Fractional spectral graph wavelets and their applications

One of the key challenges in the area of signal processing on graphs is to design transforms and dictionaries methods to identify and exploit structure in signals on weighted graphs. In this paper, we first generalize graph Fourier transform (GFT) to graph fractional Fourier transform (GFRFT), which is then used to define a novel transform named spectral graph fractional wavelet transform (SGFRWT), which is a generalized and extended version of spectral graph wavelet transform (SGWT). A fast algorithm for SGFRWT is also derived and implemented based on Fourier series approximation. The potential applications of SGFRWT are also presented.


Introduction
In traditional signal processing, the most commonly used tools are transforms, among which Fourier transform and wavelet transform play key roles [1].Fourier transform and wavelet transform were generalized and extended in many contexts in recent years.
On the one hand, Fourier transform and wavelet transform were extended to fractional domain, obtaining fractional Fourier transform (FRFT)  and fractional wavelet transform (FRWT) [24][25][26][27][28][29][30].From a mathematical viewpoint, the FRFT can be seen as a parametric Fourier transform [7], whose fractional order θ is a free parameter which offers more flexibility and generalization properties compared to the classical Fourier transform.From a signal processing point of view, the FRFT can be interpreted as a rotation in the time-frequency plane, i.e., the unified timefrequency transform.With the order θ increasing from 0 to 1, the FRFT can show the characteristics of the signal changing from the time domain to the frequency domain [8].ere are roughly three main research directions for investigating FRFT: firstly, the application of FRFT to deal with many signal processing problems [9], for example, filtering, compression, image encryption, digital watermarking, pattern recognition, edge detection, antennas, radar and sonar, and communication; secondly, the discretization algorithms of the FRFT [10][11][12][13][14][15][16][17]; and lastly, the extension of the fractional idea of FRFT to other transforms, for example, fractional cosine, sine and Hartley transforms [18][19][20][21], fractional Krawtchouk transform [22], and shorttime FRFT [23].By cascading of the FRFT and the ordinary wavelet transform, Mendlovic et al. [24] first proposed FRWT.Recently, by introducing a new structure of the fractional convolution [25] associated with the FRFT, Shi et al. [26] proposed a simplified definition of the FRWT which analyzes the signal in time-frequency-FRFD domain, where FRFD denotes fractional Fourier domain.e definition of FRWT in [26] was further improved by Prasad et al. [27], who analyzed the signal only in time-FRFD domain.More recently, Dai et al. [28] presented a new FRWT which displays the time and FRFD-frequency information jointly in the time-FRFD-frequency plane.Srivastava et al. [29] proposed a certain family of fractional wavelet transformations and Shi et al. [30] proposed a novel fractional wavelet packet transform.In general, compared to traditional transforms, fractional transforms may lead to (1) better signal filtering results due to the rotation of the time-frequency plane [9]; (2) better image compression ratios [31], image recognition results [32], and image segmentation results [33] due to the flexible choices of the fractional orders; and (3) better image encryption [34] and image watermarking [35] performances since the fractional order can be used as an additional secret key.
To the authors' knowledge, there are only few research works done for discrete signal processing on fractional graph domain, which is a combination of fractional transform domain and graph transform domain.Wang et al. [74,75] proposed a definition of fractional Fourier transform on graphs named graph fractional Fourier transform (GFRFT), which is based on algebraic signal processing theory [53,54].However, similar to Fourier transform, GFRFT is a global transform and does not provide useful localization properties in fractional graph domain.Furthermore, GFRFT does not allow multiresolution analysis of graph signals.In this paper, based on spectral graph theory, we first propose a new spectral graph fractional Fourier transform (SGFRFT), which is then used to define a new spectral graph fractional wavelet transform (SGFRWT), which can be seen as an extended version of spectral graph wavelet transform (SGWT) [44].e remainder of the paper is organized as follows.Section 2 recalls the main foundations of fractional wavelet transform, spectral graph theory, and spectral graph wavelet transform (SGWT).
e spectral graph fractional Fourier transform (SGFRFT) and the spectral graph fractional wavelet transform (SGFRWT) are defined in Section 3. Section 4 is dedicated to the Fourier series approximationbased fast algorithm for forward and inverse SGFRWT.Several applications of SGFRWT for addressing different problems are shown in Section 5. Section 6 concludes the paper.

