Model-Based Multichannel Blind Deconvolution: Mathematical Analysis and Nonconvex Optimization Algorithms

Applied and Computational Mathematics Seminar
Monday, January 22, 2018 - 13:55
1 hour (actually 50 minutes)
Skiles 005
There are numerous modern applications in data science that involve inference from incomplete data. Various geometric prior models such as sparse vectors or low-rank matrices have been employed to address the ill-posed inverse problems arising in these applications. Recently, similar ideas were adopted to tackle more challenging nonlinear inverse problems such as phase retrieval and blind deconvolution. In this talk, we consider the blind deconvolution problem where the desired information as a time series is accessed as indirect observations through a time-invariant system with uncertainty. The measurements in this case is given in the form of the convolution with an unknown kernel. Particularly, we study the mathematical theory of multichannel blind deconvolution where we observe the output of multiple channels that are all excited with the same unknown input source. From these observations, we wish to estimate the source and the impulse responses of each of the channels simultaneously. We show that this problem is well-posed if the channel impulse responses follow a simple geometric model.  Under these models, we show how the channel estimates can be found by solving corresponding non-convex optimization problems. We analyze methods for solving these non-convex programs, and provide performance guarantees for each.