We introduce the bilinear generalized vector approximate message passing (BiG-VAMP) algorithm which jointly recovers two matrices U and V from their noisy product through a probabilistic observation model. BiG-VAMP provides computationally efficient approximate implementations of both max-sum and sumproduct loopy belief propagation (BP). We show how the proposed BiG-VAMP algorithm recovers different types of structured matrices and overcomes the fundamental limitations of other state-of-the-art approaches to the bilinear recovery problem, such as BiG-AMP, BAd-VAMP and LowRAMP. In essence, BiG-VAMP applies to a broader class of practical applications which involve a general form of structured matrices. For the sake of theoretical performance prediction, we also conduct a state evolution (SE) analysis of the proposed algorithm and show its consistency with the asymptotic empirical mean-squared error (MSE). Numerical results on various applications such as matrix factorization, dictionary learning, and matrix completion demonstrate unambiguously the effectiveness of the proposed BiG-VAMP algorithm and its superiority over stateof-the-art algorithms. Using the developed SE framework, we also examine (as one example) the phase transition diagrams of the matrix completion problem, thereby unveiling a low detectability region corresponding to the low signal-to-noise ratio (SNR) regime.

中文翻译：

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双线性广义向量近似消息传递

我们介绍了双线性广义向量近似消息传递 (BiG-VAMP) 算法，该算法通过概率观察模型从它们的噪声乘积中联合恢复两个矩阵 U 和 V。BiG-VAMP 提供了 max-sum 和 sumproduct 循环置信传播 (BP) 的计算高效近似实现。我们展示了所提出的 BiG-VAMP 算法如何恢复不同类型的结构化矩阵，并克服了其他最先进的双线性恢复问题方法的基本局限性，例如 BiG-AMP、BAd-VAMP 和 LowRAMP。本质上，BiG-VAMP 适用于更广泛的实际应用类别，其中涉及结构化矩阵的一般形式。为了理论性能预测，我们还对所提出的算法进行状态演化 (SE) 分析，并显示其与渐近经验均方误差 (MSE) 的一致性。矩阵分解、字典学习和矩阵补全等各种应用的数值结果清楚地证明了所提出的 BiG-VAMP 算法的有效性及其优于最先进算法的优越性。使用开发的 SE 框架，我们还检查（作为一个例子）矩阵完成问题的相变图，从而揭示了对应于低信噪比 (SNR) 机制的低可检测性区域。和矩阵完成明确地证明了所提出的 BiG-VAMP 算法的有效性及其对最先进算法的优越性。使用开发的 SE 框架，我们还检查（作为一个例子）矩阵完成问题的相变图，从而揭示了对应于低信噪比 (SNR) 机制的低可检测性区域。和矩阵完成明确地证明了所提出的 BiG-VAMP 算法的有效性及其对最先进算法的优越性。使用开发的 SE 框架，我们还检查（作为一个例子）矩阵完成问题的相变图，从而揭示了对应于低信噪比 (SNR) 机制的低可检测性区域。