DeepSphere: towards an equivariant graph-based spherical CNN

Spherical data is found in many applications. By modeling the discretized sphere as a graph, we can accommodate non-uniformly distributed, partial, and changing samplings. Moreover, graph convolutions are computationally more efficient than spherical convolutions. As equivariance is desired to exploit rotational symmetries, we discuss how to approach rotation equivariance using the graph neural network introduced in Defferrard et al. (2016). Experiments show good performance on rotation-invariant learning problems. Code and examples are available at https://github.com/SwissDataScienceCenter/DeepSphereComment: published at the ICLR 2019 Workshop on Representation Learning on Graphs and Manifolds. arXiv admin note: text overlap with arXiv:1810.1218

Stationary Geometric Graphical Model Selection

We consider the problem of model selection in Gaussian Markov fields in the sample deficient scenario. In many practically important cases, the underlying networks are embedded into Euclidean spaces. Using the natural geometric structure, we introduce the notion of spatially stationary distributions over geometric graphs. This directly generalizes the notion of stationary time series to the multidimensional setting lacking time axis. We show that the idea of spatial stationarity leads to a dramatic decrease in the sample complexity of the model selection compared to abstract graphs with the same level of sparsity. For geometric graphs on randomly spread vertices and edges of bounded length, we develop tight information-theoretic bounds on sample complexity and show that a finite number of independent samples is sufficient for a consistent recovery. Finally, we develop an efficient technique capable of reliably and consistently reconstructing graphs with a bounded number of measurements.Comment: arXiv admin note: text overlap with arXiv:1802.0384

Accelerated Sampling of Bandlimited Graph Signals

We study the problem of sampling and reconstructing bandlimited graph signals where the objective is to select a subset of nodes of pre-specified cardinality that ensures interpolation of the original signal with the lowest possible reconstruction error. First, we consider a non-Bayesian scenario and propose an efficient iterative sampling procedure that in the noiseless case enables exact recovery of the original signal from the set of selected nodes. In the case of noisy measurements, a bound on the reconstruction error of the proposed algorithm is established. Then, we consider the Bayesian scenario where we formulate the sampling task as the problem of maximizing a monotone weak submodular function, and propose a randomized-greedy algorithm to find a sub-optimal subset. We derive a worst-case performance guarantee on the mean-square error achieved by the randomized-greedy algorithm for general non-stationary graph signals. The efficacy of the proposed methods is illustrated through extensive numerical simulations on synthetic and real-world graphs.Comment: arXiv admin note: text overlap with arXiv:1807.0718

Greedy Sampling of Graph Signals

Sampling is a fundamental topic in graph signal processing, having found applications in estimation, clustering, and video compression. In contrast to traditional signal processing, the irregularity of the signal domain makes selecting a sampling set non-trivial and hard to analyze. Indeed, though conditions for graph signal interpolation from noiseless samples exist, they do not lead to a unique sampling set. The presence of noise makes choosing among these sampling sets a hard combinatorial problem. Although greedy sampling schemes are commonly used in practice, they have no performance guarantee. This work takes a twofold approach to address this issue. First, universal performance bounds are derived for the Bayesian estimation of graph signals from noisy samples. In contrast to currently available bounds, they are not restricted to specific sampling schemes and hold for any sampling sets. Second, this paper provides near-optimal guarantees for greedy sampling by introducing the concept of approximate submodularity and updating the classical greedy bound. It then provides explicit bounds on the approximate supermodularity of the interpolation mean-square error showing that it can be optimized with worst-case guarantees using greedy search even though it is not supermodular. Simulations illustrate the derived bound for different graph models and show an application of graph signal sampling to reduce the complexity of kernel principal component analysis.Comment: 14 pages, 14 figures. Accepted for publication on IEEE Transactions on Signal Processin

Predicting Graph Signals using Kernel Regression where the Input Signal is Agnostic to a Graph

We propose a kernel regression method to predict a target signal lying over a graph when an input observation is given. The input and the output could be two different physical quantities. In particular, the input may not be a graph signal at all or it could be agnostic to an underlying graph. We use a training dataset to learn the proposed regression model by formulating it as a convex optimization problem, where we use a graph-Laplacian based regularization to enforce that the predicted target is a graph signal. Once the model is learnt, it can be directly used on a large number of test data points one-by-one independently to predict the corresponding targets. Our approach employs kernels between the various input observations, and as a result the kernels are not restricted to be functions of the graph adjacency/Laplacian matrix. We show that the proposed kernel regression exhibits a smoothing effect, while simultaneously achieving noise-reduction and graph-smoothness. We then extend our method to the case when the underlying graph may not be known apriori, by simultaneously learning an underlying graph and the regression coefficients. Using extensive experiments, we show that our method provides a good prediction performance in adverse conditions, particularly when the training data is limited in size and is noisy. In graph signal reconstruction experiments, our method is shown to provide a good performance even for a highly under-determined subsampling

