In this paper, we study a spiked Wigner problem with an inhomogeneous noise profile. Our aim in this problem is to recover the signal passed through an inhomogeneous low-rank matrix channel. While the information-theoretic performances are well-known, we focus on the algorithmic problem. We derive an approximate message-passing algorithm (AMP) for the inhomogeneous problem and show that its rigorous state evolution coincides with the information-theoretic optimal Bayes fixed-point equations. We identify in particular the existence of a statistical-to-computational gap where known algorithms require a signal-to-noise ratio bigger than the information-theoretic threshold to perform better than random. Finally, from the adapted AMP iteration we deduce a simple and efficient spectral method that can be used to recover the transition for matrices with general variance profiles. This spectral method matches the conjectured optimal computational phase transition.

翻译：本文研究具有非均匀噪声分布的尖峰维格纳问题。该问题的目标是从非均匀低秩矩阵信道中恢复通过的信号。尽管信息论性能已得到充分研究，我们重点关注算法问题。我们为这一非均匀问题推导出一种近似消息传递算法（AMP），并证明其严格的状态演化与信息论最优贝叶斯不动点方程一致。我们特别发现统计-计算差距的存在：已知算法需要信噪比高于信息论阈值才能取得优于随机猜想的性能。最后，通过改进的AMP迭代，我们推导出一种简单有效的谱方法，可用于恢复具有一般方差分布的矩阵的相变。该谱方法与猜想的最优计算相变相匹配。