Over the last decade, signal processing on graphs has become a very active area of research. Specifically, the number of applications, for instance in statistical or deep learning, using frames built from graphs, such as wavelets on graphs, has increased significantly. We consider in particular the case of signal denoising on graphs via a data-driven wavelet tight frame methodology. This adaptive approach is based on a threshold calibrated using Stein's unbiased risk estimate adapted to a tight-frame representation. We make it scalable to large graphs using Chebyshev-Jackson polynomial approximations, which allow fast computation of the wavelet coefficients, without the need to compute the Laplacian eigendecomposition. However, the overcomplete nature of the tight-frame, transforms a white noise into a correlated one. As a result, the covariance of the transformed noise appears in the divergence term of the SURE, thus requiring the computation and storage of the frame, which leads to an impractical calculation for large graphs. To estimate such covariance, we develop and analyze a Monte-Carlo strategy, based on the fast transformation of zero mean and unit variance random variables. This new data-driven denoising methodology finds a natural application in differential privacy. A comprehensive performance analysis is carried out on graphs of varying size, from real and simulated data.

中文翻译：

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应用于差分隐私的大图信号去噪

在过去十年中，图上的信号处理已成为一个非常活跃的研究领域。具体来说，使用从图构建的框架（例如图上的小波）的应用程序数量显着增加，例如在统计或深度学习中。我们特别考虑了通过数据驱动的小波紧框架方法对图进行信号去噪的情况。这种自适应方法基于使用 Stein 的无偏风险估计校准的阈值，该估计适用于紧密框架表示。我们使用 Chebyshev-Jackson 多项式近似使其可扩展到大型图，这允许快速计算小波系数，而无需计算拉普拉斯特征分解。然而，紧框架的过完备性质将白噪声转换为相关噪声。因此，变换后的噪声的协方差出现在 SURE 的散度项中，因此需要框架的计算和存储，这导致对于大图的计算不切实际。为了估计这种协方差，我们开发并分析了一种基于零均值和单位方差随机变量的快速变换的蒙特卡洛策略。这种新的数据驱动去噪方法在差分隐私中找到了自然的应用。根据真实数据和模拟数据，对不同大小的图表进行全面的性能分析。基于零均值和单位方差随机变量的快速变换。这种新的数据驱动去噪方法在差分隐私中找到了自然的应用。根据真实数据和模拟数据，对不同大小的图表进行全面的性能分析。基于零均值和单位方差随机变量的快速变换。这种新的数据驱动去噪方法在差分隐私中找到了自然的应用。根据真实数据和模拟数据，对不同大小的图表进行全面的性能分析。