Graph Filter Research Articles

Stationary Graph Processes and Spectral Estimation

Stationarity is a cornerstone property that facilitates the analysis and processing of random signals in the time domain. Although time-varying signals are abundant in nature, in many practical scenarios, the information of interest resides in more irregular graph domains. This lack of regularity hampers the generalization of the classical notion of stationarity to graph signals. This paper proposes a definition of weak stationarity for random graph signals that takes into account the structure of the graph where the random process takes place, while inheriting many of the meaningful properties of the classical time domain definition. Provided that the topology of the graph can be described by a normal matrix, stationary graph processes can be modeled as the output of a linear graph filter applied to a white input. This is shown equivalent to requiring the correlation matrix to be diagonalized by the graph Fourier transform; a fact that is leveraged to define a notion of power spectral density (PSD). Properties of the graph PSD are analyzed and a number of methods for its estimation are proposed. This includes generalizations of nonparametric approaches such as periodograms, window-based average periodograms, and filter banks, as well as parametric approaches, using moving-average, autoregressive, and ARMA processes. Graph stationarity and graph PSD estimation are investigated numerically for synthetic and real-world graph signals.

Bipartite Graph Filter Banks: Polyphase Analysis and Generalization

The work by Narang and Ortega [“Perfect reconstruction two-channel wavelet filter banks for graph structured data,” IEEE Trans. Signal Process. , vol. 60, no. 6, pp. 2786–2799, Jun. 2012], [“Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs,” IEEE Trans. Signal Process. , vol. 61, no. 19, pp. 4673–4685, Oct. 2013] laid the foundations for the two-channel critically sampled perfect reconstruction filter bank for signals defined on undirected graphs. This basic filter bank is applicable only to bipartite graphs but using the notion of separable filtering, the basic filter bank can be applied to any arbitrary undirected graphs. In this paper, several new theoretical results are presented. In particular, the proposed polyphase analysis yields filtering structures in the downsampled domain that are equivalent to those before downsampling and, thus, can be exploited for efficient implementation. These theoretical results also provide new insights that can be exploited in the design of these systems. These insights allow us to generalize these filter banks to directed graphs and to using a variety of graph base matrices, while also providing a link to the $\text{DSP}_G$ framework of Sandryhaila and Moura [“Discrete signal processing on graphs,” IEEE Trans. Signal Process. , vol. 61, no. 7, pp. 1644–1636, Apr. 2013], [“Discrete signal processing on graphs: Frequency analysis,” IEEE Trans. Signal Process. , vol. 62, no. 12, pp. 3042–3054, Jun. 2014]. Experiments show evidence that better nonlinear approximation and denoising results may be obtained by a better selection of these base matrices.

Graph Sampling for Covariance Estimation

In this paper, the focus is on subsampling as well as reconstructing the second-order statistics of signals residing on nodes of arbitrary undirected graphs. Second-order stationary graph signals may be obtained by graph filtering zero-mean white noise and they admit a well-defined power spectrum whose shape is determined by the frequency response of the graph filter. Estimating the graph power spectrum forms an important component of stationary graph signal processing and related inference tasks such as Wiener prediction or inpainting on graphs. The central result of this paper is that by sampling a significantly smaller subset of vertices and using simple least squares, we can reconstruct the second-order statistics of the graph signal from the subsampled observations, and more importantly, without any spectral priors. To this end, both a nonparametric approach as well as parametric approaches including moving average and autoregressive models for the graph power spectrum are considered. The results specialize for undirected circulant graphs in that the graph nodes leading to the best compression rates are given by the so-called minimal sparse rulers. A near-optimal greedy algorithm is developed to design the subsampling scheme for the nonparametric and the moving average models, whereas a particular subsampling scheme that allows linear estimation for the autoregressive model is proposed. Numerical experiments on synthetic as well as real datasets related to climatology and processing handwritten digits are provided to demonstrate the developed theory.

