Adaptive directional Haar tight framelets on bounded domains for digraph signal representations
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Adaptive directional Haar tight framelets on boundeddomains for digraph signal representations
Yuchen Xiao · Xiaosheng Zhuang
Received: date / Accepted: date
Abstract
Based on hierarchical partitions, we provide the construction ofHaar-type tight framelets on any compact set K ⊆ R d . In particular, on theunit block [0 , d , such tight framelets can be built to be with adaptivity and di-rectionality. We show that the adaptive directional Haar tight framelet systemscan be used for digraph signal representations. Some examples are providedto illustrate results in this paper. Keywords directional Haar tight framelets · adaptive systems · boundeddomains · graph signal processing · digraph signal · graph clustering · coarse-grained chain · network · deep learning · machine learning Harmonic analysis including Fourier analysis, frame theory, wavelet/frameletanalysis, etc., has been one of the most active research areas in mathematicsover the past two centuries [46]. Typical harmonic analysis is on theory and ap-plications related to functions defined on regular Euclidean domains [5,11,12,20,23,26,29,36]. In recent years, driven by the rapid progress of deep learningand their successful applications in solving AI (artificial intelligence) relatedtasks, such as natural language processing, autonomous systems, robotics,medical diagnostics, and so on, there has been a great interest in developingharmonic analysis for data defined on non-Euclidean domains such as man-ifold data or graph data, e.g., see [1,4,7,8,13,15,19,28,39,40,49] and manyreferences therein. For example, data in machine/statistical learning, are typi-cally from social networks, biology, physics, finance, etc., and can be naturallyobtained or organized as graphs or graph data. Such data can be regarded
Y. Xiao and X. Zhuang (Corresponding Author)Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, KowloonTong, Hong Kong. E-mail: [email protected]; [email protected] a r X i v : . [ m a t h . F A ] A ug Yuchen Xiao, Xiaosheng Zhuang as samples from an underlying manifold, where its graph Laplacian is con-nected to the manifold Laplacian encoding the essential information of thedata to be exploited by various machine/deep learning approaches [44]. Onealso refers this area as the graph signal processing (GSP) in contrast to sig-nal/image/video processing [1].For graph signal analysis, the underlying graphs are typically directedgraphs (or digraphs). For example, the citation networks modelling relationsamong papers (as nodes) are digraphs where a paper can either be cited orcite other papers, which indicates a directed edge between two nodes; theinformation networks [38] with nodes consisting of URLs of web pages are di-graphs, where an edge means that there is an URL in one web page linking toanother web page; the traffic networks [21] in modern cities with nodes rep-resenting intersections and edges representing traffic flows from one node toanother are digraphs; the human body networks, the nervous systems, and bi-ological networks, etc., are all digraphs. The interested reader can refer to [37,41,45] and many references therein. Similar to the wavelets and framelets forsignal/image processing, multiscale representation systems based on variousapproaches such as spectral theory [9], diffusion wavelets[6], non-spectral con-struction [7,8], etc., have also been developed for graph signal representationand processing.In this paper, motivated by the recent development of directional Haarframelet systems on R d [22,32,34] as well as wavelet-like systems for graphsignal processing [1,8,31,48,50], we focus on the development of directionalmultiscale representation systems for signals defined on digraphs. We are goingto investigate the following two main problems:1) How to construct directional Haar tight framelets on bounded domainswith adaptivity?2) How to efficiently represent digraph signals?In what follows, we lay out the main idea of this paper for the above twoproblems. The details are given in the later sections.For the first problem, we start with Haar wavelets. Recall that for a sepa-rable Hilbert space H , a collection X = { h j } j ∈ N ⊆ H is said to be a frame ifthere exist two positive constants 0 < C ≤ C < ∞ such that C (cid:107) f (cid:107) ≤ (cid:88) j ∈ N |(cid:104) f, h j (cid:105)| ≤ C (cid:107) f (cid:107) ∀ f ∈ H , where (cid:104)· , ·(cid:105) and (cid:107) · (cid:107) are the inner product and norm in H , respectively. If C = C , then such an X is said to be tight . If C = C = 1 and (cid:107) h j (cid:107) = 1 forall j , then such an X is an orthonormal basis for H .Haar wavelet system [18] is the first ever constructed orthonormal waveletsystem on the interval [0 , φ := χ [0 , , which is acharacteristic function defined on the unit interval I := [0 , daptive directional Haar tight framelets for digraph signal representations 3 wavelet function ψ := χ [0 , ) − χ [ , , one can show that the system X ( R ; φ, ψ ) := { φ ( · − k ) : k ∈ Z } ∪ { ψ j,k := 2 j/ ψ (2 j · − k ) : k ∈ Z } j ∈ N obtained from dilations and translations of φ and ψ , is an orthonormal waveletbasis for L ( R ) of square-integrable functions on R [12]. Here N := N ∪{ } . Thanks to their compact support property, the restriction of such anorthonormal wavelet basis on the unit interval I directly gives an orthonormalbasis on the bounded domain for L ([0 , I : X ( I ; φ, ψ ) := { φ } ∪ { ψ j,k : 0 ≤ k < j } j ∈ N . (1)This is indeed the system constructed by Haar in [18]. Fig. 1
