Abstract

In recent years, the research of wavelet frames on the graph has become a hot topic in harmonic analysis. In this paper, we mainly introduce the relevant knowledge of the wavelet frames on the graph, including relevant concepts, construction methods, and related theory. Meanwhile, because the construction of graph tight framelets is closely related to the classical wavelet framelets on , we give a new construction of tight frames on . Based on the pseudosplines of type II, we derive an MRA tight wavelet frame with three generators , and using the oblique extension principle (OEP), which generate a tight wavelet frame in . We analyze that three wavelet functions have the highest possible order of vanishing moments, which matches the order of the approximation order of the framelet system provided by the refinable function. Moreover, we introduce the construction of the Haar basis for a chain and analyze the global orthogonal bases on a graph . Based on the sequence of framelet generators in and the Haar basis for a coarse-grained chain, the decimated tight framelets on graphs can be constructed. Finally, we analyze the detailed construction process of the wavelet frame on a graph.

1. Introduction

In the past two centuries, harmonic analysis, including Fourier analysis and wavelet analysis, has been intensively researched and widely applied in areas, such as signal processing, representation theory, and number theory. Since Daubechies constructed the compactly supported orthonormal wavelet bases in 1988 [1], wavelet analysis has been extensively studied and successfully applied to more fields. For an orthonormal wavelet basis, the corresponding refinable function and its mask must satisfy the following conditions:

These two conditions enforce very restrictive constraints for the refinable function, and many refinable functions do not satisfy the conditions. By introducing redundancy into a wavelet system, a tight wavelet frame is considered as a generalization of an orthonormal wavelet basis [2–4]. It is much easier and more flexible to construct tight wavelet frames than orthonormal wavelet bases. This theory of the construction of tight wavelet frames on regular Euclidean domains is relatively mature [5–11]. In recent years, driven by the rapid progress of deep learning and their successful applications in an interdisciplinary area, data in deep learning are typically from social networks, biology, physics, finance, etc., and can be naturally organized as graphs [12–22]. There has been a great interest in developing harmonic analysis for data defined on non-Euclidean domains such as manifold data or graph data [23–26]. Such data are usually regarded as random samples from a smooth manifold. The matrix is used to organize it as an undirected graph, where the graph Laplacian approximates the manifold Laplacian. The underlying manifold encodes the geometric information of the data, which has been widely used in various machine learning and statistical models [27–32].

Different from that on Euclidean domains, the construction systems and the corresponding fast algorithm for tight framelets on graphs are less studied. The main reason is that we do not know how to define the operators of translation and dilation on the manifolds or graphs similar to the classical wavelet framelet systems. In order to further study the construction of tight framelets on non-Euclidean domains, some alternative approaches are proposed. By introducing diffusion operators, Coifman and Maggioni in [33] constructed orthogonal diffusion wavelets on a smooth manifold. Maggioni and Mhaskar in [34] further extended the construction from diffusion wavelets to diffusion polynomial frames on manifolds. In recent years, with the development of computer science and mathematics, the classical spectral theory has been introduced into the construction of wavelet framework on the graph. The eigenvalues and eigenvectors of the graph Laplacian matrix can be effectively used to define translation and dilation transformation of graph functions, and the construction of tight framelets on graphs can be obtained [23, 24, 35–37]. Based on spectral theory and graph Laplacian [38], some work about the construction of tight framelets on graphs has been derived [23–25, 35–37, 39–44]. In this paper, we focus on the review of a number of representative construction methods of tight framelets on graphs and provide a specific introduction for wavelet frame on graphs.

We organize the rest of the paper as follows: in Section 2, we introduce some basic theoretical knowledge, including the classical wavelet frame theory, graph, chain, and tight framelets on the graph, and review some construction methods of tight framelets on graphs; in Section 3, we construct three wavelet functions , and , which generate a tight wavelet frame in , and give a specific example of the construction process of wavelet frame on the graph; and in Section 4, we conclude this paper and discuss future challenges of the graph wavelet frame.

2. Wavelet Frame on the Graph

The construction method of wavelet frames on the graph is to design the wavelet function on the graph and its corresponding translation and dilation transformation of graph functions. With the spectral graph theory, by considering the spectral decomposition of the graph Laplacian, Hammond in [35] defined translation and dilation transformation on a graph and constructed the spectral graph wavelets and spectral graph scaling functions based on an indicator function. However, the spectral scaling functions in this way did not generate the graph wavelets, which is analogous to the construction of traditional wavelets through the two-scale relation. Hence, the associated construction of wavelet frames on the graph did not have the filter bank. Dong in [36] defined the quasi-affine systems on a given manifold by considering generalized translation and dilation transformation of wavelet functions in , and constructed wavelet frames on both compact manifold and discrete graphs. These wavelet functions are defined from the scaling functions, which implies that the associated construction of wavelet frames on the graph has the filter bank.

