We consider the task of estimating a low-rank matrix from non-linear and noisy observations. We prove a strong universality result showing that Bayes-optimal performances are characterized by an equivalent Gaussian model with an effective prior, whose parameters are entirely determined by an expansion of the non-linear function. In particular, we show that to reconstruct the signal accurately, one requires a signal-to-noise ratio growing as $N^{\frac 12 (1-1/k_F)}$, where $k_F$ is the first non-zero Fisher information coefficient of the function. We provide asymptotic characterization for the minimal achievable mean squared error (MMSE) and an approximate message-passing algorithm that reaches the MMSE under conditions analogous to the linear version of the problem. We also provide asymptotic errors achieved by methods such as principal component analysis combined with Bayesian denoising, and compare them with Bayes-optimal MMSE.

翻译：我们考虑从非线性且含噪声的观测中估计低秩矩阵的任务。我们证明了一个强普适性结果，表明贝叶斯最优性能可由一个具有有效先验的等效高斯模型来表征，其参数完全由非线性函数的展开确定。特别地，我们表明要精确重构信号，所需信噪比需随$N^{\frac 12 (1-1/k_F)}$增长，其中$k_F$是函数的首个非零费舍尔信息系数。我们给出了最小可达均方误差（MMSE）的渐近表征，以及一种近似消息传递算法，该算法在类似于线性版本问题的条件下能达到MMSE。我们还提供了如主成分分析结合贝叶斯去噪等方法所达到的渐近误差，并将其与贝叶斯最优MMSE进行比较。