# Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Notation

#### 2.2. Prior Art

#### 2.3. Contributions

- We provide an overview of families of graphs that accept good graph separators, i.e., graphs that can be separated into roughly equal subgraphs. Also, we discuss the characteristics of such families and how they relate to practical applications.
- Restricting ourselves to such graph families with good recursive graph separators, we propose a hierarchical decomposition for such families of graphs for approximate graph spectrum estimation.
- We propose a conquer mechanism to stitch back together the pieces of the hierarchical decomposition for approximate graph mode estimation.
- We provide a theoretical analysis, leveraging current results on graph spectrum similarity, for the proposed hierarchical decomposition. We derive asymptotic bounds for both the accuracy of the graph spectrum approximation and the computational complexity of the proposed method using the hierarchical decomposition. Through numerical simulations, we show that in practice the approximation of the graph spectrum, using such a decomposition, obeys the derived bounds.
- We employ our results to applications commonly found in the field of graph signal processing. In particular, we derive bounds for the accuracy of approximate graph filtering using the proposed decomposition for the considered families of graphs. Despite that this result shows that a straightforward application of decomposition does not provide an efficient approximation, it sheds light on the reach of such an approach. In addition, we show that the cumulative spectral density of a given graph, commonly employed for the design of graph filter banks, can be properly approximated using the proposed decomposition. This differs from other approaches used in practice which do not provide any approximation guarantees.

#### 2.4. Paper Outline

## 3. Topological Graph Theory

#### 3.1. Graph Separators

- (i)
- Finding optimal graph separators is hard.
- (ii)
- Graphs whose graph separators are sublinear in size are considered as good graph separators. That is, the number of vertices (edges) that needs to be removed to partition the graph in balanced sets is $o(n)$.
- (iii)
- Graphs with good recursive graph separators are graphs whose resulting partitions have good separators themselves. For instance, if the class $\mathcal{S}$ of graphs has good graph separators, and $\mathcal{S}$ is closed under vertex (edge) deletion, then the graphs in $\mathcal{S}$ have good recursive graph separators.

#### 3.2. Structured Graphs with Good Graph Separators

**Definition**

**1.**

## 4. Approximate Graph Spectrum

**Theorem**

**1.**

**Proof.**

## 5. Divide-And-Conquer for Fast Graph Spectrum Estimation

Algorithm 1:DC_Approx_Graph_Spectrum ($\mathcal{G},h,d$). |

`merge`, and depth d, we decided to keep the algorithm description as general as possible due to the fact that these three free parameters impact the method in the following ways.

- (a)
- The function h directly affects the computational complexity as it is the core operation of the algorithm. In addition, if h is not an exact algorithm but a randomized or approximate method, then it will impact as well the quality of the final approximation.
- (b)
- The
`merge`function is the conquer step that stitches back the solution of the leaves. As this function is called at every non-leaf node, dealing with increasing-size arguments (traversing the tree upward), it must be a low-complexity routine to not blow up the computational complexity of the method. - (c)
- The depth of the tree, i.e., the number of recursive bisections, affects both the computational complexity and approximation quality. As each bisection worsens the approximation, deeper binary trees (theoretically) worsen the quality of the approximated spectrum. In addition, as the depth parameter controls the base case of the recursion for applying h, its value also impacts the final complexity of the algorithm.

#### 5.1. Accuracy-Complexity Trade-Off

`merge`routine solely sorts its input, incurring a complexity of $O(nlogn)$ for inputs of size $n/2$ each.

**Definition**

**2.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

#### 5.2. Eigenvectors Computation

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 6. Experimental Results

#### 6.1. Depth of Hierarchical Decomposition

#### 6.2. Asymptotics for Graph Size

#### 6.3. Time Complexity

#### 6.4. Real Example: Minnesota, a Close-To-Planar Graph

## 7. Approximation of Graph Filtering

**Theorem**

**5.**

**Proof.**

#### 7.1. Results for Approximate Graph Filtering

#### 7.2. A Note on Graph Filter Bank Design

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Graph Families with Good Separators

**Planar Graphs.**A graph that can be drawn without edge crossings is called planar [49]. This type of graphs arises typically in applications related to 2-D meshes and computer graphics. For such graphs the following separator theorem is known:

**Theorem**

**A1**

**.**Planar graphs satisfy a $\mathcal{O}({n}^{1/2})$-separator theorem with constant $\alpha =2/3$ and $\beta =2\sqrt{2}.$

**Almost Planar Graphs.**Similar to planar graphs, almost planar graphs can be made planar by solely removing a small number of edges. Instances of this kind of graphs are road networks or power grids. In such (physical) networks, planarity is lost by bridges or tunnels, therefore by removing such edges, the graph can be approximated through a planar one. As a result, the $\mathcal{O}({n}^{1/2})$-separator theorem for planar graphs holds (approximately) for this family of graphs.

**Finite Element Graphs.**This family of graphs arises from finite element methods, for example through a tessellation of the space. Following the formal description of Gilbert, et al. a finite element graph can be obtained from a planar graph as follows. First, the graph is embedded in a plane. Next, we identify certain points as nodes, e.g., vertices, points on edges, points in faces. Then, edges between all nodes that share a face are drawn. From this construction, if the number of nodes per face is bounded by d, finite elements graphs satisfy a $\mathcal{O}(d{n}^{1/2})$-separator theorem [28].

**Graphs Embedded in a Low-Dimensional Space.**Interestingly enough, nearly all graphs that are used to represent connections in low dimensional spaces have small separators. For example, 3-D meshes, under certain nicety conditions, accept a $\mathcal{O}({n}^{2/3})$-separator theorem [53]. In general, unstructured meshes of dimension d allow a $\mathcal{O}({n}^{(d-1)/d})$-separator theorem [53].

**Circuits.**These are one of the families of graphs that motivated the early application of graph separators. This class and its separator are still used for VLSI design when designers want to minimize the area employed [54]. If the circuit, represented through vertices (components) and edges (connections) is drawn with few crossings, then it may be considered as an almost planar graph. Hence, the $\mathcal{O}({n}^{1/2})$-separator theorem approximately holds. Otherwise, if the circuit can be embedded on a surface of genus [9] g, then it has an exact $\mathcal{O}({(gn)}^{1/2})$-separator theorem [51].

**Geometric Graphs.**This family of graphs arises by construction through geometrical objects. As this family of graphs encloses a large variety of objects, e.g., k-nearest-neighbor graphs, meshes, etc., here we only consider the unit disk graphs subfamily. Unit disk graphs arise naturally in applications involving sensors networks. These graphs are combinatorial objects generated by the intersection of disks on the plane. They are also known as Euclidean graphs. These graphs are constructed from points in the space (vertices) when edges between points are drawn if the distance between them is smaller than some threshold, e.g., range-limit communication graph. For this family of graphs, a $\mathcal{O}({n}^{1/2})$-separator theorem holds similarly to the case of planar graphs as they are embeddable in a 2-D surface. For a more in-depth discussion of the results related to this kind of graphs, the reader is referred to [29].

**Social Graphs.**Networks such as friend, bibliographic or citation graphs have good separators in practice as they are based on communities, thus exhibiting local structure, see, e.g, [55]. Within social graphs, most links can be found within some other form of community or local domain, as the link graphs used for the web. Unfortunately, differently from the previously discussed types of graphs, social graphs cannot be guaranteed to accept good recursive separators in general. This situation can be observed in a social network represented by a power-law graph. Differently from planar or geometric graphs, despite that a power-law graph can be easily separated, there is no guarantee that its partitions accept good separators themselves as it is not (necessarily) closed under (vertex) edge deletion. However, in practice, this seldom is the case as networks of friends are, again, networks of friends [56].

## Appendix B. Proof of Theorem 1

## Appendix C. Proof of Theorem 2

## Appendix D. Proof Theorem 3

## Appendix E. Proof of Theorem 4

**Theorem**

**A2**

**.**On input an $n\times n$ symmetric diagonally dominant matrix $\mathit{X}$ with m non-zero entries and a vector $\mathit{b}$, a vector $\overline{\mathit{x}}$ satisfying $\parallel \overline{\mathit{x}}-{\mathit{X}}^{\u2020}{\mathit{b}\parallel}_{\mathit{X}}<\eta {\parallel {\mathit{X}}^{\u2020}\mathit{b}\parallel}_{\mathit{X}}$, can be computed in expected time $\tilde{\mathcal{O}}(m{log}^{2}nlog1/\eta )$.

