Recovering Structured Data From Superimposed Non-Linear Measurements

Abstract

This work deals with the problem of distributed data acquisition under non-linear communication constraints. More specifically, we consider a model setup where M distributed nodes take individual measurements of an unknown structured source vector x 0 ∈ R n , communicating their readings simultaneously to a central receiver. Since this procedure involves collisions and is usually imperfect, the receiver measures a superposition of non-linearly distorted signals. In a first step, we will show that an s-sparse vector x 0 can be successfully recovered from O(s · log(2n/s)) of such superimposed measurements, using a traditional Lasso estimator that does not rely on any knowledge about the non-linear corruptions. This direct method however fails to work for several "uncalibrated" system configurations. These blind reconstruction tasks can be easily handled with the 1,2 -Group Lasso, but coming along with an increased sampling rate of O(s · max{M, log(2n/s)}) observations-in fact, the purpose of this lifting strategy is to extend a certain class of bilinear inverse problems to non-linear acquisition. Apart from that, we shall also demonstrate how our two algorithmic approaches can be embedded into a more flexible and adaptive recovery framework which allows for sub-Gaussian measurement designs as well as general (convex) structural constraints. Finally, to illustrate the practical scope of our theoretical findings, an application to wireless sensor networks is discussed, which actually serves as the prototype example of our observation model.

Sharp Guarantees for Solving Random Equations with One-Bit Information

“…As already mentioned in Remark 3, the set of minimizers of the hinge-loss is bounded (required by Theorem III.1) only for δ > δ ε where δ ε is the value of the threshold in (15). Our numerical simulations in Figures 3 and 4 suggest Figure 1).…”

“…In particular, Corollary IV.1 is a special case of the main theorem in [29]. Several other interesting extensions of the result by Plan and Vershynin have recently appeared in the literature, e.g., [14], [16], [15], [32]. However, [29] is the only one to give results that are sharp in the flavor of this paper.…”

“…To be precise,[7] prove the statement for measurements y i , i ∈ [m] that follow a logistic model. Close inspection of their proof shows that this requirement can be relaxed by appropriately defining the random variable Y in(15).…”

Analysis of hard-thresholding for distributed compressed sensing with one-bit measurements

“…We remark that we recently became aware of the related work [15]. The authors examine recovery of structured data from superimposed non-linear measurements by group-LASSO, i.e., 2,1 -minimization.…”

“…The different analysis methods of both works are complementary, and both serve to better understand distributed compressed sensing with one-bit measurements. Our approach has the advantage of brief and elementary proofs, while the approach of [15] might give improved error bounds using more sophisticated tools.…”