  We present a refined analysis of the classic Count-Sketch streaming heavy
hitters algorithm [CCF02]. Count-Sketch uses O(k log n) linear measurements of
a vector x in R^n to give an estimate x' of x. The standard analysis shows that
this estimate x' satisfies ||x'-x||_infty^2 < ||x_tail||_2^2 / k, where x_tail
is the vector containing all but the largest k coordinates of x. Our main
result is that most of the coordinates of x' have substantially less error than
this upper bound; namely, for any c < O(log n), we show that each coordinate i
satisfies (x'_i - x_i)^2 < (c/log n) ||x_tail||_2^2/k with probability
1-2^{-Omega(c)}, as long as the hash functions are fully independent. This
subsumes the previous bound and is optimal for all c. Using these improved
point estimates, we prove a stronger concentration result for set estimates by
first analyzing the covariance matrix and then using a
median-of-median-of-medians argument to bootstrap the failure probability
bounds. These results also give improved results for l_2 recovery of exactly
k-sparse estimates x^* when x is drawn from a distribution with suitable decay,
such as a power law or lognormal. We complement our results with simulations of
Count-Sketch on a power law distribution. The empirical evidence indicates that
our theoretical bounds give a precise characterization of the algorithm's
performance: the asymptotics are correct and the associated constants are
small. Our proof shows that any symmetric random variable with finite variance
and positive Fourier transform concentrates around 0 at least as well as a
Gaussian. This result, which may be of independent interest, gives good
concentration even when the noise does not converge to a Gaussian.