Preliminary
In the following, we first recall the forward and inverse fractional Fourier transform (FRFT) [2][3][4][5] and then the forward and inverse fractional wavelet transform (FRWT) [24,[26][27][28] and also show how scaling operator may be expressed in the FRFT domain.Table 1 summarizes the definitions of various transforms for classical signal processing and also those for graph signal processing.

Fractional Fourier Transform (FRFT).
e θ-order for- e kernel in equation ( 2) can be obtained by its spectral expansion [5]: ξ k (x) in equation ( 5) is the kth-order normalized Hermite function [11,12], which is the eigenfunction of the Fourier transform.From (5), we can see that FRFT is a generalized version of the Fourier transform, since they share the same eigenfunction ξ k (x) and the eigenvalues of the FRFT are the θth root of the eigenvalues of the Fourier 2 Mathematical Problems in Engineering Discrete form of Fourier matrix: ])V T N where diag(.)denotes a diagonal matrix formed from its vector argument.
is the kth-order normalized Hermite function.Discrete form of fractional Fourier matrix: And è is the fractional order.

Graph signal processing
Algebraic signal processing (ASP) [40,41] Spectral graph theory [53,54] Graph Fourier transform is obtained by the Jordan decomposition of the adjacency matrix W as follows: W � ... c where è is the fractional order. where is obtained by the eigenvalue decomposition of the graph Laplacian matrix L as follows: , where D is the diagonal degree matrix and W is the adjacency matrix.
where è is the fractional order.
where T s g � g(sL and L is the graph Laplacian matrix.
and L è is the graph fractional Laplacian matrix Mathematical Problems in Engineering transform [2].When the rotation angle θ � π/2, the FRFT defaults to Fourier transform.e discrete form of equation ( 5) can be expressed as follows [11,12]: where diag(.)denotes a diagonal matrix formed from its vector argument.F θ N denotes the θ-order forward FRFT matrix of size N × N.
, v k is the kthorder DFT Hermite eigenvector.Note that F θ N becomes identity matrix when θ � 0 and Fourier matrix when θ � π/2.
e θ-order forward FRWT is defined as [28] W where with ψ s,a (x) � (1/s)ψ(x − a/s).e unitarity property of the FRFT applied to (7) leads to an equivalent form of FRWT: where  f θ stands for the θ-order FRFT of f.Here also, as for the classical CWT [1], the scaling is operated in the transformed domain "u" and not directly in the time domain.
e inverse of the FRWT is given by [28] f when the following admissibility condition is satisfied [28]: 2.3.Spectral Graph eory and Spectral Graph Wavelet Transform.In this section, we briefly review the spectral graph theory and also the spectral graph wavelet transform [44].

Spectral Graph eory.
We consider an undirected, connected, weighted graph G � V, E, W { }, composed of a finite set of vertices V (with Card(V) � N), a set of edges E, and an adjacency symmetric and positive-valued matrix W. A real-valued signal f defined on the vertices V of the graph G is an N × 1 vector where each entry is the value f(n) assigned to the vertex n. e (nonnormalized) graph Laplacian operator is the matrix defined by L � D − W ∈ R N×N , where D is the diagonal degree matrix defined by d m,m �  n w m,n .d m,m is the degree of vertex m.
As the matrix L is real and symmetric, then it is diagonalizable and its eigenvectors χ 0 , χ 1 , . . ., χ N− 1  , sorted according to the ascending order of the corresponding eigenvalues 0 form an orthonormal basis.erefore, with the unitary column matrix: and the diagonal matrix where the superscript H denotes the Hermitian transpose operation.e graph Fourier transform (GFT) of the signal f is then the sequence provided by the scalar products [36]: and f can be recovered (inverse GFT) by means of the reconstruction formula: Using vectors and matrices notations, equations ( 14) and ( 15) become e graph Fourier transform obeys the Parseval theorem; that is, for any signals f and h defined on the graph G, we have

Spectral Graph Wavelet Transform (SGWT).
e spectral graph wavelet transform (SGWT) of the signal f with the kernel g is defined by [44] 4 Mathematical Problems in Engineering where and the kernel g is continuous positive-valued function defined on R + satisfying Using equation ( 14), the SGWT becomes with e signal f can be recovered up to its mean value using the inverse formula [44]:

Spectral Graph Fractional Transforms
We first propose a novel spectral graph fractional Fourier transform (SGFRFT) in Section 3.1 and then give the definition of spectral graph fractional wavelet transform (SGFRWT) in Section 3.2.