Spectral Domain Sampling of Graph Signals

Sampling methods for graph signals in the graph spectral domain are presented. Though conventional sampling of graph signals can be regarded as sampling in the graph vertex domain, it does not have the desired characteristics in regard to the graph spectral domain. With the proposed methods, the down- and upsampled graph signals inherit the frequency domain characteristics of the sampled signals defined in the time/spatial domain. The properties of the sampling effects were evaluated theoretically in comparison with those obtained with the conventional sampling method in the vertex domain. Various examples of signals on simple graphs enable precise understanding of the problem considered. Fractional sampling and Laplacian pyramid representation of graph signals are potential applications of these methods.Comment: accepted to IEEE Transactions on Signal Processin

Graph Fourier Transform: A Stable Approximation

In Graph Signal Processing (GSP), data dependencies are represented by a graph whose nodes label the data and the edges capture dependencies among nodes. The graph is represented by a weighted adjacency matrix $A$ that, in GSP, generalizes the Discrete Signal Processing (DSP) shift operator $z^{-1}$. The (right) eigenvectors of the shift $A$ (graph spectral components) diagonalize $A$ and lead to a graph Fourier basis $F$ that provides a graph spectral representation of the graph signal. The inverse of the (matrix of the) graph Fourier basis $F$ is the Graph Fourier transform (GFT), $F^{-1}$. Often, including in real world examples, this diagonalization is numerically unstable. This paper develops an approach to compute an accurate approximation to $F$ and $F^{-1}$, while insuring their numerical stability, by means of solving a non convex optimization problem. To address the non-convexity, we propose an algorithm, the stable graph Fourier basis algorithm (SGFA) that we prove to exponentially increase the accuracy of the approximating $F$ per iteration. Likewise, we can apply SGFA to $A^H$ and, hence, approximate the stable left eigenvectors for the graph shift $A$ and directly compute the GFT. We evaluate empirically the quality of SGFA by applying it to graph shifts $A$ drawn from two real world problems, the 2004 US political blogs graph and the Manhattan road map, carrying out a comprehensive study on tradeoffs between different SGFA parameters. We also confirm our conclusions by applying SGFA on very sparse and very dense directed Erd\H os-R\'enyi graphs.Comment: 16 pages, 17 figures. Originally submitted in -IEEE Transactions on Signal Processing- on 01-Aug-2019. Resubmitted on 12-Jan-2020. Accept with mandatory minor revisions. Resubmitted again on 30-April-202

On the Dual Geometry of Laplacian Eigenfunctions

We discuss the geometry of Laplacian eigenfunctions $-\Delta \phi = \lambda \phi$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The 'dual' geometry of Laplacian eigenfunctions is well understood on $\mathbb{T}^d$ (identified with $\mathbb{Z}^d$) and $\mathbb{R}^n$ (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of 'similarity' $\alpha(\phi_{\lambda}, \phi_{\mu})$ between eigenfunctions $\phi_{\lambda}$ and $\phi_{\mu}$ is given by a global average of local correlations $$\alpha(\phi_{\lambda}, \phi_{\mu})^2 = \| \phi_{\lambda} \phi_{\mu} \|_{L^2}^{-2}\int_{M}{ \left( \int_{M}{ p(t,x,y)( \phi_{\lambda}(y) - \phi_{\lambda}(x))( \phi_{\mu}(y) - \phi_{\mu}(x)) dy} \right)^2 dx},$$ where $p(t,x,y)$ is the classical heat kernel and $e^{-t \lambda} + e^{-t \mu} = 1$. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result

Bayesian Design of Sampling Set for Bandlimited Graph Signals

The design of sampling set (DoS) for bandlimited graph signals (GS) has been extensively studied in recent years, but few of them exploit the benefits of the stochastic prior of GS. In this work, we introduce the optimization framework for Bayesian DoS of bandlimited GS. We also illustrate how the choice of different sampling sets affects the estimation error and how the prior knowledge influences the result of DoS compared with the non-Bayesian DoS by the aid of analyzing Gershgorin discs of error metric matrix. Finally, based on our analysis, we propose a heuristic algorithm for DoS to avoid solving the optimization problem directly.Comment: Accepted by GloalSIP201

Translation Operator in Graph Signal Processing: A Generalized Approach

The notion of translation (shift) is straightforward in classical signal processing, however, it is challenging on an irregular graph structure. In this work, we present an approach to characterize the translation operator in various signal domains. By a natural generalization from classical domains, one can characterize an abstract representation for the graph translation operator. Then we propose an isometric translation operator in joint time-vertex domain consistent with the abstract form of translation operators in other domains. We also demonstrate the connection between this notion and the Schr\"{o}dinger equation on a dynamic system which intriguingly describes the idea behind translation on graph