The capacity per unit cost, or equivalently minimum cost to transmit one bit,
is a well-studied quantity. It has been studied under the assumption of full
synchrony between the transmitter and the receiver. In many applications, such
as sensor networks, transmissions are very bursty, with small amounts of bits
arriving infrequently at random times. In such scenarios, the cost of acquiring
synchronization is significant and one is interested in the fundamental limits
on communication without assuming a priori synchronization.
We consider the recovery of a nonnegative vector x from measurements y = Ax,
where A is an m-by-n matrix whos entries are in {0, 1}. We establish that when
A corresponds to the adjacency matrix of a bipartite graph with sufficient
expansion, a simple message-passing algorithm produces an estimate \hat{x} of x
satisfying ||x-\hat{x}||_1 \leq O(n/k) ||x-x(k)||_1, where x(k) is the best
k-sparse approximation of x. The algorithm performs O(n (log(n/k))^2 log(k))
computation in total, and the number of measurements required is m = O(k
log(n/k)).
We consider asynchronous point-to-point communication. Building on a recently
developed model, we show that training based schemes, i.e., communication
strategies that separate synchronization from information transmission, perform
suboptimally at high rate.