OptShrink：一种通过最佳数据驱动奇异值收缩改进低秩信号矩阵去噪的算法
OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage
摘要：
测量矩阵的截断奇异值分解（SVD）是如何使用低秩矩阵最好地近似噪声测量矩阵的_representation_ problem的最优解。在这里，我们考虑如何通过测量矩阵的奇异向量的最佳（重新）加权来最好地近似隐藏在噪声中的低秩信号矩阵的（不可观测）_去噪_问题。我们利用随机矩阵理论的最新成果，准确地描述了最佳权系数的大矩阵限制，并表明它们可以直接从一个大类的噪声模型，包括独立同分布的数据计算。高斯噪声情况。我们的分析带来了尖锐的焦点收缩和阈值形式的最佳权重，相关的收缩函数（奇异值）的非凸性质，并解释了为什么矩阵正则化通过奇异值阈值与凸罚函数（如核范数）将始终是次优的。我们验证我们的理论预测与数值模拟，开发一个可实现的算法（OptShrink），实现了预测的性能增益，并显示我们的方法可以用来提高估计的设置中的测量矩阵有缺失的条目。
英文摘要：
The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the_representation_ problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the (unobservable)_denoising_ problem of how to best approximate a low-rank signal matrix buried in noise by optimal (re)weighting of the singular vectors of the measurement matrix. We exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the i.i.d. Gaussian noise case. Our analysis brings into sharp focus the shrinkage-and-thresholding form of the optimal weights, the non-convex nature of the associated shrinkage function (on the singular values) and explains why matrix regularization via singular value thresholding with convex penalty functions (such as the nuclear norm) will always be suboptimal. We validate our theoretical predictions with numerical simulations, develop an implementable algorithm (OptShrink) that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries.
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