Spectral Graph Fractional Fourier
Transform.Similar to (13), we define the graph fractional Laplacian operator L θ as follows: where 0 < θ ≤ 1 and γ is the power matrix given by with χ being the unitary (i.e., inverse graph Fourier transform matrix) shown in (12).Note that as χ is unitary, then c � χ θ is also unitary.And that is, , and from ( 27), we can easily get e reason why we choose L θ as the definition of graph fractional Laplacian operator will be discussed in Section 3.2.e forward spectral graph fractional Fourier transform (SGFRFT) of any signal f defined on the vertices V of the graph G is defined by where and χ ′ � V − 1 is obtained by the Jordan decomposition of the adjacency matrix W as follows [74,75]: where V is a matrix composed of eigenvector and J W is a block-diagonal matrix.For more details, please refer to Appendix A of [74].From ( 13), ( 29), (31), and ( 33)-( 36), we can see that the differences are that the proposed SGFRFTs are based on the decomposition of the graph Laplacian matrix L, while the GFRFTs are based on the decomposition of the adjacency matrix W.

Spectral Graph Fractional Wavelet Transform (SGFRWT).
Similar to (21), we can define the spectral graph fractional wavelet transform (SGFRWT) operator T s g θ as follows: where Why we choose L θ in (24) as the definition of graph fractional Laplacian operator and choose ψ θ,s,n (m) in ( 39) as the basis of SGFRWT?ere are two reasons.(1) We want to define the basis of SGFRWT ψ θ,s,n (m) based on a new function c ℓ (n) instead of the same function χ ℓ (n) as basis of SGWT ψ s,n (m) in ( 23). e SGFRWT and SGWT will have different characteristics since they are based on different functions c ℓ (n) and χ ℓ (n), respectively.(2) We want to define the basis of SGFRWT ψ θ,s,n (m) which changes directly with λ θ ℓ .Since the parameter θ acts on the eigenvalues of the Fourier matrix in the form of exponents in the spectral expansion of FRFT in (5), similarly, we also want to obtain the form that the parameter θ can directly affect λ ℓ in the proposed definition of SGFRWT.Two other possible definitions of SGFRWT are also discussed in Appendices A and B, respectively.
From (37), we can see that the proposed SGFRWT is a generalization of SGWT in (21) defined by Hammond et al. [44]; that is, when θ � 1, the proposed SGFRWT defaults to the SGWT.
e proposed SGFRWT can be seen as a parametric generalization of SGWT [7].Furthermore, the proposed SGFRWT converts a graph signal into an intermediate domain between vertex and spectral graph wavelet.With the order θ increasing from 0 to 1, the SGFRWT can provide much more domain choices of expressions for a graph signal.
e scaling functions are then obtained by and the coefficients by Stable recovery of the original signal f from the SGFRWT coefficients will be assured if the quantity G(r) � h 2 (r) +  J j�1 g 2 (s j r), where s j denotes jth scale, is bounded away from zero on the fractional Laplacian L θ .
Moreover, under the same conditions, Lemmas 5.1, 5.2, 5.3, and 5.4 and eorems 5.5 and 5.6 in [44] are verified by the SGFRWT, whose properties are shown in Appendix C.