## Appendix F. Proof Theorem 5

## References

- Shuman, D.I.; Narang, S.K.; Frossard, P.; Ortega, A.; Vandergheynst, P. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Sig. Proc. Mag.
**2013**, 30, 83–98. [Google Scholar] [CrossRef] [Green Version] - Sandryhaila, A.; Moura, J.M. Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure. IEEE Sig. Proc. Mag.
**2014**, 31, 80–90. [Google Scholar] [CrossRef] - Shuman, D.I.; Ricaud, B.; Vandergheynst, P. Vertex-frequency analysis on graphs. Appl. Comput. Harmon. Anal.
**2016**, 40, 260–291. [Google Scholar] [CrossRef] - Chung, F.R. Spectral Graph Theory; Number 92; American Mathematical Society: Providence, RI, USA, 1997. [Google Scholar]
- Le Magoarou, L.; Gribonval, R.; Tremblay, N. Approximate Fast Graph Fourier Transforms via Multilayer Sparse Approximations. IEEE Trans. Signal Inf. Process. Netw.
**2018**, 4, 407–420. [Google Scholar] [CrossRef] [Green Version] - Le Magoarou, L.; Gribonval, R. Flexible multilayer sparse approximations of matrices and applications. IEEE J. Sel. Top. Signal Process.
**2016**, 10, 688–700. [Google Scholar] [CrossRef] [Green Version] - Le Magoarou, L.; Tremblay, N.; Gribonval, R. Analyzing the approximation error of the fast graph fourier transform. In Proceedings of the 2017 51st Asilomar Conference on the Signals, Systems, and Computers, Pacific Grove, CA, USA, 29 October–1 November 2017; pp. 45–49. [Google Scholar]
- Golub, G.H.; Van Loan, C.F. Matrix Computations; Johns Hopkins Universtiy Press: Baltimore, MD, USA, 2012; Volume 3. [Google Scholar]
- Diestel, R. Graph Theory (Graduate Texts in Mathematics); Springer: Heidelberg, Germany, 2005; Volume 173. [Google Scholar]
- Spielman, D.A. Algorithms, graph theory, and linear equations in Laplacian matrices. In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures; Springer: Berlin, Germany, 2010; pp. 2698–2722. [Google Scholar]
- Vaidya, P.M. Solving linear equations with symmetric diagonally dominant matrices by constructing good preconditioners. Talk Based This Manuscr.
**1991**, 2, 2–4. [Google Scholar] - Koutis, I.; Miller, G.L.; Peng, R. Approaching optimality for solving SDD linear systems. In Proceedings of the 2010 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), Las Vegas, NV, USA, 23–26 October 2010; pp. 235–244. [Google Scholar]
- Karypis, G.; Kumar, V. A fast and high quality multilevel scheme for partitioning irregular graphs. Siam J. Sci. Comput.
**1998**, 20, 359–392. [Google Scholar] [CrossRef] - Loukas, A.; Vandergheynst, P. Spectrally approximating large graphs with smaller graphs. arXiv
**2018**, arXiv:1802.07510. [Google Scholar] - Candes, E.J. The restricted isometry property and its implications for compressed sensing. Comptes Rendus Math.
**2008**, 346, 589–592. [Google Scholar] [CrossRef] - Narang, S.K.; Ortega, A. Downsampling graphs using spectral theory. In Proceedings of the 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, Czech Republic, 22–27 May 2011; pp. 4208–4211. [Google Scholar]
- Narang, S.K.; Ortega, A. Perfect reconstruction two-channel wavelet filter banks for graph structured data. IEEE Trans. Signal Process
**2012**, 60, 2786–2799. [Google Scholar] [CrossRef] [Green Version] - Saad, Y. Iterative Methods for Sparse Linear Systems; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2003. [Google Scholar]
- Irion, J.; Saito, N. Hierarchical graph Laplacian eigen transforms. JSIAM Lett.
