Random sampling of bandlimited signals on graphs

Abstract

International audienceWe study the problem of sampling k-bandlimited signals on graphs. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure an accurate and stable recovery of all k-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we propose a computationally efficient decoder to reconstruct k-bandlimited signals from their samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we conduct several experiments to test these techniques. [Code available at http://grsamplingbox.gforge.inria.fr/

Online Graph Completion: Multivariate Signal Recovery in Computer Vision

“…Therefore, to evaluate our algorithm, we computed the number of errors that SR makes as well as mean precision of the prediction. We compared our results with two other baseline methods 1) Puy et a.l [27] and 2) Rao et al [28], which are the state-of-art methods for graph completion. For the signal recovery step, we used α = 1, ξ = 0.01 and only 60% of the total bases in our algorithm for estimation while other methods required all of them.…”

“…7, our result (in red) shows much lower error than the baseline methods. When we used these estimation results to identify whether each participants had elevated amyloid burden (i.e., whether mean of PIB measures over all ROIs is > 1.18), our estimation offered 91.1% accuracy while [27] and [28] provided 88.6% and 87.6%.…”

“…The optimal solution Ẑ * here is an estimation of sparse encoding of the original signal f in the frequency space, and its representation in the original space can be empirically recovered by performing the inverse graph Fourier transform as $Z^{\ast }false(nfalse)={\sum }_{l=1}^k{\hat{Z}}_k^{\ast }false(lfalse){\chi }_{l}false(nfalse)$. Note that in (7), we avoid imposing a smoothness constraint that has been used in other approaches [28, 27], since our solution is already smooth due to its low-rank and bandlimited properties. However, the smoothness criteria may still be useful when our assumption (i.e., sparsity) in the dual domain does not hold.…”

“…In [28], the idea of collaborative filtering was generalized to graphs where a smoothness assumption was imposed using the Laplacian of the graph. Separately, a random sampling scheme with a bandlimited assumption was introduced in [27] where the authors define a probability distribution for sampling at each node of a graph by analyzing the eigenvalues/eigenvectors of the Laplacian of the graph. These methods essentially model the graph completion problem (an example demonstrated in Fig.…”

Adaptive Signal Recovery on Graphs via Harmonic Analysis for Experimental Design in Neuroimaging

“…This clearly yields a matrix completion problem; unfortunately, the setup lies far from incoherent sampling and the matrix versions of restricted isometry property (RIP) that make the low-rank completion argument work in practice [27, 28]. This observation has been made in recent works where collaborative filtering was generalized to the graph domain [29] and where random sampling was introduced for graphs in [30]. However, these approaches, which will serve as excellent baselines, do not exploit the band-limited nature of measurements in frequency space.…”