Streaming Symmetric Norms via Measure Concentration

Abstract

We characterize the streaming space complexity of every symmetric norm $l$ (a norm on $\mathbb{R}^n$ invariant under sign-flips and coordinate-permutations), by relating this space complexity to the measure-concentration characteristics of $l$. Specifically, we provide nearly matching upper and lower bounds on the space complexity of calculating a $(1\pm\epsilon)$-approximation to the norm of the stream, for every $0<\epsilon\leq 1/2$. (The bounds match up to $poly(\epsilon^{-1} \log n)$ factors.) We further extend those bounds to any large approximation ratio $D\geq 1.1$, showing that the decrease in space complexity is proportional to $D^2$, and that this factor the best possible. All of the bounds depend on the median of $l(x)$ when $x$ is drawn uniformly from the $l_2$ unit sphere. The same median governs many phenomena in high-dimensional spaces, such as large-deviation bounds and the critical dimension in Dvoretzky's Theorem. The family of symmetric norms contains several well-studied norms, such as all $l_p$~norms, and indeed we provide a new explanation for the disparity in space complexity between $p\le 2$ and $p>2$. In addition, we apply our general results to easily derive bounds for several norms that were not studied before in the streaming model, including the top-$k$ norm and the $k$-support norm, which was recently employed for machine learning tasks. Overall, these results make progress on two outstanding problems in the area of sublinear algorithms (Problems 5 and 30 in~\url{http://sublinear.info}).