Fourier Series Approximation and
Fast SGFRWT If we compute the SGFRWT by using equation ( 37) directly, we have to compute the entire set of eigenvectors and eigenvalues of L θ shown in (24).Both the computational complexity and the memory consumption become unacceptable when the data is larger than hundreds of thousands or millions of dimensions.erefore, in this section, we propose a fast algorithm for computing the SGFRWT based on approximating the scaled generating kernels g θ by loworder Fourier series.It should be noted that the proposed fast SGFRWT algorithm is an extension of the fast SGWT algorithm from real domain to complex domain.e proposed fast SGFRWT algorithm is different from the fast graph Fourier transform proposed by Magoarou et al. [39], who use the modified Jacobi eigenvalue algorithm for approximating the graph Laplacian matrix.Note that the proposed SGFRWT algorithm can approximate an arbitrary matrix while the algorithm in [39] is only practicable for approximating symmetric matrix.
e following lemma shows that the polynomial approximation of g(sx) may be taken over a finite range containing the spectrum of L θ .Lemma 1.Let r max ≥ r N− 1 be any upper bound on the fractional spectrum of L θ .For fixed s > 0, let p(x) be a polynomial approximant of g(sx e proof of Lemma 1 is similar to Lemma 6.1 in [44].In [44], Hammond et al. approximated g(sL) shown in (19) by truncated Chebyshev expansions.e reason is that g(sL) is real as the original L shown in ( 13) is a real matrix.However, the Chebyshev polynomial approximation method is not suitable for our problem of approximating g(sL θ ) shown in (38), because g(sL θ ) may be complex as Mathematical Problems in Engineering g(sL θ ) shown in (33) may be a complex matrix.erefore, in this part, we will use the truncated Fourier series expansion instead of Chebyshev expansions to approximate g(sL θ ) shown in (38).
e Fourier series of a general function f(x) is given by [78] where We now assume a fixed set of wavelet scales s n .For each n, approximating g(s n x) for x ∈ [0, r max ] can be done by scaling the domain of exp(− iky), y ∈ [0, 2π], using the transformation x � yr max /(2π).en, g(s n x) can be written by with Note that e above equation is due to the fact that For each scale s j , g(s j x) can be approximated by the truncated Fourier expansion (42) to 2M j + 1 terms; that is, We may choose an appropriate M j in practice to achieve a balance between accuracy and computational complexity.We can also use exactly the same scheme to approximate the scaling function kernel h θ shown in (40) by p 0 .en, the approximated SGFRWT wavelet and scaling function coefficients are, respectively, given by where Note that ( 52) is easily obtained by applying the function F k (.) in (46) to the matrix L θ , and we have e efficiency of this algorithm depends on the following recursive formula: As the signal f ∈ R N , taking equation ( 53) into consideration, equation (54) can only be computed for k ∈ [1, M j ], and the other k ∈ [− M j , − 1] can be obtained by conjugate operation.e computational complexity of the proposed fast SGFRWT algorithm is analyzed as follows: (1) e computation of c from χ in (25) by using Schur-Padé algorithm [76,77] needs , where |E| is the number of nonzero edges in the graph, and 6|E| because F 1 (L θ ) is a complex matrix and a complex multiplication requires 4 real multiplications and 2 real additions (6 operations).erefore, the computation of all (3) It requires O(6N) operations at scale j for each k ≤ M j in ( 50) and (51).erefore, the computation of ( 50) and (51) needs O(6N  J j�0 M j ) operations.erefore, the total computational complexity of the fast SGFRWT algorithm is O(N 3 + 6M j |E| + 6N  J j�0 M j ).

Mathematical Problems in Engineering
Implementation of (54) requires memory of size 2N.e total memory requirement for the proposed fast SGFRWT is 2N(J + 1) + 2N.

