旋转不变矩形矩阵的最优去噪
Optimal denoising of rotationally invariant rectangular matrices
摘要：
在这篇手稿中，我们考虑去噪的大型矩形矩阵：给定一个嘈杂的观察信号矩阵，什么是最好的方式来恢复信号矩阵本身？对于高斯噪声和旋转不变信号先验，我们完全描述了最佳降噪器及其在高维极限中的性能，其中信号矩阵的大小以固定的纵横比变为无穷大，并且在贝叶斯最佳设置下，即统计学家知道信号和观测是如何生成的。我们的研究结果概括了以前的作品，认为只有对称矩阵的非对称和矩形的更一般的情况下。我们探讨分析和数值分解信号的特定选择之前，模型互协方差矩阵和矩阵分解问题。作为我们的分析的副产品，我们提供了一个明确的矩形Harish-Chandra-Itzykson-Zuber积分在特殊情况下的渐近评价。
英文摘要：
In this manuscript we consider denoising of large rectangular matrices: given a noisy observation of a signal matrix, what is the best way of recovering the signal matrix itself? For Gaussian noise and rotationally-invariant signal priors, we completely characterize the optimal denoiser and its performance in the high-dimensional limit, in which the size of the signal matrix goes to infinity with fixed aspects ratio, and under the Bayes optimal setting, that is when the statistician knows how the signal and the observations were generated. Our results generalise previous works that considered only symmetric matrices to the more general case of non-symmetric and rectangular ones. We explore analytically and numerically a particular choice of factorized signal prior that models cross-covariance matrices and the matrix factorization problem. As a byproduct of our analysis, we provide an explicit asymptotic evaluation of the rectangular Harish-Chandra-Itzykson-Zuber integral in a special case.
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