Parametric Bilinear Generalized Approximate Message Passing

We propose a scheme to estimate the parameters b(i) and c(j) of the bilinear form z(m) = Sigma(i,j) b(i)z(m)((i,j))c(j) from noisy measurements {y(m)}(m=1)(M), where y(m) and z(m) are related through an arbitrary likelihood function and z(m)((i,j)) are known. Our scheme is based on generalized approximate message passing (G-AMP): it treats b(i) and c(j) as random variables and z(m)((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.