Fast Computation of Adjoint.
Given a fixed set of wavelet scales s j   J j�1 , and including the scaling functions ϕ n , the SGFRWT W θ f � ((T h θ f) T , (T s 1 g θ f) T , . . ., (T s J g θ f) T ) T can be seen as a linear map W θ : R N ⟶ C N(J+1) , where C denotes the complex domain.Let  W θ f � (((p 0 (L θ ))f) T , (p 1 (L θ )f) T , . . ., (p J (L θ )f) T ) T denote the approximated fractional wavelet transform by using the fast SGFRWT algorithm.In this section, we show that both the adjoint W * θ : C N(J+1) ⟶ R N and the composition W * θ W θ : R N ⟶ R N can be computed efficiently by using the Fourier series approximation.
For any η ∈ C N(J+1) , we consider η as the concatenation Considering 〈η, W θ f〉 N(J+1) � 〈W * θ η, f〉 N , we have Similar to ( 50) and ( 51), the adjoint operator W * θ in (56) can be approximated by In the following, we derive a fast algorithm for the computation of  W * θ  W θ , which is an approximation of the composition W * W. In general, when computing  W * θ  W θ , we first apply  W θ and then apply  W * θ by the fast SGFRWT, which needs 2J Fourier series expansions.However, the computational complexity of  W * θ  W θ can further be reduced.Note that In the following, we derive a fast algorithm for the computation of  W * θ  W θ f.Lemma 2. Set P(x) �  J j�0 p * j (x)p j (x), which has degree M * � 2 max M j  .en, P(x) can be computed by where d k �  J j�0 d j,k , and Note that F * k (x)F l (x) � F l− k (x) is used in the above derivation.erefore, e proof of Lemma 2 is now completed.Combining ( 58) and (59), we have From ( 62 In this part, we show the reconstruction of proposed SGFRWT by using the fast adjoint algorithm in Section 4.1.e SGFRWT is an overcomplete transform since it maps an input signal f of size N to the N(J + 1) factional wavelets coefficients e pseudoinverse of the above equation can be obtained by solving the following square matrix equation: Equation ( 64) can be solved by the conjugate gradients method [79], whose computation in each step is dominated by applying W * θ W θ to a single vector.What is more, W * θ W θ f can be fast approximated by  W * θ  W θ f shown in (62).erefore, we can reconstruct input signal f from c.

Application Examples
Similar to [44], in this section, we will show several application examples of the SGFRWT on different real and synthetic datasets by modifying the GSPBOX toolbox [80].
e first and the third experiments are implemented in MATLAB programming language on a PC machine, which sets up Microsoft Windows 7 operating system and has an Intel ® Core ™ i7-2600 CPU with speed of 3.40 GHz and 16 GB RAM. e second experiment is implemented using PyTorch on a PC machine, which sets up Ubuntu 16.04 operating system and has an Intel ® Core ™ i7-4790 CPU with speed of 4.00 GHz and 32 GB RAM and has also two NVIDIA GTX1080-Ti GPUs.

SGFRWT Design Details.
We should design two kernels in SGFRWT.One is the wavelet kernel g(x), and the other is the scaling function h(x).
For the wavelet kernel g(x), we choose the same cubic spline in [44] as follows: where α and β are integers and they are used to determine the decay rate of spline.x 1 and x 2 are boundary of transitions region.Note that g is normalized such that g(x 1 ) � g(x 2 ) � 1. e coefficients of the cubic polynomial s(x) are determined by s(x 1 ) � s(x 2 ) � 1, s ′ (x 1 ) � α/x 1 , and s ′ (x 2 ) � − β/x 2 .All of the examples in this paper were produced using α � β � 2, x 1 � 1, and x 2 � 2; in this case, s(x) � − 5 + 11x − 6x 2 + x 3 .For the SGFRWT, we can easily substitute x in (65) by L θ shown in (24) to get g(L θ ).