Filtering Random Graph Processes Over Random Time-Varying Graphs

Graph filters play a key role in processing the graph spectra of signals supported on the vertices of a graph. However, despite their widespread use, graph filters have been analyzed only in the deterministic setting, ignoring the impact of stochastic- ity in both the graph topology as well as the signal itself. To bridge this gap, we examine the statistical behavior of the two key filter types, finite impulse response (FIR) and autoregressive moving average (ARMA) graph filters, when operating on random time- varying graph signals (or random graph processes) over random time-varying graphs. Our analysis shows that (i) in expectation, the filters behave as the same deterministic filters operating on a deterministic graph, being the expected graph, having as input signal a deterministic signal, being the expected signal, and (ii) there are meaningful upper bounds for the variance of the filter output. We conclude the paper by proposing two novel ways of exploiting randomness to improve (joint graph-time) noise cancellation, as well as to reduce the computational complexity of graph filtering. As demonstrated by numerical results, these methods outperform the disjoint average and denoise algorithm, and yield a (up to) four times complexity redution, with very little difference from the optimal solution.

Optimal Graph-Filter Design and Applications to Distributed Linear Network Operators

We study the optimal design of graph filters (GFs) to implement arbitrary linear transformations between graph signals. GFs can be represented by matrix polynomials of the graph-shift operator (GSO). Since this operator captures the local structure of the graph, GFs naturally give rise to distributed linear network operators. In most setups, the GSO is given so that GF design consists fundamentally in choosing the (filter) coefficients of the matrix polynomial to resemble desired linear transformations. We determine spectral conditions under which a specific linear transformation can be implemented perfectly using GFs. For the cases where perfect implementation is infeasible, we address the optimization of the filter coefficients to approximate the desired transformation. Additionally, for settings where the GSO itself can be modified, we study its optimal design as well. After this, we introduce the notion of a node-variant GF, which allows the simultaneous implementation of multiple (regular) GFs in different nodes of the graph. This additional flexibility enables the design of more general operators without undermining the locality in implementation. Perfect and approximate designs are also studied for this new type of GFs. To showcase the relevance of the results in the context of distributed linear network operators, this paper closes with the application of our framework to two particular distributed problems: finite-time consensus and analog network coding.

Graph Filters and the <italic>Z</italic>-Laplacian

In network science, the interplay between dynamical processes and the underlying topologies of complex systems has led to a diverse family of models with different interpretations. In graph signal processing, this is manifested in the form of different graph shifts and their induced algebraic systems. In this paper, we propose the unifying Z-Laplacian framework, whose instances can act as graph shift operators. As a generalization of the traditional graph Laplacian, the Z-Laplacian spans the space of all possible Z-matrices, i.e., real square matrices with nonpositive off-diagonal entries. We show that the Z-Laplacian can model general continuous-time dynamical processes, including information flows and epidemic spreading on a given graph. It is also closely related to general nonnegative graph filters in the discrete time domain. We showcase its flexibility by considering two applications. First, we consider a wireless communications networking problem modeled with a graph, where the framework can be applied to model the effects of the underlying communications protocol and traffic. Second, we examine a structural brain network from the perspective of low- to high-frequency connectivity.

Fast Resampling of Three-Dimensional Point Clouds via Graphs

To reduce cost in storing, processing and visualizing a large-scale point cloud, we consider a randomized resampling strategy to select a representative subset of points while preserving application-dependent features. The proposed strategy is based on graphs, which can represent underlying surfaces and lend themselves well to efficient computation. We use a general feature-extraction operator to represent application-dependent features and propose a general reconstruction error to evaluate the quality of resampling. We obtain a general form of optimal resampling distribution by minimizing the reconstruction error. The proposed optimal resampling distribution is guaranteed to be shift, rotation and scale-invariant in the 3D space. We next specify the feature-extraction operator to be a graph filter and study specific resampling strategies based on all-pass, low-pass, high-pass graph filtering and graph filter banks. We finally apply the proposed methods to three applications: large-scale visualization, accurate registration and robust shape modeling. The empirical performance validates the effectiveness and efficiency of the proposed resampling methods.

