Identifiability for Bilinear Inverse Problems with (Eventual) Applications to Blind Deconvolution, Matrix Factorization and Dictionary Finding
Motivated by channel estimation problems in underwater acoustic communications, sparse channel estimation is investigated. For a particular problem relevant to cooperative communications, we see that our estimation problem of interest can be mapped to a bilinear inverse problem. In fact, a number of important inverse problems in signal processing, such as blind deconvolution, matrix factorization, dictionary learning and blind source separation share the common characteristic of being bilinear inverse problems. In such problems, the observation model is a function of two variables and conditioned on one variable being known, the observation is a linear function of the other variable. A key question is that of identifiability: can one unambiguously recover the pair of inputs from the output? We shall consider both deterministic conditions for identifiability as well as probabilistic statements that result in new scaling laws under cone constraints. Our approach is based on a unifying and flexible approach to identifiability that exploits a connection to low-rank matrix recovery.