Blind sparse recovery from superimposed non-linear sensor measurements

In this work, we study the problem of sparse recovery from superimposed, non-linearly distorted measurements. This challenge is particularly relevant to wireless sensor networks that consist of autonomous and spatially distributed sensor units. Here, each of the M wireless sensors acquires m individual measurements of an s-sparse source vector x
₀
∈ ℝ
ⁿ
. All devices transmit simultaneously to a central receiver, causing collisions. Since this process is imperfect, e.g., caused by low-quality sensors and the wireless channel, the receiver measures a superposition of corrupted signals. First, we will show that the source vector can be successfully recovered from m = O(s log(2n/s)) coherently communicated measurements via the vanilla Lasso. The more general situation of non-coherent communication can be approximated by a bilinear compressed sensing problem. Even in the non-linear setting, it will turn out that m = O(s · max{M, log(2n/s)}) measurements are already sufficient for reconstruction using the (group) ℓ
^1,2
-Lasso. In particular, as long as M = O(log(2n/s)) sensors are used, there is no substantial increase in performance when building a coherently communicating network. Finally, we shall discuss several practical implications and extensions of our approach.