The tensor product of 1D Haar wavelet scaling function φ and wavelet function ψ .All are supported on the unit square [0 , . The 4 big squares are (from left to right): φ ⊗ φ , φ ⊗ ψ , ψ ⊗ φ , ψ ⊗ ψ . Each colored sub-block represents either 1 (blue) or − In higher dimensions, the tensor product approach is usually employedto obtain orthonormal wavelets, e.g., see Fig. 1 for the 4 generators φ ⊗ φ , φ ⊗ ψ , ψ ⊗ φ , and ψ ⊗ ψ of the 2D orthonormal Haar wavelets. However, itis well-known that the tensor product orthonormal real-valued wavelets lackdirectionality [2], which hinders the sparsity representation of such systemsfor high-dimensional data analysis and their applications in image/video pro-cessing. Various approaches including curvelets [2], shearlets [3,14,27,29,51],dual-tree complex wavelets [43], TP- C TFs [24,25,52], etc., on increasing direc-tionality of multiscale representation systems have been proposed over the pasttwo decades, which we will not get into much of such developments but drawour attentions only to the main focus of this paper:
Haar-type multiscale repre-sentation systems with directionality on bounded domains . Note that in orderto increase directionality, one necessarily needs to consider wavelet frames orframelet systems, which are more redundant representation systems than theorthonormal systems.In [32], the authors proposed a new and simple Haar-type directional sys-tem, the directional Haar tight framelets (DHF) for L ( R ), whose generatorshave 4 directions (0 ◦ , ◦ , and ± ◦ ). The system is generated from the scalingfunction ϕ = χ I ⊗ χ I and 6 generators in Ψ := { ψ (1 , , . . . , ψ (3 , } defined by ψ (1 , = χ B − χ B , ψ (1 , = χ B − χ B , ψ (1 , = χ B − χ B ,ψ (2 , = χ B − χ B , ψ (2 , = χ B − χ B , ψ (3 , = χ B − χ B , (2) Yuchen Xiao, Xiaosheng Zhuang
Fig. 2
The 6 functions in Ψ . Left to right: ψ (1 , , ψ (1 , , ψ (1 , , ψ (2 , , ψ (2 , , ψ (3 , . Eachof them is supported on the unit square [0 , , which is split to 4 sub-blocks B , . . . , B .Each colored sub-block represents either 1 (blue) or − where B := [0 , ) × [0 , ), B := [ , × [0 , ), B := [0 , ) × [ , B := [ , × [ ,
1] are the 4 sub-blocks obtained from refining the unit square I = [0 , × [0 ,
1] = ∪ (cid:96) =1 B (cid:96) , see Fig. 2. Clearly, compared to the 2D tensorproduct Haar wavelets (see Fig. 1), the DHF system has more directional-ity: the generators ψ (1 , and ψ (3 , can be used for vertical edge informationextraction, the generators ψ (1 , and ψ (2 , can be used for horizontal edge in-formation extraction, and the generators ψ (1 , and ψ (2 , can be used for ± ◦ edge information extraction. Note that the labelling ( (cid:96) , (cid:96) ) , ≤ (cid:96) < (cid:96) ≤ , ( (cid:0) (cid:1) = 6 , see Theorem 1 for more general results).The system defined by X ( R ; ϕ, Ψ ) := { ϕ ( · − k ) : k ∈ Z } ∪ { ψ j,k : k ∈ Z , ψ ∈ Ψ } j ∈ N , where ψ j,k := 2 j/ ψ (2 j · − k ), is a tight frame for L ( R ). Its restriction to theunit square I = [0 , can be shown as X ( I ; ϕ, Ψ ) := { ϕ } ∪ { ψ j,k : k = ( k , k ) , ≤ k , k < j ; ψ ∈ Ψ } j ∈ N . (3)This system X ( I ; ϕ, Ψ ) is indeed also a tight frame for L ([0 , ) (see The-orem 1). Such a tightness property on [0 , is not explicitly shown in [32]nor in [22]. In [22], the authors further generalized such directional Haar tightframelets to arbitrary dimension R d .By inspecting the structure of the system, we can regroup X ( I ; ϕ, Ψ ) as X ( I ; ϕ, Ψ ) = { ϕ } ∪ ∞ (cid:91) j =0 2 j − (cid:91) k ,k =0 Ψ j, ( k ,k ) , (4)where each Ψ j,k := { j/ ψ ( (cid:96) ,(cid:96) ) (2 j · − k ) : 1 ≤ (cid:96) < (cid:96) ≤ } has 6 framelet functions supported on a sub-block B j, ( k ,k ) = [2 − j k , − j ( k + 1)] × [2 − j k , − j ( k + 1)] ⊆ I at level j . Each B j, ( k ,k ) is further refined to 4 sub-blocks B j +1 , (2 k , k ) , B j +1 , (2 k +1 , k ) , B j +1 , (2 k , k +1) , and B j +1 , (2 k +1 , k +1) at level j + 1, seeFig. 3 for the illustration. In other words, the system in (4) is based on a daptive directional Haar tight framelets for digraph signal representations 5 hierarchical partition of the unit square I . This point of view together withhow to sparsely represent digraph signals motivates us the main result in The-orem 1, where the question boils down to the construction of directional Haartight framelet systems with adaptivity . Here, by “adaptivity” we mean thatthe blocks are not necessary square sub-blocks. We affirmatively show thata system X ( {B j } j ∈ N ), associated with a sequence B j with each B j being acollection of subsets of a compact set K ⊆ R d from a refining process, couldbe built to be a tight frame for L ( K ). When K = I = [0 , , such a systemis our adaptive directional Haar tight framelets and it plays a key role in oursecond problem of efficient representations of digraph signals. Fig. 3
The unit square I = [0 , is refined to 4 sub-blocks B , . . . , B . Each block B (cid:96) isfurther refined to 4 sub-blocks, and so on. Left: the unit square I is associated with ϕ and ψ ∈ Ψ at level j = 0. Middle: 4 refined blocks B , . . . , B are associated with Ψ ,k at level j = 1. Right: 16 refined blocks are associated with Ψ ,k at level j = 2. Now, continue to the second problem of efficient representations of signalson digraphs (directed graphs). We begin with undirected graphs. Recall thata graph is an ordered pair G = ( V, W ) with a nonempty set V = { v , . . . , v n } of vertices and an (weighted) adjacency matrix W : V × V → [0 , ∞ ) of size n × n indicating edges between vertices ( W ( v i , v j ) (cid:54) = 0 if there is an edgefrom the vertex v i to v j ; otherwise 0). If the edges are unordered, that is, theedge from v i to v j is considered to be the same as the edge from v j to v i ,in which case, the matrix W is symmetric, then G is said to be undirected ;otherwise, it is called a directed graph or digraph , see Fig. 4 for an example ofundirected graph and digraph. A signal defined on a graph G (or graph signal)is a function f : V → C .For a signal f on an undirected graph G = ( V, W ), one could identify it witha function on I = [0 ,
