Bilinear Inversion Theory for Self-calibration and Security – BITS2

Bilinear inverse problems (BIPs) are ubiquitous in engineering and applied sciences as they are naturally fit to model unknown linear systems. Nonetheless, BIPs come with many statistical and algorithmic challenges. They demand stringent structural assumptions on each variable for identifiability, such as sparsity or low-rank model, and are recoverable only up to an unresolvable ambiguity class. Theoretical guarantees for BIPs are derived today on a case-by-case basis, and algorithmic approaches rely on discretizing the input variables, yielding to sub-optimal solutions, and reconstructing spurious artifacts. Moreover, noise stability remains widely unexplored.

Of particular interest in imaging and telecommunication is the blind super-resolution problem, a continuous BIP with sparse input prior and where the bilinear map is convolutive, as those systems are generally assumed to be shift-invariant. The projects will address the identifiability and stability of the blind deconvolution problem under different sampling schemes, such as direct sampling, Fourier measurement, or time-frequency samples. Specifically, a comprehensive theory of the Rayleigh limit will be investigated in the bilinear settings. Furthermore, efficient algorithms capable of solving BIPs without relying on discretization will be investigated.

The BITS2 project has two main application focuses:
1) The realization of self-calibrating imaging and sensing systems, enabling an experimenter to skip the tedious task of calibrating measurement devices by harnessing the diversity of multi-channel observations.
2) The prototyping of BIPs-based physical layer security protocols to protect both the messages and physical parameters of the wireless transmitter, such as its physical position.

Finally, the mathematical analyses and tools developed in the BITS2 project are anticipated to foster the interplay between a broad range of fields, including harmonic analysis, signal processing, telecommunication, machine learning, control engineering, and operation research.

The project includes the organization of a workshop on “structured signal processing for data privacy.”