, without distorting some inner products (and distances) by more than $\varepsilon$. On the positive side, we provide a randomized polynomial time algorithm for a bipartite variant of the Johnson-Lindenstrauss lemma in which scalar products are approximated up to an additive error of at most $\varepsilon$. This variant allows a reduction of the dimension down to $O(\frac{\log (2+\varepsilon^2 n)}{\varepsilon^2})$, where $n$ is the number of points.