1] by associating each vertex v ∈ V a suitable subinterval I v ⊆ I . In [7], the paper uses the concept of a filtration , which a weighttree, for identifying vertices as subintervals in I as well as building an (Haar-type) orthonormal basis on the filtration to represent signals on the underlyinggraph G . In this paper, we use the concept of the coarse-grained chains ([31,48,50]). Roughly speaking, a coarse-grained chain G J → := ( G J , G J − , . . . , G )of G ≡ G J is a sequence of graphs such that G j − is from the clustering result of G j . When G has only one node, the coarse-grained chain is actually equivalentto a filtration in [7]. See Fig. 5 for an example of a coarse-grained chain G → :=( G , . . . , G ) of G . Each vertex in G j − is a cluster of vertices in G j . Based onsuch a coarse-grained chain, one can give a hierarchical representation {I j } j =0 Yuchen Xiao, Xiaosheng Zhuang a b c d e f G a b c d e f G x Fig. 4
An undirected graph G x = ( V, W x ) (Top) and a digraph G = ( V, W ) (Bottom) withthe same vertex set V = { a, b, c, d, e, f } . Note that W x (cid:54) = W and W x is symmetric. of the interval I , where each I j = { I j,k } is the collection of subintervals I j,k of I such that ∪ k I j,k = I , see Fig. 5.Based on such a hierarchical sequence, one could build a Haar-type or-thonormal basis [7] for the function space span { χ I ,k : k = 1 , . . . , } , whichis the space for the signal defined on the graph. See Section 4 for more details. a b c d e f G [0 , ) [ , ) [ , ) [ , )[ , ) [ , a b c d e f G [0 , ) [ , ) [ , a b c d e f
12 8 G [0 , ) [ , a b c d e f G [0 , Fig. 5
A coarse-grained chain of G . G is the underlying graph G . G j − isfrom clustering of G j for j = 1 , ,
3. Note that G has one vertex only. Hereeach box represents a node (or cluster) in the graph, the lines representedges between vertices, and the arc on a same node indicates a self-loop. G can be identified as the root interval I = [0 , G as [0 , ) ∪ [ , G as [0 , ) ∪ [ , ) ∪ [ , G as [0 , ) ∪ [ , ) ∪ [ , ) ∪ [ , ) ∪ [ , ) ∪ [ , Returning to digraph signal representations, can one use similar approachesfor undirected graph to represent digraph signals? The answer is yes and no .For “no” it is because most of the clustering algorithms are developed basedon the symmetry property of the adjacency matrix W or the well-defined andwell-understood operator on the undirected graph: the graph Laplacian [9]. Itis not trivial to directly use them for digraph cases. For “yes” it is becausethere are undirect ways to circumvent such difficulties. In fact, one typicalapproach is to define a counterpart graph Laplacian on digraph, such as the Hodge Laplacian in [35], the weighted adjacency matrix in [10], the so-called dilaplacian in [33], and so on. In this paper, we use the idea developed in [8]:to lift the dimension from one to two by using a pair of undirected graphsto represent a given digraph. In a nutshell, a digraph signal is identified as asignal defined on I = [0 , × [0 ,
1] through the following steps.1) In view of the singular value decomposition, the adjacency matrix W ina digrah G = ( V, W ) is uniquely determined by
W W (cid:62) and W (cid:62) W fromwhich one could construct a pair of undirected graphs G x = ( V, W W (cid:62) )and G y = ( V, W (cid:62) W ).2) Applying well-known techniques, e.g., [7], for undirected graphs, one canrepresent vertices in each graph of G x and G y as subintervals in I = [0 , v is identified as a subinterval I xv = [ a, b ) on G x and I yv = [ c, d ) on G y , then v in the original digraph G is identified as a block[ a, b ) × [ c, d ) ⊆ I . Consequently, the vertices in the digraph are sub-blocksin the unit square.4) Then, signals on G can be viewed as functions defined on the unit square[0 , .In [8], once orthonormal bases are built for G x and G y , then the tensorproduct approach is used to construct orthonormal bases for G . As we pointedout, the tensor product approach lacks directionality. Since directional systemsprovide better sparse representations than those by the tensor product ones,in this paper, we use our adaptive directional Haar tight framelets in 2D forthe digraph signal representations based on the above digraph representations G ↔ ( G x , G y ).The contribution of the paper is threefold. First, based on a hierarchicalpartition, we provide a simply yet flexible construction of Haar-type tightframelets on any compact set K ⊆ R d . Second, such framelet systems includedirectional Haar systems in [22,32] as special cases and lead to the adaptivedirectional Haar tight framelet systems for non-dyadic partitions of the unitblock [0 , d . Last but not least, we demonstrate that digraph signals can beidentified as signals defined on the unit square [0 , and hence could beefficiently represented by the adaptive directional Haar tight framelet systemswhere the directionality is a really desired property.The structure of the paper is as follows. In Section 2, we present our mainresults on the construction of tight frames for L ( K ) for some compact set K ⊆ R d . Then, adaptive directional Haar tight framelets on bounded domainsare deduced. In Section 3, we show how to represent digraph signals using the Yuchen Xiao, Xiaosheng Zhuang developed adaptive directional Haar tight framelets. In Section 4, we providesome examples to illustrate our main results. Conclusion and further remarksare given in Section 5. Some proofs are postponed to the last section.