Kernel-Based Reconstruction of Graph Signals

A number of applications in engineering, social sciences, physics, and biology involve inference over networks. In this context, graph signals are widely encountered as descriptors of vertex attributes or features in graph-structured data. Estimating such signals in all vertices given noisy observations of their values on a subset of vertices has been extensively analyzed in the literature of signal processing on graphs (SPoG). This paper advocates kernel regression as a framework generalizing popular SPoG modeling and reconstruction and expanding their capabilities. Formulating signal reconstruction as a regression task on reproducing kernel Hilbert spaces of graph signals permeates benefits from statistical learning, offers fresh insights, and allows for estimators to leverage richer forms of prior information than existing alternatives. A number of SPoG notions such as bandlimitedness, graph filters, and the graph Fourier transform are naturally accommodated in the kernel framework. Additionally, this paper capitalizes on the so-called representer theorem to devise simpler versions of existing Thikhonov regularized estimators, and offers a novel probabilistic interpretation of kernel methods on graphs based on graphical models. Motivated by the challenges of selecting the bandwidth parameter in SPoG estimators or the kernel map in kernel-based methods, the present paper further proposes two multi-kernel approaches with complementary strengths. Whereas the first enables estimation of the unknown bandwidth of bandlimited signals, the second allows for efficient graph filter selection. Numerical tests with synthetic as well as real data demonstrate the merits of the proposed methods relative to state-of-the-art alternatives.

Autoregressive Moving Average Graph Filtering

One of the cornerstones of the field of signal processing on graphs are graph filters, direct analogues of classical filters, but intended for signals defined on graphs. This work brings forth new insights on the distributed graph filtering problem. We design a family of autoregressive moving average (ARMA) recursions, which (i) are able to approximate any desired graph frequency response, and (ii) give exact solutions for tasks such as graph signal denoising and interpolation. The design philosophy, which allows us to design the ARMA coefficients independently from the underlying graph, renders the ARMA graph filters suitable in static and, particularly, time-varying settings. The latter occur when the graph signal and/or graph are changing over time. We show that in case of a time-varying graph signal our approach extends naturally to a two-dimensional filter, operating concurrently in the graph and regular time domains. We also derive sufficient conditions for filter stability when the graph and signal are time-varying. The analytical and numerical results presented in this paper illustrate that ARMA graph filters are practically appealing for static and time-varying settings, as predicted by theoretical derivations.

Extending Classical Multirate Signal Processing Theory to Graphs—Part II: M-Channel Filter Banks

This paper builds upon the basic theory of multirate systems for graph signals developed in the companion paper (Part I) and studies $M$ -channel polynomial filter banks on graphs. The behavior of such graph filter banks differs from that of classical filter banks in many ways, the precise details depending on the eigenstructure of the adjacency matrix $\boldsymbol {A}$ . It is shown that graph filter banks represent, (linear and) periodically, shift-variant systems only when $\boldsymbol {A}$ satisfies the noble identity conditions developed in Part I. It is then shown that perfect reconstruction graph filter banks can always be developed when $\boldsymbol {A}$ satisfies the eigenvector structure satisfied by $M$ -block cyclic graphs and has distinct eigenvalues (further restrictions on eigenvalues being unnecessary for this). If $\boldsymbol {A}$ is actually $M$ -block cyclic then these PR filter banks indeed become practical, i.e., arbitrary filter polynomial orders are possible, and there are robustness advantages. In this case, the PR condition is identical to PR in classical filter banks—any classical PR example can be converted to a graph PR filter bank on an $M$ -block cyclic graph. It is shown that for $M$ -block cyclic graphs with all eigenvalues on the unit circle, the frequency responses of filters have meaningful correspondence with classical filter banks. Polyphase representations are then developed for graph filter banks and utilized to develop alternate conditions for alias cancellation and perfect reconstruction, again for graphs with specific eigenstructures. It is then shown that the eigenvector condition on the graph can be relaxed by using similarity transforms.