**2014**, 6, 21–24. [Google Scholar] [CrossRef] [Green Version] - Tremblay, N.; Borgnat, P. Subgraph-based filterbanks for graph signals. IEEE Trans. Signal Process.
**2016**, 64, 3827–3840. [Google Scholar] [CrossRef] [Green Version] - Szlam, A.D.; Maggioni, M.; Coifman, R.R.; Bremer, J.C., Jr. Diffusion-driven multiscale analysis on manifolds and graphs: Top-down and bottom-up constructions. In Proceedings of the Wavelets XI. International Society for Optics and Photonics, San Diego, CA, USA, 21 September 2005; Volume 5914, p. 59141D. [Google Scholar]
- Kondor, R.; Teneva, N.; Garg, V. Multiresolution matrix factorization. In Proceedings of the International Conference on Machine Learning, Beijing, China, 21–26 June 2014; pp. 1620–1628. [Google Scholar]
- Teneva, N.; Mudrakarta, P.K.; Kondor, R. Multiresolution matrix compression. In Proceedings of the Artificial Intelligence and Statistics, Cadiz, Spain, 9–11 May 2016; pp. 1441–1449. [Google Scholar]
- Ithapu, V.K.; Kondor, R.; Johnson, S.C.; Singh, V. The incremental multiresolution matrix factorization algorithm. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Hawaii, HI, USA, 21–26 July 2017; pp. 2951–2960. [Google Scholar]
- Gross, J.L.; Tucker, T.W. Topological Graph Theory; Courier Corporation: Chelmsford, MA, USA, 1987. [Google Scholar]
- Beineke, L.; Wilson, R. Topics in Topological Graph Theory; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Rosenberg, A.L.; Heath, L.S. Graph Separators, With Applications; Springer Science & Business Media: Berlin, Germany, 2001. [Google Scholar]
- Lipton, R.J.; Tarjan, R.E. A separator theorem for planar graphs. Siam J. Appl. Math.
**1979**, 36, 177–189. [Google Scholar] [CrossRef] - Smith, W.D.; Wormald, N.C. Geometric separator theorems and applications. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), FOCS. IEEE, Palo Alto, CA, USA, 8–11 November 1998; p. 232. [Google Scholar]
- Rutter, J.D. A Serial Implementation of Cuppen’s Divide and Conquer Algorithm For the Symmetric Eigenvalue Problem; University of California: Berkeley, CA, USA, 1994. [Google Scholar]
- Bauer, F.L.; Fike, C.T. Norms and exclusion theorems. Numer. Math.
**1960**, 2, 137–141. [Google Scholar] [CrossRef] - Koutis, I.; Miller, G.L. A linear work, O (n 1/6) time, parallel algorithm for solving planar Laplacians. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, New Orleans, Louisiana, 7–9 January 2007; pp. 1002–1011. [Google Scholar]
- Matlab Mesh Partitioning and Graph Separator Toolbox. Available online: https://sites.cs.ucsb.edu/~gilbert/cs219/cs219Spr2018/Matlab/meshpart/meshpart.htm (accessed on 31 August 2020).
- MatlabBGL: A Matlab Graph Library. Available online: https://www.cs.purdue.edu/homes/dgleich/packages/matlab_bgl/ (accessed on 5 July 2020).
- MatlabBGL. Planar graphs in MatlabBGL. 2020. Available online: https://www.cs.purdue.edu/homes/dgleich/packages/matlab_bgl/planar_graphs/planar_graphs.html (accessed on 31 August 2020).
- Van Mieghem, P. Graph Spectra for Complex Networks; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Marques, A.G.; Segarra, S.; Leus, G.; Ribeiro, A. Stationary graph processes and spectral estimation. IEEE Trans. Signal Process
**2017**, 65, 5911–5926. [Google Scholar] [CrossRef] - Perraudin, N.; Paratte, J.; Shuman, D.; Martin, L.; Kalofolias, V.; Vandergheynst, P.; Hammond, D.K. GSPBOX: A toolbox for signal processing on graphs. arXiv
**2014**, arXiv:cs.IT/1408.5781. [Google Scholar] - Coutino, M.; Isufi, E.; Leus, G. Advances in Distributed Graph Filtering. arXiv
**2018**, arXiv:1808.03004. [Google Scholar] [CrossRef] [Green Version] - Liu, J.; Isufi, E.; Leus, G. Autoregressive moving average graph filter design. In Proceedings of the 2017 IEEE Global Conference on. IEEE Signal and Information Processing (GlobalSIP), Montreal, QC, Canada, 14–16 November 2017; pp. 593–597. [Google Scholar]
- Segarra, S.; Marques, A.; Ribeiro, A. Optimal Graph-Filter Design and Applications to Distributed Linear Network Operators. IEEE Trans. Signal Process