Let K ⊆ R d be a compact set and consider the Hilbert space L ( K ) := { f : (cid:107) f (cid:107) := ( (cid:82) K | f ( x ) | dx ) < ∞} of square-integrable functions f on K . Theinner product on L ( K ) is defined by (cid:104) f, g (cid:105) := (cid:82) K f ( x ) g ( x ) dx for f, g ∈ L ( K ).In this section, based on a hierarchical partition of K , we construct a system X = { ϕ } ∪ { Ψ j } j ∈ N of elements in L ( K ) and show that it is a tight frame for L ( K ). Such a system X leads to our adaptive directional Haar tight framelets (AdaDHF) on K .Before we present and prove our main result in Theorem 1, let us introducesome necessary notation, definitions, and auxiliary results first. For a Hilbertspace H , the collection X = { h j } j ∈ N ⊆ H is a tight frame for H if (cid:107) f (cid:107) = (cid:88) j ∈ N |(cid:104) f, h j (cid:105)| ∀ f ∈ H . (5)Using the polarization identity, one can show that it is equivalent to f = (cid:88) j ∈ N (cid:104) f, h j (cid:105) h j ∀ f ∈ H . (6)We denote I m the identity matrix of size m × m . The matrix A in thefollowing lemma is used to connect functions on two scales supported on asame block B ⊆ K . Its proof is postponed to Section 6. Lemma 1
Let m ∈ N and b , . . . , b m be m positive constants such that (cid:80) m(cid:96) =1 b (cid:96) = 1 . Let n = (cid:0) m (cid:1) and A = ( a i,(cid:96) ) ≤ i ≤ n ;1 ≤ (cid:96) ≤ m be a matrix of size ( n + 1) × m of the form: A = √ b √ b √ b · · · (cid:112) b m − √ b m √ b −√ b · · · √ b −√ b · · · ... ... ... . . . ... ... · · · √ b m − (cid:112) b m − . (7) That is, the first row of A (with respect to i = 0 ) is ( a ,(cid:96) ) m(cid:96) =1 = (cid:16)(cid:112) b (cid:96) (cid:17) m(cid:96) =1 , and the row ( a i,(cid:96) ) m(cid:96) =1 for i (cid:54) = 0 is given by a i,(cid:96) = a ( i ,i ) ,(cid:96) = (cid:112) b i if (cid:96) = i , − (cid:112) b i if (cid:96) = i , otherwise, daptive directional Haar tight framelets for digraph signal representations 9 where for each i (cid:54) = 0 , the index i is uniquely determined by a pair ( i , i ) satisfying ≤ i < i ≤ m through ( i , i ) (cid:55)→ i = (2 m − i )( i − + i − i . Then, A satisfies A (cid:62) A = I m . For a measurable set B ⊆ R d , we denote | B | as its Lebesgue measure and χ B as the characteristic function on B . The following lemma shows that wecan construct a set of functions supported on B so that it is tight. Lemma 2
Let B ⊆ K ⊆ R d be a measurable subset in the compact set K satisfying | B | > and B (cid:96) , (cid:96) = 1 , . . . , m with m ≥ be measurable sub-blocksof B such that B = ∪ m(cid:96) =1 B (cid:96) , | B (cid:96) | > for all (cid:96) = 1 , . . . , m , and | B (cid:96) ∩ B (cid:96) | = 0 for (cid:96) (cid:54) = (cid:96) . Define the set Ψ B := { ψ ( (cid:96) ,(cid:96) ) : 1 ≤ (cid:96) < (cid:96) ≤ m } of functions by ψ ( (cid:96) ,(cid:96) ) := (cid:112) b (cid:96) γ (cid:96) − (cid:112) b (cid:96) γ (cid:96) , ≤ (cid:96) < (cid:96) ≤ m, (8) where γ (cid:96) := χ B(cid:96) √ | B (cid:96) | and b (cid:96) := | B (cid:96) || B | . Then Ψ B is a tight frame for W B := span { ψ ( (cid:96) ,(cid:96) ) : 1 ≤ (cid:96) < (cid:96) ≤ m } . That is, f = (cid:88) ≤ (cid:96) <(cid:96) ≤ m (cid:68) f, ψ ( (cid:96) ,(cid:96) ) (cid:69) ψ ( (cid:96) ,(cid:96) ) ∀ f ∈ W B . The proof of Lemma 2 uses results in Lemma 1 and it is one of the keysteps in our proof of the main theorem. We postpone it to Section 6.Note that ψ ( (cid:96) ,(cid:96) ) in Lemma 2 are constructed from χ B (cid:96) , (cid:96) = 1 , . . . , m .The following lemma demonstrates that those χ B (cid:96) ’s can be constructed from ψ ( (cid:96) ,(cid:96) ) ’s together with χ B as well. Lemma 3
Let B , B (cid:96) , γ (cid:96) , (cid:96) = 1 , . . . , m , and Ψ B := { ψ ( (cid:96) ,(cid:96) ) , ≤ (cid:96) < (cid:96) ≤ m } be defined as in Lemma 2. Define vectors Γ B , Φ B of functions by Γ B := ( γ (cid:96) ) m(cid:96) =1 and Φ B := ( γ B , Ψ B ) = ( γ B , ψ , . . . , ψ n ) , where γ B := χ B √ | B | and ψ , . . . , ψ n are from enumerating the elements in Ψ B through ( (cid:96) , (cid:96) ) (cid:55)→ (2 m − (cid:96) )( (cid:96) − + ( (cid:96) − (cid:96) ) with n = (cid:0) m (cid:1) = m × ( m − . Then Γ B = A (cid:62) Φ B . Consequently, the space V := span { γ (cid:96) : (cid:96) = 1 , . . . , m } = V B ⊕ W B where V B := span { χ B } and W B := span { ψ : ψ ∈ Ψ B } .Proof Note that Γ B is a vector of size m while Φ B is a vector of size n = (cid:0) m (cid:1) +1.From the definition of ψ ( (cid:96) ,(cid:96) ) in (8) and B = ∪ (cid:96) B (cid:96) , it is easy to verify that Φ B = AΓ B , where A = ( a i,(cid:96) ) ≤ i ≤ n ;1 ≤ (cid:96) ≤ m is a matrix of size ( n + 1) × m defined as inLemma 1. By Lemma 1, we have A (cid:62) A = I m , which implies that Γ B = A (cid:62) Φ B .Hence, V ⊆ V B + W B . Now the fact that V = V B ⊕ W B follows directly from V B ⊆ V , W B ⊆ V and V B ⊥ W B . This completes the proof. (cid:4) By splitting the compact set K , one can obtain subsets B (cid:96) of K . For eachsubset B (cid:96) , one can further refine it to have smaller subsets. Such a processcould continue and one could obtain a hierarchical partition of K . We saythat the sequence {B j } j ∈ N is a hierarchical partition of K if it satisfies thefollowing conditions:a) Root property : each B j is a collection of finite number of measurable subsetsof K with B = { K } , ∪ B ∈B j B = K , | B | > B ∈ B j , and | B ∩ B | =0 for any B (cid:54) = B in B j .b) Nested property : {B j } j ∈ N is nested in the sense that for each B ∈ B j − , B = ∪ c B (cid:96) =1 B (cid:96) with B (cid:96) ∈ B j . That is B (cid:96) ’s are children of B in B j and thepositive integer c B ≥ B in B j . In otherwords, the sets in B j are obtained from the splitting of sets in B j − .c) Density property : lim j →∞ diam( B j ) = 0 where diam( B j ) := max { diam( B ) : B ∈ B j } and diam( B ) := sup {| x − y | : x, y ∈ B } is the diameter of theset B .We are now ready to introduce and prove our main result. Theorem 1
Let K ⊆ R d be a compact set in R d with | K | > and {B j } j ∈ N be a hierarchical partition of K . Define the set X ( {B j } j ∈ N ) := { ϕ } ∪ { Ψ j,B : B ∈ B j } j ∈ N of functions by ϕ := χ K √ | K | and Ψ j,B := { ψ ( (cid:96) ,(cid:96) ) j,B : 1 ≤ (cid:96) < (cid:96) ≤ c B } ∞ j =0 with ψ ( (cid:96) ,(cid:96) ) j,B := (cid:112) b (cid:96) γ (cid:96) − (cid:112) b (cid:96) γ (cid:96) , ≤ (cid:96) < (cid:96) ≤ c B , (9) where B (cid:96) ∈ B j +1 , (cid:96) = 1 , . . . , c B are the children sub-blocks of B , γ (cid:96) := χ B(cid:96) √ | B (cid:96) | ,and b (cid:96) := | B (cid:96) || B | . Then, X ( {B j } j ∈ N ) is a tight frame for L ( K ) .Proof By (5), we need to prove that (cid:107) f (cid:107) = | (cid:104) f, ϕ (cid:105) | + ∞ (cid:88) j =0 (cid:88) B ∈B j (cid:88) ≤ (cid:96) <(cid:96) ≤ c B (cid:12)(cid:12)(cid:12)(cid:68) f, ψ ( (cid:96) ,(cid:96) ) j,B (cid:69)(cid:12)(cid:12)(cid:12) ∀ f ∈ L ( K ) . We proceed through the following steps.1) First, let V := span { χ K } = span { ϕ } and V j := span { χ B : B ∈ B j } (10)for j ∈ N . Then by the nested property of {B j } j ∈ N , we have V ⊆ V ⊆ · · · ⊆ V j ⊆ V j +1 ⊆ · · · . By the density property of {B j } j ∈ N , we see that ∪ j ∈ N V j is dense in L ( K ). daptive directional Haar tight framelets for digraph signal representations 11
2) Let W j := span { ψ : ψ ∈ Ψ j,B , B ∈ B j } (11)and W j,B := span { ψ : ψ ∈ Ψ j,B } , j ∈ N . Thanks to the nested property of {B j } j ∈ N and our construction of ψ ( (cid:96) ,(cid:96) ) j,B ,we have that for any B ∈ B j (cid:68) χ B , ψ ( (cid:96) ,(cid:96) ) j,B (cid:69) = 0 , ≤ (cid:96) < (cid:96) ≤ c B . Hence, we see that V j ⊥ W j and W j ⊥ W j (cid:48) for all j, j (cid:48) ∈ N and j (cid:54) = j (cid:48) .Moreover, we claim that V j +1 = V j ⊕ W j ∀ j ∈ N . Obviously, V j ⊆ V j +1 and W j ⊆ V j +1 . Hence, we only need to show that V j +1 ⊆ ( V j + W j ), which by the nested property and noticing W j = ⊕ B ∈B j W j,B for all j ∈ N , it suffices to show that for each B ∈ B j , func-tions in { χ B (cid:96) : B (cid:96) ∈ B j +1 are children of B } ⊆ V j +1 are the linear combinations of functions in { χ B } ∪ { ψ ( (cid:96) ,(cid:96) ) j,B : 1 ≤ (cid:96) < (cid:96) ≤ c B } ⊆ ( V j + W j ) , which follows from Lemma 3. Therefore V j +1 = V j ⊕ W j for all j ∈ N .3) Consequently, V ⊕ (cid:76) j ∈ N ,B ∈B j W j,B is dense in L ( K ). Hence, for each f ∈ L ( K ), there exists a sequence { c ϕ } ∪ { c ( (cid:96) ,(cid:96) ) j,B : B ∈ B j , ≤ (cid:96) , (cid:96) ≤ c B } j ∈ N of constants such that f = c ϕ ϕ + ∞ (cid:88) j =0 (cid:88) B ∈B j (cid:88) ≤ (cid:96) <(cid:96) ≤ c B c ( (cid:96) ,(cid:96) ) j,B ψ ( (cid:96) ,(cid:96) ) j,B , where the equality holds in the L -sense. Define f j,B := (cid:88) ≤ (cid:96) <(cid:96) ≤ c B c ( (cid:96) ,(cid:96) ) j,B ψ ( (cid:96) ,(cid:96) ) j,B ∈ W j,B . Then f = c ϕ ϕ + (cid:80) ∞ j =0 (cid:80) B ∈B j f j,B and we have (cid:107) f (cid:107) = (cid:104) f, f (cid:105) = | c ϕ | + ∞ (cid:88) j =0 (cid:88) B ∈B j (cid:107) f j,B (cid:107) , where the series converges absolutely. On the other hand, we have (cid:88) h ∈ X ( {B j } j ∈ N | (cid:104) f, h (cid:105) | = | (cid:104) f, ϕ (cid:105) | + ∞ (cid:88) j =0 (cid:88) B ∈B j (cid:88) ≤ (cid:96) <(cid:96) ≤ c B (cid:12)(cid:12)(cid:12)(cid:68) f, ψ ( (cid:96) ,(cid:96) ) j,B (cid:69)(cid:12)(cid:12)(cid:12) = | (cid:104) f, ϕ (cid:105) | + ∞ (cid:88) j =0 (cid:88) B ∈B j (cid:88) ≤ (cid:96) <(cid:96) ≤ c B (cid:12)(cid:12)(cid:12)(cid:68) f j,B , ψ ( (cid:96) ,(cid:96) ) j,B (cid:69)(cid:12)(cid:12)(cid:12) . Hence, to prove that (cid:107) f (cid:107) = (cid:80) h ∈ X ( {B j } j ∈ N ) |(cid:104) f, h (cid:105)| , it suffices to showthat for each j ∈ N and B ∈ B j , we have (cid:107) f j,B (cid:107) = (cid:88) ≤ (cid:96) <(cid:96) ≤ c B (cid:12)(cid:12)(cid:12)(cid:68) f j,B , ψ ( (cid:96) ,(cid:96) ) j,B (cid:69)(cid:12)(cid:12)(cid:12) . This is equivalent to showing that Ψ j,B = { ψ ( (cid:96) ,(cid:96) ) j,B : 1 ≤ (cid:96) , (cid:96) ≤ c B } is atight frame for W j,B , which follows from Lemma 2.Consequently, we prove that X ( {B j } j ∈ N ) is a tight frame for L ( K ). (cid:4) The system X ( {B j } j ∈ N ) in Theorem 1, which depends only on the hier-archical partition of K , is very flexible. Next, we discuss some of its specialcases.In practice, signals usually lie in finite-dimensional spaces. Hence, it isuseful to study the cut-off system X ( {B j } Jj =0 ) up to some scale J ∈ N . Corollary 1
Retaining all assumptions and notation in Theorem 1. Given J ∈ N , define the cut-off system X ( {B j } Jj =0 ) by X ( {B j } Jj =0 ) := { ϕ } ∪ { Ψ j,B : B ∈ B j } J − j =0 . (12) Then, X ( {B j } Jj =0 ) is a tight frame for V J defined as in (10) .Proof Note that V J = V ⊕ (cid:76) J − j =0 W j . The conclusion follows similarly to theproof of Theorem 1 by showing that (cid:107) f (cid:107) = (cid:80) h ∈ X ( {B j } Jj =0 ) | (cid:104) f, h (cid:105) | for all f ∈ V J .We immediately have the following corollary if each splitting of a block B has at most two children (sub-blocks). Corollary 2
Retaining all assumptions in Theorem 1. In addition, if c B ≤ for each B ∈ B j and j ∈ N , that is, the number of children of each block B isat most , then X ( {B j } j ∈ N ) is an orthonormal basis for L ( K ) .Proof Note that L ( K ) = V ⊕ j ∈ N ,B ∈B j W j,B from the proof of Theorem 1.If c B ≤
2, then there is at most one element ψ j,B = √ b γ − √ b γ in Ψ j,B and (cid:107) ψ j,B (cid:107) = 1 for all j, B . Note that (cid:107) ϕ (cid:107) = 1 also. Consequently, each W j,B is at most one-dimensional. Hence, X ( {B j } j ∈ N ) is orthonormal. (cid:4) The Haar orthonormal wavelets on I = [0 ,
1] is a special case of the conse-quence of Corollary 2. It is with respect to the hierarchical partition {I j } j ∈ N of the unit interval I with I j := { I j,k := [2 − j k, − j ( k +1)] : k = 0 , . . . , j − } .Note that each I j,k has exactly two children subintervals I j +1 , k and I j +1 , k +1 with the same size. Such a type of partition is called dyadic . More generally,in dimension d , we have the following result, which includes directional Haartight framelets in [22,32] as special cases. daptive directional Haar tight framelets for digraph signal representations 13 Corollary 3
For j ∈ N , let I j := { I j,k := [2 − j k, − j ( k + 1)] : k =0 , . . . , j − } and define B j := ⊗ d I j := { I j,k × I j,k × · · · × I j,k d : 0 ≤ k , . . . , k d < j } . (13) Then, the system X ( {B j } j ∈ N ) defined as in Theorem 1 is a tight frame for L ([0 , d ) . In particular, when d = 1 , it is the Haar orthonormal wavelets X ( I ; φ, ψ ) defined as in (1) and when d = 2 , it is the directional Haar tightframelets X ( I ; ϕ, ψ ) defined as in (3) .Proof It is easy to show that {B j } j ∈ N is a hierarchical partition of the unitblock I d := [0 , d in d -dimension. Each B ∈ B j has exactly 2 d children sub-blocks in B j +1 . The conclusions follow directly from Theorem 1 and Corol-lary 2. (cid:4) The blocks in B j defined as in (13) are dyadic. In practice, as demonstratedin Fig. 5, intervals used to identify nodes on a graph are not necessarily dyadic.Hence, we introduce the adaptive directional Haar tight framelets (AdaDHF)based on the following hierarchical partition of the unit square I d = [0 , d ,whose sub-blocks are not necessarily dyadic. Corollary 4