SGFRWT on Synthetic Graph Data.
In the first example, we perform the proposed SGFRWT on "Swiss roll," which is a dataset whose points are sampled randomly from a two-dimensional (2D) manifold as follows: Figure 1 shows the Swiss roll data cloud.We take 500 points sampled uniformly on the manifold and then build a weighted graph, whose edge weight is defined as follows: Note that the definition of the edge weight is a critical issue in graph construction.e most direct way to compute the edge weight is based on a given (dis)similarity measure.In practice, the edge weight is generally defined using different measures for better interpretability.As discussed in [81], Gaussian kernel and its variants (i.e., heat kernel or RBF kernel) are one of the most popular schemes to assign edge weights for a graph.erefore, we take the Gaussian kernel in (67) to calculate the edge weight to simplify our discussion in this paper.We perform SGFRWT with fractional order θ ∈ [0.1, 1.0] on Swiss roll data cloud with J � 5 wavelet scales, and the results of some fractional orders θ ∈ {0.1, 0.5, 0.8, 1.0} are shown in Figure 2. e last column of Figure 2 shows the results of performing SGWT [44] on Swiss roll, from which we can see that SGWT [44] searches for the optimal expression domain only along the dimension of scale.However, the proposed SGFRWT searches for the optimal expression domain not only along the dimension of scale but also along the dimension of fractional order.at is, compared with SGWT, the proposed SGFRWT can express Swiss roll in higher dimensions, which makes the expression space larger and is therefore more likely to obtain a better expressive domain for the original problem.Just like [31][32][33], the fractional order θ is a hyperparameter depending on the original problem and you should search for a suitable value of θ.For example, the fractional order θ � 0.4 is an appropriate value for gland segmentation when using fractional wavelet scattering network [33].
As a comparison, we also give the results of the proposed spectral graph fractional Fourier transform (SGFRFT) and graph fractional Fourier transform (GFRFT) proposed in [74,75] on the "Swiss roll" in Figure 3, respectively.As we can see from Figure 3, the results of the proposed SGFRFT are different from those of GFRFT in [74,75].Furthermore, both SGFRFT and GFRFT are global transform and are not easily to be used for local analysis.However, let us see the third column of Figure 2, that is, the results of SGFRWT with J � 5 and fractional order θ � 0.8; the localization is more and more obvious along scale dimension.Comparing the third column of Figure 2 with the third column of Figure 3, we can Mathematical Problems in Engineering see that, compared with SGFRFT and GFRFT, the proposed SGFRWT allows localization in fractional graph domain, which may lead to better local analysis in spectral graph domain.e experiment results of Figure 3 verify the local properties of SGFRWT shown in Appendix C (i.e., Lemmas C2-C4 and eorem C5) to a certain extent.

SGFRWT on MNIST Database.
In the second example, we will see that the proposed SGFRWT can be used as a data augmentation method in the preprocessing of convolutional neural networks, which require as many input images as possible.Because, as shown in the first example, the proposed SGFRWT allows deriving many images from the original image by considering several fractional orders.In the following, a series of experiments are performed on the MNIST dataset to verify the effectiveness of the proposed SGFRWT as a method of data augmentation.
We construct the graph for each image in MNIST dataset.e vertices of the graph are made up of pixels in the image and each pixel is connected with its vertical and horizontal neighbor pixel.e edge weight is calculated by a Gauss kernel weighting function [36] as follows: where w i,j is the weight on the edge connection of the vertex v i and v j , and dist (i, j) represents the distance between pixel i and pixel j.
In order to prove the effectiveness of SGWT and SGFRWT, we augment MNIST dataset by the following strategies: (i) e original MNIST database has a training set of 60,000 images and a testing set of 10,000 images, which are used as a control group.(ii) For each image of the MNIST training set, we perform SGWT [44] with J � 5 wavelet scales and generate a total of 360,000 images.e original 10,000 test images are still used as the testing set.(iii) For each image of the MNIST training set, we apply the proposed SGFRWT with J � 5 wavelet scales and fractional order θ ∈ {0.2, 0.4, 0.6, 0.8, 1.0}, generating a total of 1,800,000 training images.e details of the generated four datasets are shown in Table 1. Figure 4 shows all sample images by processing an image of the digit "zero" by SGFRWT with J � 5 wavelet scales and fractional order θ ∈ {0.2, 0.4, 0.6, 0.8, 1.0}.
To evaluate the data augmentation performance of the proposed approach, we train and test a simple convolutional neural network, which is shown in Figure 5, using the three datasets mentioned above.e CNN model consists of 2 convolutional layers, 2 pooling layers, and a fully connected layer with a 10-way Softmax classifier.
e inputs of the CNN are images of size 28 × 28. e first convolutional layer has 32 kernels with size 5 × 5, while the other layer has 64 kernels with size 3 × 3. e Rectified Linear Units (ReLUs) are chosen as the activation function.Every convolutional layer is followed by a nonoverlapping max-pooling with filter of size 2 × 2. Stochastic gradient descent algorithm is employed to optimize the network with a learning rate of 0.001.When the network converges, we evaluate the classification accuracy on the testing set with an average value of 10 times.e last column of Table 2 shows the classification performance comparison of the three different datasets.As we can see from Table 2, if we use traditional data augmentation methods (flipping, rotating, and adding noise), then the recognition rate can achieve 98.41%.When the training dataset of MNIST is augmented by SGWT without traditional data augmentation, the recognition rate can increase to 98.71%.From the last row of Table 2, we can see that our proposed SGFRWT can further improve the accuracy.

