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.

翻译：我们提出并分析了一种适用于矩阵张量积模型的近似消息传递算法，该模型是标准尖峰矩阵模型的推广，允许在隐变量集合上存在多种类型的成对观测。该算法的核心创新在于提出了一种在每次迭代中最优加权并融合多个估计量的方法。基于非可分函数的AMP收敛定理，我们证明了非可分函数的状态演化，该演化在高维极限下提供了其性能的渐近精确描述。利用这一状态演化结果，我们给出了信号恢复的充要条件。这些条件依赖于由模型信噪比的适当推广所导出的线性算子的奇异值。我们的结果不仅包含了近期提出的若干情境模型（如协变量辅助聚类）方法作为特例，也涵盖了非齐次噪声模型。