For each s = 1 , . . . , d , let {I sj } j ∈ N be a hierarchical partition ofthe unit interval I = [0 , , where I sj := { I sj,k : k = 1 , . . . , n j,s } . Define B j := ⊗ d I sj = { I j,k × · · · × I j,k d : k s = 1 , . . . , n j,s , s = 1 , . . . , d } . (14) Then, {B j } j ∈ N is a hierarchical partition of the unit block I d = [0 , d and thesystem X ( {B j } j ∈ N ) defined as in Theorem 1 is a tight frame for L ([0 , d ) .In particular, for any J ∈ N , the cut-off system X ( {B j } Jj =0 ) as defined in (12) is a tight frame for V J .Proof Since {I sj } j ∈ N is a hierarchical partition of I , by the definition of B j , {B j } j ∈ N is a hierarchical partition of the unit square I d . The conclusionsfollow directly from Theorem 1 and Corollary 1. (cid:4) In this section, we use the AdaDHF systems developed in Corollary 4 of Sec-tion 2 to investigate digraph signal representations. We study representationsof signals on undirected graphs first and then turn to digraph signal represen-tations.3.1 The coarse-grained chain of an undirected graphWe start with undirected graphs and the coarse-grained chain of an undirectedgraph. For an undirected graph G = ( V, W ) with vertex (or node) set V = { v , . . . , v n } and adjacency matrix W : V × V → [0 , ∞ ). We use | V | (abuse of notation) to denote the number of vertices of G . The degree of a vertex v i isdenoted by deg( v i ) := (cid:80) nj =1 W ( v i , v j ). If W ( v i , v j ) >
0, then it corresponds toan edge ( v i , v j ) (unordered pair). Two vertices v i , v j are said to be connected if there exists a path between them, that is, [ W m ]( v i , v j ) (cid:54) = 0 for some positiveinteger m . The graph G is said to be connected if there exists a path betweenany two vertices. Throughout the paper, we only consider connected graphs.Let G = ( V, W ) and G cg = ( V cg , W cg ) be two undirected graphs. We saythat G cg is a coarse-grained graph of G if V cg is a partition of V ; i.e., thereexists subsets U , . . . , U m of V for some m ∈ N such that V cg = { U , . . . , U m } , U ∪ · · · ∪ U m = V, U i ∩ U j = ∅ , ≤ i < j ≤ m. In such a case, each node U i of G cg is called a cluster from G . The edgesof G cg are edges between clusters. Clusters U , . . . , U m define an equivalencerelation on G : two vertices u, v ∈ G are equivalent, denoted by u ∼ v , if u and v belong to the same cluster. An equivalent class (cluster) in G , whichis a node in G cg , associated with a vertex v ∈ V , then can be denoted as[ v ] G cg := { u ∈ G : u ∼ v } , and we have V cg = V / ∼ = { [ v ] G cg : v ∈ V } . If noconfusion arises, we will drop the subscript G cg and simply use [ v ] to denote acluster from G . Note that a vertex v in G can be viewed as [ v ] G = { v } , whichis a singleton.Given an undirected graph G = ( V, W ), there are many clustering algo-rithms can be used to obtain clusters from G , see e.g., [8,16,17,30,47]. Oncewe obtain the set { U , . . . , U m } =: V cg of clusters from G , we can define theweighted adjacency matrix W cg on V cg × V cg by W cg ([ u ] , [ v ]) := (cid:88) u ∈ [ u ] (cid:88) v ∈ [ v ] W ( u, v ) , [ u ] , [ v ] ∈ V cg . (15)Then, the new graph G cg := ( V cg , W cg ) is a coarse-grained graph of G . Giventhe new graph G cg , we can further apply clustering process on it and obtaina coarse-grained graph of G cg . Recursively doing such clustering processes, wewould obtain a chain of graphs from the original graph G . More precisely, let J ≥ G J → := ( G J , G J − , . . . , G )with G J ≡ G is a coarse-grained chain of G if G j = ( V j , W j ) is a coarse-grainedgraph of G for all 0 ≤ j ≤ J and [ v ] G j ⊆ [ v ] G j − for each j = 1 , . . . , J and forall v ∈ V . Note that, we treat each vertex v of the finest level graph G J ≡ G asa cluster of singleton. See Fig. 5 and Fig. 7 for illustrations of coarse-grainedchains.Once we have a coarse-grained chain G J → of G , we next discuss how toassociate it with a hierarchical partition of I = [0 , G = ( V , W ) has only one node, which is a clusterconsisting of all vertices of G . If not, we simply add such a graph to the chain.Now we define I j recursively as follows (c.f. Fig. 5 and Fig. 7).1) I = { I = [0 , } is the root node. daptive directional Haar tight framelets for digraph signal representations 15
2) Suppose I j − = { I j − ,k = [ a k , b k ] : k = 1 , . . . , | V j − |} has been de-fined and associated with the graph G j − = ( V j − , W j − ). Then, I j − ,k isassociated with a node u k ∈ V j − .3) For each u k ∈ V j − , denote (and order) the children of u k in G j = ( V j , W j )as u k, , . . . , u k,m ∈ V j and define subintervals I j,k, , . . . , I j,k,m by I j,k,s = [ a k + w s − , a k + w s ] , s = 1 , . . . , m, (16)where w s := ( b k − a k ) × (cid:80) si =1 deg( u k,i ) (cid:80) mi =1 deg( u k,i ) . Note that [ a k , b k ] = ∪ s I j,k,s . Collectall such subintervals I j,k,s as the collection I j := { I j,k (cid:48) : k (cid:48) = 1 , . . . , | V j |} .Then I j is associated with the graph G j .Given a hierarchical partition {I j } Jj =0 that is associated with a coarse-grained chain of the graph G = ( V, W ), then the vertex v ∈ V is associatedwith a subinterval I v ∈ I J . A signal f : V → C on the graph can be identifiedas a function f = (cid:88) v ∈ V f ( v ) χ I v defined on I = [0 , L ( G ) := L ( G|G J → ) := span { χ I v : v ∈ V } ⊆ L ([0 , (cid:107)·(cid:107) and inner product (cid:104)· , ·(cid:105) for L ([0 , Theorem 2
