In this paper, we extend the bilinear generalized approximate message passing (BiG-AMP) approach, originally proposed for high-dimensional generalized bilinear regression, to the multi-layer case for the handling of cascaded problem such as matrix-factorization problem arising in relay communication among others. Assuming statistically independent matrix entries with known priors, the new algorithm called ML-BiGAMP could approximate the general sum-product loopy belief propagation (LBP) in the high-dimensional limit enjoying a substantial reduction in computational complexity. We demonstrate that, in large system limit, the asymptotic MSE performance of ML-BiGAMP could be fully characterized via a set of simple one-dimensional equations termed state evolution (SE). We establish that the asymptotic MSE predicted by ML-BiGAMP’ SE matches perfectly the exact MMSE predicted by the replica method, which is well-known to be Bayes-optimal but infeasible in practice. This consistency indicates that the ML-BiGAMP may still retain the same Bayes-optimal performance as the MMSE estimator in high-dimensional applications, although ML-BiGAMP's computational burden is far lower. As an illustrative example of the general ML-BiGAMP, we provide a detector design that could estimate the channel fading and the data symbols jointly with high precision for the two-hop amplify-and-forward relay communication systems.

中文翻译：

---

多层双线性广义近似消息传递

在本文中，我们将最初为高维广义双线性回归提出的双线性广义近似消息传递（BiG-AMP）方法扩展到多层情况，以处理级联问题，例如中继中出现的矩阵分解问题之间的交流。假设具有已知先验的统计独立矩阵条目，称为 ML-BiGAMP 的新算法可以在高维限制中近似一般和积循环置信传播 (LBP)，从而显着降低计算复杂度。我们证明，在大系统限制下，ML-BiGAMP 的渐近 MSE 性能可以通过一组称为状态演化 (SE) 的简单一维方程来完全表征。我们确定 ML-BiGAMP' SE 预测的渐近 MSE 与副本方法预测的精确 MMSE 完全匹配，众所周知，这是贝叶斯最优但在实践中不可行。这种一致性表明，尽管 ML-BiGAMP 的计算负担要低得多，但 ML-BiGAMP 在高维应用中仍可能保持与 MMSE 估计器相同的贝叶斯最优性能。作为一般 ML-BiGAMP 的一个说明性示例，我们提供了一种检测器设计，可以为两跳放大转发中继通信系统以高精度联合估计信道衰落和数据符号。这种一致性表明，尽管 ML-BiGAMP 的计算负担要低得多，但 ML-BiGAMP 在高维应用中仍可能保持与 MMSE 估计器相同的贝叶斯最优性能。作为一般 ML-BiGAMP 的一个说明性示例，我们提供了一种检测器设计，可以为两跳放大转发中继通信系统以高精度联合估计信道衰落和数据符号。这种一致性表明，尽管 ML-BiGAMP 的计算负担要低得多，但 ML-BiGAMP 在高维应用中仍可能保持与 MMSE 估计器相同的贝叶斯最优性能。作为一般 ML-BiGAMP 的一个说明性示例，我们提供了一种检测器设计，可以为两跳放大转发中继通信系统以高精度联合估计信道衰落和数据符号。