Tight Lower Bound for Linear Sketches of Moments

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 prm th moment, for p∈(0,2] has been settled [KNW10], for p>2 the exact complexity remains open. For p>2 the current best algorithm uses O(n1-2/plog n) words of space [AKO11,BO10], whereas the lower bound is of Omega(n1-2/p) [BJKS04]. In this paper, we show a tight lower bound of Omega(n1-2/plog 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.