Tight Lower Bound for Linear Sketches of Moments

Abstract: The problem of estimating frequency moments of a data stream has attracted a lot of attention since the onset of streaming algorithms [AMS99]. While the space complexity for approximately computing the $p^{\rm th}$ moment, for $p\in(0,2]$ has been settled [KNW10], for $p>2$ the exact complexity remains open. For $p>2$ the current best algorithm uses $O(n^{1-2/p}\log n)$ words of space [AKO11,BO10], whereas the lower bound is of $\Omega(n^{1-2/p})$ [BJKS04]. In this paper, we show a tight lower bound of $\Omega(n^{1-2/p}\log n)$ words for the class of algorithms based on linear sketches, which store only a sketch $Ax$ of input vector $x$ and some (possibly randomized) matrix $A$. We note that all known algorithms for this problem are linear sketches.