Large Graph Signal Denoising with Application to Differential Privacy

TL;DR

This paper introduces a scalable graph signal denoising method using wavelet tight frames and Stein's risk estimate, with applications to differential privacy, handling large graphs efficiently.

Contribution

It develops a data-driven wavelet frame denoising approach for large graphs using Chebyshev-Jackson approximations and a Monte-Carlo covariance estimation strategy.

Findings

Effective denoising on large graphs demonstrated.

Scalable approach avoids eigendecomposition.

Application to differential privacy shown.

Abstract

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.