**2017**, 65, 4117–4131. [Google Scholar] [CrossRef] - Girault, B.; Gonçalves, P.; Fleury, E.; Mor, A.S. Semi-supervised learning for graph to signal mapping: A graph signal wiener filter interpretation. In Proceedings of the 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, Italy, 4–9 May 2014; pp. 1115–1119. [Google Scholar]
- Isufi, E.; Loukas, A.; Simonetto, A.; Leus, G. Autoregressive Moving Average Graph Filtering. IEEE Trans. Signal Process
**2017**, 65, 274–288. [Google Scholar] [CrossRef] [Green Version] - Teke, O.; Vaidyanathan, P.P. Extending classical multirate signal processing theory to graphs Part II: M-channel filter banks. IEEE Trans. Signal Process
**2017**, 65, 423–437. [Google Scholar] [CrossRef] - Tay, D.B.; Lin, Z. Design of near orthogonal graph filter banks. IEEE Sig. Proc. Lett.
**2015**, 22, 701–704. [Google Scholar] [CrossRef] - Li, S.; Jin, Y.; Shuman, D.I. A Scalable M-Channel Critically Sampled Filter Bank for Graph Signals. arXiv
**2016**, arXiv:1608.03171. [Google Scholar] [CrossRef] [Green Version] - Lin, L.; Saad, Y.; Yang, C. Approximating spectral densities of large matrices. SIAM Rev.
**2016**, 58, 34–65. [Google Scholar] [CrossRef] - Avron, H.; Toledo, S. Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix. J. ACM (JACM)
**2011**, 58, 1–34. [Google Scholar] [CrossRef] - Trudeau, R.J. Introduction to Graph Theory; Courier Corporation: Chelmsford, MA, USA, 2013. [Google Scholar]
- Gilbert, J.R. Graph Separator Theorems and Sparse Gaussian Elimination. Ph.D. Thesis, Stanford University, Stanford, CA, USA, 1981. [Google Scholar]
- Gilbert, J.R.; Hutchinson, J.P.; Tarjan, R.E. A separator theorem for graphs of bounded genus. In Technical Report; Cornell University: Ithaca, NY, USA, 1982. [Google Scholar]
- Gilbert, J.R.; Rose, D.J.; Edenbrandt, A. A separator theorem for chordal graphs. Siam J. Algebr. Discret. Methods
**1984**, 5, 306–313. [Google Scholar] [CrossRef] [Green Version] - Miller, G.L.; Teng, S.H.; Thurston, W.; Vavasis, S.A. Separators for sphere-packings and nearest neighbor graphs. J. ACM (JACM)
**1997**, 44, 1–29. [Google Scholar] [CrossRef] - Leiserson, C.E. Area-Efficient Graph Layouts (for VLSI). In Technical Report; Carnegie-Mellon University: Pittsburgh, PA, USA, 1980. [Google Scholar]
- Blandford, D.K.; Blelloch, G.E.; Kash, I.A. An experimental analysis of a compact graph representation. In ALENEX/ANALC; Arge, L., Italiano, G.F., Sedgewick, R., Eds.; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2004. [Google Scholar]
- Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small-world’networks. Nature
**1998**, 393, 440–442. [Google Scholar] [CrossRef] - Cohen-Steiner, D.; Kong, W.; Sohler, C.; Valiant, G. Approximating the spectrum of a graph. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, London, UK, 19–23 August 2018; pp. 1263–1271. [Google Scholar]
- Sanfeliu, A.; Fu, K.S. A distance measure between attributed relational graphs for pattern recognition. IEEE Trans. Syst. Man Cybern.
**1983**, 353–362. [Google Scholar] [CrossRef] - Horn, R.A.; Johnson, C.R. Topics in Matrix Analysis; Cambridge University Presss: Cambridge, UK, 1991; Volume 37, p. 39. [Google Scholar]

