Parametric Bilinear Generalized Approximate Message Passing

We propose a scheme to estimate the parameters $b_i$ and $c_j$ of the bilinear form $z_m={\sum }_{i,j}{b}_i{z}_m^{(i,j)}{c}_j$ 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 $\mathit{B}$ and $\mathit{C}$ from a noisy measurement of $\mathit{Z}=$, 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.