Extending Classical Multirate Signal Processing Theory to Graphs—Part I: Fundamentals

Signal processing on graphs finds applications in many areas. In recent years, renewed interest on this topic was kindled by two groups of researchers. Narang and Ortega constructed two-channel filter banks on bipartitie graphs described by Laplacians. Sandryhaila and Moura developed the theory of linear systems, filtering, and frequency responses for the case of graphs with arbitrary adjacency matrices, and showed applications in signal compression, prediction, etc. Inspired by these contributions, this paper extends classical multirate signal processing ideas to graphs. The graphs are assumed to be general with a possibly nonsymmetric and complex adjacency matrix. The paper revisits ideas, such as noble identities, aliasing, and polyphase decompositions in graph multirate systems. Drawing such a parallel to classical systems allows one to design filter banks with polynomial filters, with lower complexity than arbitrary graph filters. It is shown that the extension of classical multirate theory to graphs is nontrivial, and requires certain mathematical restrictions on the graph. Thus, classical noble identities cannot be taken for granted. Similarly, one cannot claim that the so-called delay chain system is a perfect reconstruction system (as in classical filter banks). It will also be shown that M-partite extensions of the bipartite filter bank results will not work for M-channel filter banks, but a more restrictive condition called M-block cyclic property should be imposed. Such graphs are studied in detail. A detailed theory for M-channel filter banks is developed in a companion paper.

Reconstruction of Graph Signals Through Percolation from Seeding Nodes

New schemes to recover signals defined in the nodes of a graph are proposed. Our focus is on reconstructing bandlimited graph signals, which are signals that admit a sparse representation in a frequency domain related to the structure of the graph. Most existing formulations focus on estimating an unknown graph signal by observing its value on a subset of nodes. By contrast, in this paper, we study the problem of reconstructing a known graph signal using as input a graph signal that is non-zero only for a small subset of nodes (seeding nodes). The sparse signal is then percolated (interpolated) across the graph using a graph filter. Graph filters are a generalization of classical time-invariant systems and represent linear transformations that can be implemented distributedly across the nodes of the graph. Three setups are investigated. In the first one, a single simultaneous injection takes place on several nodes in the graph. In the second one, successive value injections take place on a single node. The third one is a generalization where multiple nodes inject multiple signal values. For noiseless settings, conditions under which perfect reconstruction is feasible are given, and the corresponding schemes to recover the desired signal are specified. Scenarios leading to imperfect reconstruction, either due to insufficient or noisy signal value injections, are also analyzed. Moreover, connections with classical interpolation in the time domain are discussed. The last part of the paper presents numerical experiments that illustrate the results developed through synthetic graph signals and two real-world signal reconstruction problems: influencing opinions in a social network and inducing a desired brain state in humans.

Lifting‐based design of two‐channel biorthogonal graph filter bank

In this study, the lifting scheme is first employed to design two-channel biorthogonal graph filter bank. The biorthogonal condition is parameterised by imposing a single-level lifting structure on the analysis and synthesis graph kernels. Based on the parametric structure, the two kernels are separately optimised by constrained quadratic programming. The obtained two-channel biorthogonal graph filter banks are of structurally perfect reconstruction. Numerical results and comparison are included to show the proposed algorithm can lead to biorthogonal graph filter banks with improved performance.

Critically sampled graph filter banks with polynomial filters from regular domain filter banks

Graph signal processing deals with the processing of signals defined on irregular domains and is an emerging area of research. Graph filter banks allow the wavelet transform to be extended for processing graph signals. Sakiyama and Tanaka (2015) [22] recently proposed a technique to convert linear-phase biorthogonal filter banks for regular domain signals to biorthogonal graph filter banks. Perfect reconstruction is preserved using the technique but the resulting spectral filter functions are transcendental and not polynomial. Polynomial function filters are desired for the localization property and implementation efficiency. In this work we present alternative techniques to perform the conversion. Perfect reconstruction is preserved with the proposed techniques and the resulting spectral filters are polynomial functions.

Highly localised near orthogonal graph wavelets

The processing of signals defined over domains modelled as graphs is becoming a fast emerging area of research. A graph filter bank allows for the wavelet transform on graph signals that is similar to the discrete wavelet transform (DWT) for regular domain signals. Localisation is an important property of the transform and the most localised DWT is with the Haar wavelets. Spectral graph filters that are highly localised is constructed. These filters can be considered as the counterparts to the Haar filters in DWT.

