Compressed Sensing on the Image of Bilinear Maps

Abstract

For several communication models, the dispersive part of a communication channel is described by a bilinear operation $T$ between the possible sets of input signals and channel parameters. The received channel output has then to be identified from the image $T(X,Y)$ of the input signal difference sets $X$ and the channel state sets $Y$. The main goal in this contribution is to characterize the compressibility of $T(X,Y)$ with respect to an ambient dimension $N$. In this paper we show that a restricted norm multiplicativity of $T$ on all canonical subspaces $X$ and $Y$ with dimension $S$ resp. $F$ is sufficient for the reconstruction of output signals with an overwhelming probability from $\mathcal{O}((S+F)\log N)$ random sub-Gaussian measurements.