  We develop a framework that we call compressive rate estimation. We assume
that the composite channel gain matrix (i.e. the matrix of all channel gains
between all network nodes) is compressible which means it can be approximated
by a sparse or low rank representation. We develop and study a novel sensing
and reconstruction protocol for the estimation of achievable rates. We develop
a sensing protocol that exploits the superposition principle of the wireless
channel and enables the receiving nodes to obtain non-adaptive random
measurements of columns of the composite channel matrix. The random
measurements are fed back to a central controller that decodes the composite
channel gain matrix (or parts of it) and estimates individual user rates. We
analyze the rate loss for a linear and a non-linear decoder and find the
scaling laws according to the number of non-adaptive measurements. In
particular if we consider a system with $N$ nodes and assume that each column
of the composite channel matrix is $k$ sparse, our findings can be summarized
as follows. For a certain class of non-linear decoders we show that if the
number of pilot signals $M$ scales like $M \sim k \log(N/k)$, then the rate
loss compared to perfect channel state information remains bounded. For a
certain class of linear decoders we show that the rate loss compared to perfect
channel state information scales like $1/\sqrt{M}$.
