We study the performance of a Bayesian statistician who estimates a rank-one signal corrupted by non-symmetric rotationally invariant noise with a generic distribution of singular values. As the signal-to-noise ratio and the noise structure are unknown, a Gaussian setup is incorrectly assumed. We derive the exact analytic expression for the error of the mismatched Bayes estimator and also provide the analysis of an approximate message passing (AMP) algorithm. The first result exploits the asymptotic behavior of spherical integrals for rectangular matrices and of low-rank matrix perturbations; the second one relies on the design and analysis of an auxiliary AMP. The numerical experiments show that there is a performance gap between the AMP and Bayes estimators, which is due to the incorrect estimation of the signal norm.

翻译：我们研究了一位贝叶斯统计学家在估计被非对称旋转不变噪声（具有奇异值的一般分布）污染的秩一信号时的性能。由于信噪比和噪声结构未知，该统计学家错误地假定了高斯设置。我们推导了失配贝叶斯估计器误差的精确解析表达式，并提供了一种近似消息传递（AMP）算法的分析。第一个结果利用了矩形矩阵的球面积分和低秩矩阵扰动的渐近行为；第二个结果依赖于辅助AMP的设计与分析。数值实验表明，AMP与贝叶斯估计器之间存在性能差距，这是由于信号范数的估计不准确所致。