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   ${B}$   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.
['Jason T. Parker', 'Philip Schniter']
Parametric Bilinear Generalized Approximate Message Passing