We consider the additive version of the matrix denoising problem, where a random symmetric matrix $S$ of size $n$ has to be inferred from the observation of $Y=S+Z$, with $Z$ an independent random matrix modeling a noise. For prior distributions of $S$ and $Z$ that are invariant under conjugation by orthogonal matrices we determine, using results from first and second order free probability theory, the Bayes-optimal (in terms of the mean square error) polynomial estimators of degree at most $D$, asymptotically in $n$, and show that as $D$ increases they converge towards the estimator introduced by Bun, Allez, Bouchaud and Potters in [IEEE Transactions on Information Theory 62, 7475 (2016)]. We conjecture that this optimality holds beyond strictly orthogonally invariant priors, and provide partial evidences of this universality phenomenon when $S$ is an arbitrary Wishart matrix and $Z$ is drawn from the Gaussian Orthogonal Ensemble, a case motivated by the related extensive rank matrix factorization problem.

翻译：我们考虑矩阵去噪问题的加性版本，即需要从观测数据 $Y=S+Z$ 中推断出尺寸为 $n$ 的随机对称矩阵 $S$，其中 $Z$ 是建模噪声的独立随机矩阵。对于在正交矩阵共轭作用下不变的 $S$ 和 $Z$ 先验分布，我们利用一阶和二阶自由概率理论的结果，渐近地确定了不超过 $D$ 阶的多项式估计器在均方误差意义下的贝叶斯最优解，并证明当 $D$ 增大时，这些估计器收敛于 Bun、Allez、Bouchaud 和 Potters 在 [IEEE Transactions on Information Theory 62, 7475 (2016)] 中提出的估计器。我们推测这种最优性在严格正交不变先验之外仍然成立，并在 $S$ 为任意 Wishart 矩阵且 $Z$ 取自高斯正交系综的情形下，为这种普适性现象提供了部分证据，该案例的动机来源于相关的扩展秩矩阵分解问题。