  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 $\boldsymbol{B}$ and $\boldsymbol{C}$ from a
noisy measurement of $\boldsymbol{Z}=\boldsymbol{BC}$, 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.