**Figure 1.**Illustration of a vertex separator. The vertex set $\mathcal{V}$ is divided in three subsets $\mathcal{A},\phantom{\rule{0.166667em}{0ex}}\mathcal{B},\phantom{\rule{0.166667em}{0ex}}\mathcal{C}$ such that $\mathcal{A}$ and $\mathcal{B}$ are disconnected, while minimizing the size of $\mathcal{C}$ and maintaining a balance (in terms of the number of vertices) between $\mathcal{A}$ and $\mathcal{B}$.

**Figure 2.**Illustration of an edge separator. The vertex set $\mathcal{V}$ is divided in balanced subsets $\mathcal{A}$ and $\mathcal{B}$ such that the number of edges between them are few.

**Figure 3.**Illustration of the divide and conquer approach described in Algorithm 1. First, a binary tree with successive graph bisections is constructed (divide). The graph spectrum is reconstructed by stitching back together the pieces of the spectra of the reduced-size graphs (conquer).

**Figure 4.**(

**a**) Approximation quality of the graph spectrum for the family ${\mathcal{S}}_{1/2}^{l}$, with N = 1000, for different depths of the hierarchical decomposition. (

**b**) Approximation quality of the graph spectrum for the family ${\mathcal{S}}_{1/2}^{l}$, with decomposition depth $d=\{4,5,6\}$, for graphs of different sizes.

**Figure 5.**Comparison of the time required by the built-in MATLAB routine for obtaining the eigenvalues of a matrix, and the time required by the proposed algorithm levering a hierarchical decomposition.

**Figure 6.**Illustration of the method for the Minnesota graph, a close-to-planar graph. (

**a**) Comparison of the true (red) and approximate (green) spectrum of its Laplacian matrix. (

**b**) Comparison of the true (red) and approximate (green) cumulative spectral density of the Laplacian spectrum. (

**c**) Minnesota graph with partition obtained by the graph separator.

**Figure 7.**(

**a**) Absolute value of inner products between the approximate, $\widehat{\mathit{U}}$, and the true graph modes, i.e., $|{[{\widehat{\mathit{U}}}^{T}\mathit{U}]}_{i,j}|$. (

**b**) Absolute value of the approximate diagonalization of the original Laplacian, i.e., $|{[{\widehat{\mathit{U}}}^{T}\mathit{L}\widehat{\mathit{U}}]}_{i,j}|$. (

**c**) Comparison of the original permuted graph (red and blue) and its block diagonal approximation (blue).

**Figure 8.**Approximation error for graph filtering using the proposed hierarchical decomposition for different kinds of inputs. (

**a**) Comparison of the approximation error for graph with varying number of nodes and filter order K = 30. Comparison of the approximation error for different filter orders for (

**b**) Gaussian and bipolar signals and (

**c**) positive and unipolar signals.

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**MDPI and ACS Style**

Coutino, M.; Chepuri, S.P.; Maehara, T.; Leus, G.
Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering. *Algorithms* **2020**, *13*, 214.
https://doi.org/10.3390/a13090214

**AMA Style**

Coutino M, Chepuri SP, Maehara T, Leus G.
Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering. *Algorithms*. 2020; 13(9):214.
https://doi.org/10.3390/a13090214

**Chicago/Turabian Style**

Coutino, Mario, Sundeep Prabhakar Chepuri, Takanori Maehara, and Geert Leus.
2020. "Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering" *Algorithms* 13, no. 9: 214.
https://doi.org/10.3390/a13090214