Streaming Symmetric Norms via Measure Concentration

Understand

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.