Face Recognition by
Fusing SGFRWT Features.In the third example, the application of SGFRWT to provide features for face recognition is investigated.We use four classical face databases, whose MATLAB formats are provided in [82,83].2) AR Database [85].We use a nonoccluded subset of AR database containing 700 images (50 male subjects and 50 female subjects, 7 images per subject).( 3) CMU PIE Face Database [86].We also use a nonoccluded subset of CMU PIE database which contains 11560 images (68 persons, 170 images per person).( 4 e experiment is carried out by the following three steps: (1) For each image of the four face databases, we apply the propoJ � tr(S b )/tr(S w ),sed SGFRWT with   wavelet scale J � 10 and fractional order θ ∈ {0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8, 0.9, 1.0}, obtaining 100 times the images of original datasets.
(2) We then extract the SGFRWT features of three fractional orders which have the highest trace ration [88] as follows: where tr(.) denotes the trace.S b and S w are betweenclass scatter matrix and the within-class scatter matrix, respectively,      (3) e SGFRWT features of three fractional orders are fused by canonical correlation analysis (CCA) [89] to obtain the feature vector, whose dimension is then reduced by principal component analysis (PCA) and Linear Discriminant Analysis (LDA).Finally, we feed the feature vector to the support vector machine (SVM).
We conduct experiments on 18 different datasets which are constructed by four classical face databases mentioned above.Each set of datasets is compared with three different methods (baseline, SGWT, and SGFRWT), and we evaluate the classification accuracy on the testing set with an average value of 10 times.e recognition comparison results of the proposed SGFRWT feature building, the SGWT feature, and the baseline (only PCA + LDA + SVM used) on four face databases are shown in Table 3, from which we can see that the proposed SGFRWT feature building approach can achieve promising results and perform better than SGWT feature and also the SVM baseline.

Conclusion
is paper investigates the issue of extension of spectral graph wavelet transform (SGWT) to fractional domain.e main contributions of this paper can be summarized as follows.(1) A novel transform named spectral graph fractional wavelet transform (SGFRWT) is defined.(2) A Fourier series approximation-based fast algorithm for SGFRWT is derived and implemented since the SGFRWT includes complex domain computations compared to SGWT.(3) Applications of SGFRWT to synthetic and real datasets are also given to highlight its potential usefulness.In summary, the proposed fractional spectral graph wavelets provide a new choice for the graph signal processing.Further research may include the extension of the proposed SGFRWT for dealing with the directed graphs [40,41] and the extension of the idea of SGFRWT to critically sampled graph wavelets like GraphBio [46].

A. Definition of SGFRWT by Using the Idea of Traditional FRWT
According to (8), we can define the SGFRWT in (37) in another form by using the idea of traditional FRWT as follows: where

B. Definition of SGFRWT by Using Another Graph Fractional Laplacian Operator L θ
Different from the definition in (24), the graph fractional Laplacian operator can also be defined as follows: Comparing (B.1) and ( 13), we can see that the decomposition of L and L θ shares the same χ.at is, if we define L θ as the graph fractional Laplacian operator, then, similar to (21), we can define the spectral graph fractional wavelet transform (SGFRWT) operator T s g θ as follows:

