Bilinear lifting for applications in signal processing
We will start with an overview of recent results on convex relaxations for solving systems of bilinear equations. We will show how certain problems that are prevalent in imaging (blind deconvolution, auto-calibration) can be recast as low rank recovery problems, and discuss the theoretical conditions under which these problems can be practically solved. We will show how this theory applies to problems including coil calibration in MRI, blind deblurring in coded imaging, and passive acoustic imaging. We will also discuss recent results on estimating matrices that are simultaneously sparse a low rank, and their applications to coded imaging, phase retrieval, and sparse covariance estimation.