Blind Identification of Graph Filters

Network processes are often represented as signals defined on the vertices of a graph. To untangle the latent structure of such signals, one can view them as outputs of linear graph filters modeling underlying network dynamics. This paper deals with the problem of joint identification of a graph filter and its input signal, thus broadening the scope of classical blind deconvolution of temporal and spatial signals to the less-structured graph domain. Given a graph signal $\mathbf{y}$ modeled as the output of a graph filter, the goal is to recover the vector of filter coefficients $\mathbf{h}$, and the input signal $\mathbf{x}$ which is assumed to be sparse. While $\mathbf{y}$ is a bilinear function of $\mathbf{x}$ and $\mathbf{h}$, the filtered graph signal is also a linear combination of the entries of the lifted rank-one, row-sparse matrix $\mathbf{x} \mathbf{h}^T$. The blind graph-filter identification problem can thus be tackled via rank and sparsity minimization subject to linear constraints, an inverse problem amenable to convex relaxations offering provable recovery guarantees under simplifying assumptions. Numerical tests using both synthetic and real-world networks illustrate the merits of the proposed algorithms, as well as the benefits of leveraging multiple signals to aid the blind identification task.

Distributed Autoregressive Moving Average Graph Filters

We introduce the concept of autoregressive moving average (ARMA) filters on a graph and show how they can be implemented in a distributed fashion. Our graph filter design philosophy is independent of the particular graph, meaning that the filter coefficients are derived irrespective of the graph. In contrast to finite-impulse response (FIR) graph filters, ARMA graph filters are robust against changes in the signal and/or graph. In addition, when time-varying signals are considered, we prove that the proposed graph filters behave as ARMA filters in the graph domain and, depending on the implementation, as first or higher ARMA filters in the time domain.

Techniques for Constructing Biorthogonal Bipartite Graph Filter Banks

The processing of data defined on irregular discrete domains, i.e., graph signals, is becoming an emerging area with great application potential. Using spectral graph theory, Narang and Ortega (2013) laid the framework for two channel filter banks with critical sampling for bipartite graph signals. The bipartite graph filter bank can be extended to any arbitrary graph using the notion of separable filtering. The design of the biorthogonal filter banks by Narang and Ortega (2013) is based on the factorization of a maximally flat polynomial. The factorization technique does not allow much control of the spectral response of the graph filters, resulting in response asymmetry. In this paper, we present a generic framework for constructing biorthogonal graph filter banks that does not require factorization. We introduce the notion of polyphase representation and ladder structures for graph filter banks. We show that filters having virtual spectral symmetry and almost energy preservation can be constructed without any sophisticated optimization. Fine control of the spectral response can also be achieved with ease.

Infinite Impulse Response Graph Filters in Wireless Sensor Networks

Many signal processing problems in wireless sensor networks can be solved by graph filtering techniques. Finite impulse response (FIR) graph filters (GFs) have received more attention in the literature because they enable distributed computation by the sensors. However, FIR GFs are limited in their ability to represent the global information of the network. This letter proposes a family of GFs with infinite impulse response (IIR) and provides algorithms for their distributed realization in wireless sensor networks. IIR GFs bring more flexibility to GF designers, as they can be designed and realized even when the graph spectrum is unknown. Numerical results show that IIR GFs are more accurate in approximating ideal GFs and more robust against network variation than FIR GFs.

Design of Near Orthogonal Graph Filter Banks

The processing of signal on graphs is becoming an important emerging area that has great potential in a wide variety of applications, e.g. social network. The work by Narang and Ortega (2012) laid the framework for critically sampled orthogonal two-channel filter bank for signal on undirected bipartite graphs. The design method presented by Narang and Ortega (2012) does not allow the tailoring of the filters spectral response and the control of the reconstruction error of the filter bank. This work present a method to design spectral filters that is based on Bernstein polynomial approximation and constrained optimization. The method allows the trade-off between transition band sharpness, ripple magnitude and reconstruction error.