论文标题
旋转不变的矩形矩阵的最佳降解
Optimal denoising of rotationally invariant rectangular matrices
论文作者
论文摘要
在本手稿中，我们考虑对大型矩形矩阵进行降级：鉴于对信号矩阵的嘈杂观察，恢复信号矩阵本身的最佳方法是什么？对于高斯噪声和旋转不变的信号先验，我们完全表征了最佳的DeNoiser及其在高维限制中的性能，在高维极限中，信号矩阵的大小以固定的纵横比为无穷大，而在贝叶斯的最佳设置下，即统计学家知道信号和观测值是如何生成的。我们的结果概括了以前的作品，仅将对称矩阵视为非对称和矩形矩阵的情况。我们在分析和数值上探索了特定的分解信号选择，该选择是模型交叉协方差矩阵和矩阵分解问题。作为我们分析的副产品，我们在特殊情况下对矩形Harish-Chandra-Itzykson-Zuber积分进行了明确的渐近评估。
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.