However, the graph wavelet frames constructed by the above two methods are the undecimated wavelet frames. When the level number of framelet transforms is big, the undecimated wavelet frames may result in a high redundancy rate. In contrast, using clustering techniques, the decimated framelet system can be constructed by embedding edge relation and clustering feature in a chain-based orthonormal system and would achieve a low redundancy rate. Chui [23] first defined an orthogonal system based on a local filtration and constructed the different orthogonal system at the vertices at each level. Later, Chui [24] further extended the work to directed graphs and established an orthogonal system with localization properties on the tree. Based on graph clustering algorithms [45–50], a suitable orthonormal eigenpair can be constructed for a coarse-grained chain on the graph, and decimated tight framelets can be constructed based on the orthonormal eigenpair in [37]. By introducing a Haar basis for a coarse-grained chain on a graph, the graph neural networks and graph pooling under the Haar basis can be defined in [25, 39], and achieved low computational cost when the graph size is large.

The construction of decimated tight framelets on the graph first utilizes graph clustering to obtain the coarse-grained chain. Once a chain of nested graphs is obtained, we can build the multiscale structure on a graph by framelet filter banks, which bridges with the classical wavelet framelets systems on . In this paper, we focus on the construction of decimated tight framelets on the graph. In the following, we will review relevant knowledge for the construction methods of decimated tight framelets on the graph.

2.1. Tight Wavelet Frames on

In this subsection, we first recall some basic theories of the tight wavelet frame on the , which are the basis for the construction of decimated tight framelets [5, 8, 9]. A function is a refinable function if it satisfies the following refinement equation:where is called the mask for the refinable function , which is a finitely supported sequence on . The Fourier series of a sequence is defined to be

The Fourier transform of a function is defined byand can be naturally extended to functions. In terms of the Fourier transform, the refinement equation in (3) can be rewritten as follows:

Throughout this paper, we assume . According to (6), we have .

We say that a set of functions in generates a tight wavelet frame in ifwhere and . The set is called a set of generators for the corresponding tight wavelet frame. For any function , we have the wavelet expansion

By introducing redundancy into a wavelet system, it has a lot of freedom in the construction of tight wavelet frames derived from refinable function. Petukhov [7] showed that if the mask satisfies for all , then a symmetric tight wavelet frame with three generators can be obtained. Jiang [8] systematically analyzed the construction of hexagonal tight wavelet frame filter banks with three high-pass filters. For the construction of the wavelet frame, the oblique extension principle is considered as a popular approach, which was first proposed in the literature [5], and the specific contents are as follows.

Theorem 1. Let be a compactly supported refinable function in with a finitely supported mask on such that and . Suppose that there exist finitely supported sequences and a -periodic trigonometric polynomial such that andwhere

The wavelet functions is defined as follows:

Then, generates a tight wavelet frame in . Moreover, has vanishing moments if and only if

The construction in Theorem 1 is called the oblique extension principle. Daubechies et al. in [5] discussed the importance of the nonconstant in this principle, which provides a tight wavelet frame with good vanishing moments. The order of vanishing moments is an important property of a tight wavelet frame. For a set of compactly supported functions in , it has vanishing moments of order if

The related theory of compactly supported functions with good vanishing moments can be found in references [9, 51]. Moreover, some scholars have also studied the construction of multiwavelet and multiwavelet frames [52–56]. Based on two-direction refinable functions, Li and Yang [52] studied the construction of dual multiwavelet frames with symmetry and discussed the vanishing moment of the constructed multiwavelet frames. Atreas et al. [53] discussed homogeneous dual multiwavelet frames from a pair of refinable function vectors and derived that the mixed oblique extension principle can be described by dual multiwavelet frames. For OEP-based tight framelets, Han [54] also extended to high dimensional case and constructed compactly supported tight -wavelet frames from compactly supported -refinable functions for any dilation matrix . Based on the approach of polyphase matrix extension of multiscaling vectors, Cen et al. [55] presented a novel approach for the construction of symmetric compactly supported bi-orthogonal multiwavelets with multiplicity of 2. Li and Peng [56] analyzed the sampling property of bi-orthogonal multiwavelets and discussed the application of bi-orthogonal multiwavelets in image compression filter bank. In addition, some researchers investigated the construction and application of multivariate wavelet frames. The interested readers are referred to the literature [57–65].

2.2. Graph and Chain

In this subsection, we introduce some basic conceptions about a graph and chain [6, 37].

