We propose a scheme to estimate the parameters $b_i$ and $c_j$ of thebilinear 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 arbitrarylikelihood function and $z_m^{(i,j)}$ are known. Our scheme is based ongeneralized approximate message passing (G-AMP): it treats $b_i$ and $c_j$ asrandom variables and $z_m^{(i,j)}$ as an i.i.d.\ Gaussian 3-way tensor in orderto derive a tractable simplification of the sum-product algorithm in thelarge-system limit. It generalizes previous instances of bilinear G-AMP, suchas those that estimate matrices $\boldsymbol{B}$ and $\boldsymbol{C}$ from anoisy measurement of $\boldsymbol{Z}=\boldsymbol{BC}$, allowing the applicationof AMP methods to problems such as self-calibration, blind deconvolution, andmatrix compressive sensing. Numerical experiments confirm the accuracy andcomputational efficiency of the proposed approach.
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