We propose and analyze an approximate message passing (AMP) algorithm for the
matrix tensor product model, which is a generalization of the standard spiked
matrix models that allows for multiple types of pairwise observations over a
collection of latent variables. A key innovation for this algorithm is a method
for optimally weighing and combining multiple estimates in each iteration.
Building upon an AMP convergence theorem for non-separable functions, we prove
a state evolution for non-separable functions that provides an asymptotically
exact description of its performance in the high-dimensional limit. We leverage
this state evolution result to provide necessary and sufficient conditions for
recovery of the signal of interest. Such conditions depend on the singular
values of a linear operator derived from an appropriate generalization of a
signal-to-noise ratio for our model. Our results recover as special cases a
number of recently proposed methods for contextual models (e.g., covariate
assisted clustering) as well as inhomogeneous noise models.

提出并分析一种近似信息传递 (AMP) 算法，用于矩阵张量乘积模型，其中使用一种新方法在每次迭代中优化加权和组合多个估计；利用非可分函数的 AMP 收敛定理，证明了非可分函数的状态演变，提供了其在高维极限下性能的渐近精确描述。