Compressed sensing on the image of bilinear maps

摘要

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 O((S + F) log N) random sub-Gaussian measurements. Thus, in this case, the number of degrees of freedom of each output grows only additively instead of multiplicatively with the input dimensions (sparsity) S and F. This is a relevant improvement in the output compressibility and suggests a substantially reduced rate in compressed sampling algorithms.