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 {\bf 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，即需要从观测信号 $Y=S+Z$ 中推断规模为 $n$ 的随机对称矩阵 $S$，其中 $Z$ 是独立随机矩阵，用于建模噪声。针对在正交矩阵共轭变换下保持不变的 $S$ 与 $Z$ 的先验分布，我们利用一阶和二阶自由概率论的结果，确定了在 $n$ 渐近意义下、次数不超过 $D$ 的贝叶斯最优（以均方误差为指标）多项式估计量，并证明随着 $D$ 增大，这些估计量趋近于 Bun、Allez、Bouchaud 和 Potters 在 [IEEE Transactions on Information Theory {\bf 62}, 7475 (2016)] 中提出的估计量。我们推测该最优性在严格正交不变先验之外仍成立，并针对 $S$ 为任意 Wishart 矩阵、$Z$ 服从高斯正交系综这一与相关的大规模低秩矩阵分解问题相关的情形，提供了这一普适性现象的部分证据。