C. Some Properties of SGFRWT
In this appendix, we give some properties of proposed SGFRWT without proof, since the proof of these properties is very similar that of those properties of SGWT in [44].where f # � f − 〈c 0 , f〉c 0 .In particular, the complete reconstruction is given by f � f # +  f(0)c 0 .
Lemma c.1 shows that the mean of f may not be recovered from the zero-mean fractional graph wavelets.Lemma c.1 is a generalization of Lemma 5.1 in [44].Lemma c.3 shows that if two fractional kernels g θ and  g θ are close to each other in some sense, then the resulting graph fractional wavelets should be close to each other.Lemma 3.3 is a generalization of Lemma c.3 in [44].
Lemma C.4.Let g θ be K + 1 times continuously differentiable, satisfying g(0) � 0, g (i) (0) � 0 for all i < K, and g (K) (0) � C ≠ 0. Assume that there is some s ′ > 0 such that Lemma c.4 shows that if a fractional kernel g θ has a zero of integer multiplicity at the origin, then g θ can be approximated by a single monomial for small scales.Lemma c.4 is a generalization of Lemma 5.4 in [44].
Theorem C.1.Let G be a weighted graph with fractional Laplacian L θ .Let gθ be a kernel satisfying the hypothesis of Lemma 3.4, with constants s ′ and B. Let m and n be vertices of G such that d G (m, n) > K. en, there exist constants D and s ″ , such that ψ θ,s,n (m)/‖ψ θ,s,n ‖ ≤ Ds for all s < min(s ′ , s ″ ).eorem c.1 shows that the localization of the fractional wavelet ψ θ,s,n must include a renormalization factor in the limit of small scales.In general, ψ θ,s,n (m) ⟶ 0 as s ⟶ 0 for all m and n, due to the normalization chosen for the graph fractional wavelets.eorem c.1 is a generalization of eorem 5.5 in [44].Note that the basic definition of a frame is as follows: Given a Hilbert space H, a set of vectors Γ k ∈ H form a frame with frame bounds A and B if the inequality A‖f‖ 2 ≤  k |〈f, Γ k 〉| 2 ≤ B‖f‖ 2 holds for all f ∈ H. eorem C.6 shows that the numerical stability of recovering the vector f from the inner product 〈f, F k 〉 depends on the frame bounds A and B. eorem c.2 is a generalization of eorem 5.6 in [44].

32 Figure 1 :
Figure 1: Spectral fractional graph wavelets on Swiss roll data cloud.e figure shows vertex at which SGFRWT is centered; that is, we will apply SGFRWT on the signal value of centered vertex.

Figure 5 :
Figure 5: e convolutional neural network used in our experiment.

Lemma C. 2 .
Let G be a weighted graph, L θ the graph fractional Laplacian operator, and t > 0 an integer.For any two vertices m and n, if d G (m, n) > t then ((L θ ) t ) m,n � 0. Note that d G (m, n) is the shortest-path distance, i.e., the minimum number of edges for any paths connecting m and n.Lemma c.2 shows the localization result for integer powers of the fractional Laplacian L θ .Lemma c.2 is a generalization of Lemma 5.2 in [44].Lemma C.3.Let ψ θ,s,n � T s g θ δ n and  ψ θ,s,n � T s  g θ δ n be the fractional wavelets at scale s generated by the fractional kernels g θ and  g θ .If |g(sr) −  g(sr)| ≤ M(s) for all r ∈ [0, r N− 1 ], then |ψ θ,s,n (m) −  ψ θ,s,n (m)| ≤ M(s) for each vertex m.Additionally, ‖ψ θ,s,n −  ψ θ,s,n ‖ 2 ≤ �� N √ M(s).

Table 1 :
e definitions of various transforms for classical signal processing and for graph signal processing.

Table 2 :
Dataset generation implementation details.Comparison of classification performance of three different datasets with three different augmentation methods on the same CNN network.

Table 3 :
e recognition results of the proposed SGFRWT feature, the SGWT feature, and the baseline on four face databases (Yale, AR, ORL, and CMU PIE).We randomly selected several images per individual as training set and the remaining images as testing set."itrain"means that we randomly choose i images in each class for training.eaccuracyis the average results of ten random testing experiments.Mathematical Problems in Engineeringwhere the superscript T denotes transposition.C denotes the number of classes.A is the mean of A i and A i is the mean image of class X i and N i is the number of images in class X i .|X (θ, j) | the modulus of SGFRWT coefficient with the fractional order θ and the wavelet scale j.