具有旋转不变先验的贝叶斯广秩矩阵分解
Bayesian Extensive-Rank Matrix Factorization with Rotational Invariant Priors
AI总结：
本文研究高秩、加性噪声下的贝叶斯矩阵分解，利用随机矩阵理论、球面积分与副本方法推导旋转不变估计器，并通过与 Oracle 估计器的数值比较验证其大维极限最优性。
摘要：
我们考虑一种矩阵分解的统计模型，其所处情形是两个隐藏矩阵因子的秩随其维度线性增长，并且它们的乘积被加性噪声破坏。尽管已有多种方法，但这类问题的统计极限和算法极限仍不清楚。我们研究一个贝叶斯设定，并假设：(a) 其中一个矩阵因子是对称的；(b) 两个因子以及加性噪声都具有旋转不变先验；(c) 这些先验为统计学家所知。我们推导了用于重构两个矩阵因子的 Rotation Invariant Estimators（旋转不变估计器）的解析公式，并推测这些公式在大维度极限下是最优的，即它们使平均均方误差最小化。我们提供了数值检验；在与按定义即为最优、但需要使用真值的 Oracle Estimators（Oracle 估计器）比较时，这些检验证实了最优性猜想。我们的推导依赖于多种工具的结合，即随机矩阵理论变换、球面积分公式，以及来自统计力学的副本方法。
英文摘要：
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.
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