A graph is made up of a set of vertices and a set of edges between vertices. The non-negative function indicates weight of edges between vertices. if there is an edge from the vertex to vertex ; otherwise, 0. If we ignore the directionality of the edges, the graph is called an undirected graph. In this case, weight is symmetric, that is, for all . Otherwise, the graph is said to be a directed graph. The degree of a vertex is denoted as . The sum of degrees of all vertices of is denoted as vol vol , which is the volume of the graph [6, 37]. For a subset of , the volume is the sum of degrees of all nodes in .

Let be a sequence of edges in . If there exist distinct vertices in such that any pair of consecutive nodes is connected by the edges of , that is, for , then the sequence is called a path of the graph between and . The length of the path is defined to be . If there exists a path between the vertex and vertex , the length of the shortest possible path is defined as the distance between them, denoted as . If there is no path between vertices and , we define the distance . A graph is said to be connected if any two distinct vertices of are connected [6, 37].

Let and be two graphs; we say that is the coarse-grained graph of if is a partition of . In this case, there exists subsets of for some such that

That is, each vertex of is called a cluster for . The edges of are the links between clusters of . We define that two vertices and are equivalent, if and are in the same cluster, denoted by [6]. Generally, we use to denote a cluster in with respect to a vertex in the coarse-grained graph .

Let be two integers; a coarse-grained chain of is a sequence of graphs with . Each is a coarse-grained graph of for all , and for all and all . The graph is the level graph of the chain , and the can be viewed as a coarse-grained graph of for . If for all , the chain is called as an undecimated chain of . Otherwise, is called an decimated chain of . For convenience of discussion, it is usually assumed each vertex of the finest level graph as a cluster of singleton. When there is only one vertex in the coarsest graph , we call a tree [6].

2.3. Orthonormal Bases on Graphs

In this subsection, we introduce the orthonormal bases for the coarse-grained chain on a graph [6, 35–37].

Let be the Hilbert space of vectors on the graph with the inner product , which is defined as follows:where is the complex conjugate to . The norm is given by for . Let be the Kronecker delta satisfying if and 0 otherwise, and is the number of vertices. A set of vectors in is an orthonormal basis for if

The generalized Fourier coefficient of degree for with respect to is defined to be . Then, for all , we have . Let be a nondecreasing sequence of non-negative numbers satisfying , if is an orthonormal basis for with ; we say is an orthonormal eigenpair for [35]. Meanwhile, Hammond [35] also gives the definition of the graph Laplacian

If , the equation corresponding to the eigenvalues and eigenvectors is non-negative, and satisfies with [37].

Based on the properties of the eigenvalues and eigenvectors, Wang and Zhuang in [37] gave an orthonormal basis for the chain. Let be a chain on the graph with vertices; be the set of all vectors defined on the union of vertices on all levels . If the restriction on the th-level graphs is an orthonormal basis for at each level , a set pairs of vectors and complex numbers in are called an orthonormal basis for the chain .

2.4. Decimated Tight Framelets on Graphs

In this subsection, we introduce the construction methods of a decimated tight framelet on a graph. The conclusion of this part mainly comes from reference [6]. Let be a graph and be a chain on the graph . For each vertex in , a weight is defined. For the bottom level with , let . Let be the set of weights on , and be the sequence of weights for the coarse-grained chain . Zheng et al. in [6] gave the construction method of the decimated framelets on the graph, and the details are as follows.

Definition 1. Let be a tight frame in at scale for . The decimated framelets and , at scale for the chain on the graph are defined bywhere for , we let and . We call and low-pass and high-pass framelets at scale .
The decimated tight framelets in Definition 1 are constructed based on framelet generators in and the orthonormal basis associated with the chain . The function can be defined by . In order to obtain the decimated tight framelets on the graph , first, we study the construction methods of tight frames on . Then, in next section, we focus on the specific construction process of the graph wavelet frame.

3. Construction of Tight Wavelet Frames on the Graph

In this section, we introduce the construction process of decimated tight framelets on the graph in detail. First, we start from the construction of the tight wavelet frames on by a given scaling function.

3.1. Construction of Tight Wavelet Frames on

For the construction of classical tight wavelet frames on , there have been many creative results. These works are generally based on the multiresolution analysis viewpoint. Chui and He in [11] demonstrated that a tight wavelet frame with 3 symmetric generators can be derived from the B-spline functions However, this construction only has single-order vanishing moments. It is desirable to construct symmetric tight wavelet frames with high vanishing moments in applications. In order to achieve high order of vanishing moments, Daubechies et al. [5] considered a tight wavelet frame with 2 compactly supported generators from the B-spline functions which have order vanishing moments. Unfortunately, this tight wavelet frames are not symmetry. For studying symmetric tight wavelet frames, based on symmetric compactly supported refinable functions, Petukhov [66] discussed the symmetry of tight wavelet frames using the unitary extension principle and obtained the existence criterion of the symmetric or antisymmetric compactly supported framelets. For any compactly supported symmetric real-valued refinable function, Bin and Mo [67] showed that symmetric tight wavelet frames with 3 generators and high vanishing moments can be derived.

