Parametric Bilinear Generalized Approximate Message Passing

We propose a scheme to estimate the parameters bi and cj of the bilinear form zm= Σi,jbizm(i,j)cjfrom noisy measurements {ym}mM=1, where ymand zmare related through an arbitrary likelihood function and zm(i,j)are known. Our scheme is based on generalized approximate message passing (G-AMP): it treats biand cjas random variables and zm(i,j)as an i.i.d. Gaussian 3-way tensor in order to derive a tractable simplification of the sum-product algorithm in the large-system limit. It generalizes previous instances of bilinear G-AMP, such as those that estimate matrices B and C from a noisy measurement of Z = BC, allowing the application of AMP methods to problems such as self-calibration, blind deconvolution, and matrix compressive sensing. Numerical experiments confirm the accuracy and computational efficiency of the proposed approach.