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.
eess.SP↗
arXiv subjects
Baptiste Olivier
Publications and source records attributed to Baptiste Olivier.
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.
eess.SP↗
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 $g(\mathcal{L})f$.
eess.SP↗
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.
eess.SP↗
Privacy in trajectory micro-data publishing : a survey
We survey the literature on the privacy of trajectory micro-data, i.e., spatiotemporal information about the mobility of individuals, whose collection is becoming increasingly simple and frequent thanks to emerging information and communication technologies. The focus of our review is on privacy-preserving data publishing (PPDP), i.e., the publication of databases of trajectory micro-data that preserve the privacy of the monitored individuals. We classify and present the literature of attacks against trajectory micro-data, as well as solutions proposed to date for protecting databases from such attacks. This paper serves as an introductory reading on a critical subject in an era of growing awareness about privacy risks connected to digital services, and provides insights into open problems and future directions for research.
cs.CR↗
Fixed-point spectrum for group actions by affine isometries on Lp-spaces
The fixed-point spectrum of a locally compact second countable group G on lp is defined to be the set of real numbers p such that every action by affine isometries of G on lp admits a fixed-point. We show that this set is either empty, or is equal to a set of one of the following forms : [1,\pc[, [1,\pc[\{2} for some \pc<\infty or \pc=\infty, or [1,\pc], [1,\pc]\{2} for some pc<infty. This answers a question closely related to a conjecture of C. Drutu which asserts that the fixed-point spectrum is connected for isometric actions on Lp(0,1). We also study the topological properties of the fixed-point spectrum on Lp(X,μ) for general measure spaces (X,μ), and show partial results toward the conjecture for actions on Lp(0,1). In particular, we prove that the spectrum F_{L^{\infty}(X,μ)(G,π) of actions with linear part πis either empty, or an interval of the form [1,\pc] or [1,\infty[, whenever πis an orthogonal representation associated to a measure-preserving ergodic action on a finite measure space (X,μ).
math.GR↗
Variants of a-T-menability for actions on non-commutative Lp spaces
We investigate various characterizations of the Haagerup property (H) for a second countable locally compact group G, in terms of orthogonal representations of G on non-commutative Lp spaces. We introduce a variant (H_Lp) for orthogonal representations with vanishing coefficients on Lp, and study its relationships with property (H). We also give a characterization of (H) by the means of strongly mixing actions on a non-commutative Lp space. We construct proper actions of groups with (H) by affine isometries on some non-commutative Lp space, such as the Lp space associated to the hyperfinite II infinite factor.
math.GR↗
On groups with Property (T_lp)
Let p be a real number with 1<p and different from 2. We study Property (T_lp) for a second countable locally compact group G. Property (T_lp) is a weak version of Kazhdan's Property (T), defined in terms of the orthogonal representations of G on the sequence space lp. We show that Property (T_lp) for a totally disconnected group G is characterized by an isolation property of the trivial representation from the quasi-regular representations associated to open subgroups of G. Groups with Property (T_lp) share some important properties with Kazhdan groups (compact generation, compact abelianization, ...). Simple algebraic groups over non-archimedean local fields as well as automorphism groups of regular trees have Property (T_lp). In the case of discrete groups, Property (T_lp) implies Lubotzky's Property tau and is implied by Property (F) of Glasner and Monod. We show that an irreducible lattice in a product of two locally compact groups G and H have Property (T_lp), whenever G has Property (T) and H is connected and minimally almost periodic.
math.GR↗
Property (T) with respect to non-commutative Lp-spaces
We show that a group with Kazhdan's property $(T)$ has property $(T_{B})$ for $B$ the Haagerup non-commutative $L_{p}(\mathcal{M})$-space associated with a von Neumann algebra $\mathcal{M}$, $1
math.GR↗