Gasper: GrAph Signal ProcEssing in R
We present a short tutorial on to the use of the R gasper package. Gasper is a package dedicated to signal processing on graphs. It also provides an interface to the SuiteSparse Matrix Collection.
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arXiv subjects
Basile de Loynes
Publications and source records attributed to Basile de Loynes.
Large Graph Signal Denoising with Application to Differential Privacy
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.
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Localized Fourier Analysis for Graph Signal Processing
We propose a new point of view in the study of Fourier analysis on graphs, taking advantage of localization in the Fourier domain. For a signal $f$ on vertices of a weighted graph $\mathcal{G}$ with Laplacian matrix $\mathcal{L}$, standard Fourier analysis of $f$ relies on the study of functions $g(\mathcal{L})f$ for some filters $g$ on $I_\mathcal{L}$, the smallest interval containing the Laplacian spectrum ${\mathrm sp}(\mathcal{L}) \subset I_\mathcal{L}$. We show that for carefully chosen partitions $I_\mathcal{L} = \sqcup_{1\leq k\leq K} I_k$ ($I_k \subset I_\mathcal{L}$), there are many advantages in understanding the collection $(g(\mathcal{L}_{I_k})f)_{1\leq k\leq K}$ instead of $g(\mathcal{L})f$ directly, where $\mathcal{L}_I$ is the projected matrix $P_I(\mathcal{L})\mathcal{L}$. First, the partition provides a convenient modelling for the study of theoretical properties of Fourier analysis and allows for new results in graph signal analysis (\emph{e.g.} noise level estimation, Fourier support approximation). We extend the study of spectral graph wavelets to wavelets localized in the Fourier domain, called LocLets, and we show that well-known frames can be written in terms of LocLets. From a practical perspective, we highlight the interest of the proposed localized Fourier analysis through many experiments that show significant improvements in two different tasks on large graphs, noise level estimation and signal denoising. Moreover, efficient strategies permit to compute sequence $(g(\mathcal{L}_{I_k})f)_{1\leq k\leq K}$ with the same time complexity as for the computation of $g(\mathcal{L})f$.
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Data-driven Thresholding in Denoising with Spectral Graph Wavelet Transform
This paper is devoted to adaptive signal denoising in the context of Graph Signal Processing (GSP) using Spectral Graph Wavelet Transform (SGWT). This issue is addressed \emph{via} a data-driven thresholding process in the transformed domain by optimizing the parameters in the sense of the Mean Square Error (MSE) using the Stein's Unbiased Risk Estimator (SURE). The SGWT considered is built upon a partition of unity making the transform semi-orthogonal so that the optimization can be performed in the transformed domain. However, since the SGWT is over-complete, the divergence term in the SURE needs to be computed in the context of correlated noise. Two thresholding strategies called coordinatewise and block thresholding process are investigated. For each of them, the SURE is derived for a whole family of elementary thresholding functions among which the soft threshold and the James-Stein threshold. This multi-scales analysis shows better performance than the most recent methods from the literature. That is illustrated numerically for a series of signals on different graphs.
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Random walk on a directed graph and Martin boundary
The Martin boundary associated with the simple random walk on an example of partially oriented lattice is shown to be trivial by computing fine estimates of the Green kernel.
math.PR↗
A note on the Martin boundary of the simple random walk on an example of directed graph
In this note, we study the Martin boundary of the simple random walk on an example of directed graph. More precisely, the Martin boundary is shown to be trivial, i.e. there is no non-constant positive harmonic functions. The proof involves fine estimates of the Green function which are summarized in this note.
math.PR↗