We consider the estimation of an n-dimensional vector s from the noisy element-wise measurements of $\mathbf{s}\mathbf{s}^T$, a generic problem that arises in statistics and machine learning. We study a mismatched Bayesian inference setting, where some of the parameters are not known to the statistician. We derive the full exact analytic expression of the asymptotic mean squared error (MSE) in the large system size limit for the particular case of Gaussian priors and additive noise. From our formulas, we see that estimation is still possible in the mismatched case; and also that the minimum MSE (MMSE) can be achieved if the statistician chooses suitable parameters. Our technique relies on the asymptotics of the spherical integrals and can be applied as long as the statistician chooses a rotationally invariant prior.

中文翻译：

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高斯噪声下秩一对称矩阵的失配估计

我们考虑从 $\mathbf{s}\mathbf{s}^T$ 的噪声元素级测量中估计 n 维向量 s，这是统计学和机器学习中出现的一般问题。我们研究了一个不匹配的贝叶斯推理设置，其中一些参数对于统计学家来说是未知的。对于高斯先验和加性噪声的特殊情况，我们推导出大系统大小限制中渐近均方误差 (MSE) 的完整精确解析表达式。从我们的公式中，我们看到在不匹配的情况下仍然可以进行估计；并且如果统计学家选择合适的参数，则可以实现最小 MSE (MMSE)。我们的技术依赖于球积分的渐近性，只要统计学家选择旋转不变的先验，就可以应用。