Codes and Expansions (CodEx) Seminar

Felix Krahmer (Technische Universität München)
The convex geometry of blind deconvolution revisited

Blind deconvolution problems are ubiquitous in many areas of imaging and technology and have been the object of study for several decades. Recently, motivated by the theory of compressed sensing, a new viewpoint has been introduced, motivated by applications in wireless application, where a signal is transmitted through an unknown channel. Namely, the idea is to randomly embed the signal into a higher dimensional space before transmission. Due to the resulting redundancy, one can hope to recover both the signal and the channel parameters. In this talk we analyze convex approaches based on lifting as they have first been studied by Ahmed et al. (2014). We show that one encounters a fundamentally different geometric behavior as compared to generic bilinear measurements. Namely, for very small levels of deterministic noise, the error bounds based on common paradigms can only scale linearly in the noise level if one admits a constant that grows with the dimension. As an alternative, we establish a bound for the reconstruction error that initially scales like the square root of the noise level and eventually transitions to a linear scaling, but without additional dimensional constants.

This is joint work with Yulia Kostina (TUM) and Dominik Stöger (KU Eichstätt-Ingolstadt).