  Counting is among the most fundamental operations in computing. For example,
counting the pth frequency moment has been a very active area of research, in
theoretical computer science, databases, and data mining. When p=1, the task
(i.e., counting the sum) can be accomplished using a simple counter.
  Compressed Counting (CC) is proposed for efficiently computing the pth
frequency moment of a data stream signal A_t, where 0<p<=2. CC is applicable if
the streaming data follow the Turnstile model, with the restriction that at the
time t for the evaluation, A_t[i]>= 0, which includes the strict Turnstile
model as a special case. For natural data streams encountered in practice, this
restriction is minor.
  The underly technique for CC is what we call skewed stable random
projections, which captures the intuition that, when p=1 a simple counter
suffices, and when p = 1+/\Delta with small \Delta, the sample complexity of a
counter system should be low (continuously as a function of \Delta). We show at
small \Delta the sample complexity (number of projections) k = O(1/\epsilon)
instead of O(1/\epsilon^2).
  Compressed Counting can serve a basic building block for other tasks in
statistics and computing, for example, estimation entropies of data streams,
parameter estimations using the method of moments and maximum likelihood.
  Finally, another contribution is an algorithm for approximating the
logarithmic norm, \sum_{i=1}^D\log A_t[i], and logarithmic distance. The
logarithmic distance is useful in machine learning practice with heavy-tailed
data.