The tight wavelet frames with desired approximation orders are very critical in practical applications. Dong and Shen in [68] proved that the tight frame system derived from a pseudospline normally have better approximation order than that derived from B-splines. Later, Dong showed that the shifts of an arbitrarily given pseudosplines are linearly independent [69]. The pseudosplines are considered an important family of refinable functions and provide a wide variety of choices of refinable functions. By selecting different parameters, pseudosplines with various orders fill in the gaps between the B-splines and orthogonal refinable functions for the first type and between B-splines and interpolatory refinable functions for the second type [69]. Hence, pseudosplines have large flexibilities in wavelet and framelet construction. In this subsection, we focus on the construction of tight wavelet frames on based on the pseudosplines.

In many applications, such as computational cost and storage concern [70–75], we hope a symmetric tight wavelet frame with as small as possible number of generators; that is, the symmetric tight wavelet frame is generated by a single wavelet function. Yet, except the tight frames generated by the discontinuous Haar wavelet function or its dilated version, it is impossible to exist an MRA compactly supported real-valued symmetric tight wavelet frame with one continuous generator [25]. Therefore, one is interested in considering a symmetric tight wavelet frame with two generators. They have been extensively studied in [13, 68, 76–80]. As shown in [68], a necessary and sufficient condition has been derived for the existence of a symmetric tight wavelet frame with two generators, and details are as follows: , (1)There exists a -periodic trigonometric polynomial with real coefficients such that and for all ;(2)There exists a real-valued sequence on with symmetry such that , where the matrix is defined in (10);(3)The greatest common factor of all the entries of the matrix satisfies a technical “gcd” condition. The technical “gcd” condition is shown in [68].

However, it is difficult to obtain a -periodic trigonometric polynomial satisfying all above the conditions, since there exist nonlinear equations in these conditions. Therefore, the symmetric tight wavelet frame with three generators is usually considered. Here, we will derive a new construction method of a symmetric tight wavelet frame with three generators based on pseudosplines.

Pseudosplines are defined in terms of their refinement masks [5, 68]. The refinement mask of the first type of pseudosplines with order is given byand the refinement mask of the second type of pseudosplines with order is given bywhere .

In this subsection, we only consider the construction of tight wavelet frame from the second type of pseudosplines using the oblique extension principle. In order to state the main results, we review the following results in [67].

Theorem 2 (see [67]). Let be a finitely supported real-valued mask on such that and . Then, there does not exist a-periodic trigonometric rational polynomial with real coefficient such that(i) and , a.e.;(ii)can be regarded as a -periodic trigonometric polynomial and for all .

Consequently, for any positive integer , there do not exist finitely supported real-valued sequences and a -periodic trigonometric rational polynomial with real coefficient such that all the conditions in Theorem 2 are satisfied.

Theorem 3 (see [67]). Let be a finitely supported real-valued mask on such that for some positive integer and some finitely supported sequence on . Let be the compactly supported real-valued refinable function associated with mask . Suppose that and the shifts of are stable. Then, there exists a-periodic trigonometric polynomialsuch that

Theorem 4 (see [67]). Let be a finitely supported mask in such that for all and for some . Then, there does not exist a-periodic trigonometric polynomial such that and

Symmetry is an important property in various purposes. For a Laurent polynomial with real coefficients, we say that is symmetric (or antisymmetric) about for some if (or ). For a nonzero Laurent polynomial , we introduce an operator to beThe following result can be given, as shown in [76].

Theorem 5 (see [76]). Let and be two Laurent polynomials with real coefficients. Then,(1) is (anti)symmetric about for some if and only if .(2) .(3) and for .(4)If and are (anti)symmetric such that , then is (anti)symmetric and .

Next, we give the construction of symmetric tight wavelet frame. For a given refinable function with mask , the key is to find a -periodic , such that OEP condition is satisfied. We have the following the main result.

Theorem 6. Let denote the second type of pseudosplines of order with a finitely supported mask , which is defined in (20). Suppose that there is a-periodic trigonometric polynomial with real coefficients such that , is symmetric, and for all . In addition, assume the following:where. By the Fej é r–Rieszlemma, there exists a-periodic trigonometric polynomial with real coefficients such that , defined by