We consider a statistical model for matrix factorization in a regime where the rank of the two hidden matrix factors grows linearly with their dimension and their product is corrupted by additive noise. Despite various approaches, statistical and algorithmic limits of such problems have remained elusive. We study a Bayesian setting with the assumptions that (a) one of the matrix factors is symmetric, (b) both factors as well as the additive noise have rotational invariant priors, (c) the priors are known to the statistician. We derive analytical formulas for Rotation Invariant Estimators to reconstruct the two matrix factors, and conjecture that these are optimal in the large-dimension limit, in the sense that they minimize the average mean-square-error. We provide numerical checks which confirm the optimality conjecture when confronted to Oracle Estimators which are optimal by definition, but involve the ground-truth. Our derivation relies on a combination of tools, namely random matrix theory transforms, spherical integral formulas, and the replica method from statistical mechanics.

翻译：我们考虑一种矩阵分解的统计模型，其中两个隐藏矩阵因子的秩随其维度线性增长，且它们的乘积受加性噪声干扰。尽管已有多种方法，这类问题的统计与算法极限仍难以捉摸。我们研究一个贝叶斯设定，并假设：(a) 其中一个矩阵因子对称；(b) 两个因子及加性噪声均具有旋转不变先验；(c) 统计学家已知这些先验。我们推导了旋转不变估计量（Rotation Invariant Estimators）的解析公式以重构两个矩阵因子，并推测在大维极限下这些估计量是最优的，即它们能最小化平均均方误差。我们通过数值检验证实了这一最优性猜想——当与本质上最优的“神谕估计量”（Oracle Estimators）对比时（该估计量依赖于真实参数），结果吻合。我们的推导融合了多种工具：随机矩阵变换、球面积分公式以及统计力学中的复制方法。