The system X ( {I j } Jj =0 ) = { ϕ } ∪ { Ψ j,I : I ∈ I j } J − j =0 defined asin (12) is a tight frame for L ( G ) . We remark that, from Corollary 2, when each I j,k has at most two children,such a system X ( {I j } Jj =0 ) is an orthonormal basis for L ( G ) (c.f. [7,8]).3.2 Digraph signal representationsNow continue to the digraph case. For a digraph ( V, W ), the underlying undi-rected graph is given by (
V, W ), where W = ( W + W (cid:62) ) /
2. A digraph is weaklyconnected if the underlying undirected graph is connected. For simplicity, werestrict ourselves in this paper to weakly connected digraph while results inthe paper can be easily extended to general digraphs. To represent signals onthe digraph G , we use the following steps to produce a pair ( G x , G y ) of twoundirected graphs:1) Extension : we define the extended graph G e = ( V, W e ) by W e = I + W ,which is the same graph as G = ( V, W ) except for a new (or enhanced)self-loop inserted at each vertex. This increases the connectivity of theundirected graphs obtained in the next step.2)
Symmetrization : define the pre-symmetrized graph G = ( V, W ) and the post-symmetrized graph ([42]) G = ( V, W ) for the digraph G e = ( V, W e )by W := W e W (cid:62) e and W := W (cid:62) e W e . Post-processing : remove the self–loops of G and G by W x := W − diag( W ) and W y := W − diag( W ). Define G x = ( V, W x ) and G y =( V, W y ).It is not difficult to show that if G is weakly connected, then G x and G y areconnected (undirected) graphs.We next use the pair ( G x , G y ) to study signals defined on G . As discussedin previous subsections, using various clustering algorithms, we can obtaincoarse-grained chains G xJ x → and G yJ y → of G x and G y , respectively for some J x , J y ∈ N . Without loss of generality, we can assume J x = J y =: J . In fact,if J x (cid:54) = J y , say J x < J y , then we simply extend the chain G xJ x → as G xJ y → byappending G x : G xJ y → := ( G xJ y , . . . , G xJ x +1 , G xJ x , . . . , G x ) , where G xj ≡ G x for all j ≥ J x .For each of the coarse-grained chains G xJ → and G yJ → , it is associated witha hierarchical partition {I xj } Jj =0 and {I yj } Jj =0 , respectively. Define B j := I xj ⊗ I yj := { I x × I y : I x ∈ I xj , I y ∈ I yj } , j = 0 , . . . , J. (17)Then, by Corollary 4, we immediately have the following result. Theorem 3
Let B j , j = 0 , . . . , J be defined as in (17) from I xj and I yj as-sociating with the coarse-grained chains G xJ → and G yJ → for graphs G x , G y ,respectively. Then, the system X ( {B j } Jj =0 ) defined as in (12) is a tight framefor L ( G x , G y ) := L ( G x |G xJ → , G y |G yJ → ) := V J = span { χ B : B ∈ B J } . For signals f : V → C defined on the digraph G , we can define the digraphsignal space L ( G ) as follows. For v ∈ V , there are I xv ∈ I xJ and I yv ∈ I yJ . B v := I xv × I yv is then a block in B J . Thus, f can be identified as a functiondefined on [0 , : f = (cid:88) v ∈ V f ( v ) χ B v , B v = I xv × I yv , v ∈ V. (18)Hence, we can define L ( G ) as L ( G ) := L ( G| ( G xJ → , G yJ → )) := span { χ B v : v ∈ V } . with the usual norm (cid:107) · (cid:107) and inner product (cid:104)· , ·(cid:105) for L ([0 , ).Since f ∈ L ( G ) is supported on ∪ v ∈ V B v , we can conclude this section bythe following result. Corollary 5
Let X ( {B j } Jj =0 ) be defined as in Theorem 3 and B v , v ∈ V beblocks defined as in (18) associated with the digraph G . Define X ( {B j } Jj =0 | G ) := { ϕ }∪{ ψ : | supp ψ ∩ B v | > for some v ∈ V, ψ ∈ Ψ j,B , B ∈ B j } J − j =0 . (19) Then X ( {B j } Jj =0 | G ) is a tight frame for L ( G ) . daptive directional Haar tight framelets for digraph signal representations 17 Proof
Note that L ( G ) ⊆ L ( G x , G y ) ⊆ L ([0 , ) . Hence, by Theorem 3, any f ∈ L ( G ) can be represented by the tight framesystem X ( {B j } Nj =0 ) as f = (cid:104) f, ϕ (cid:105) ϕ + J − (cid:88) j =0 (cid:88) B ∈B j (cid:88) ≤ (cid:96) <(cid:96) ≤ c B (cid:68) f, ψ ( (cid:96) ,(cid:96) ) j,B (cid:69) ψ ( (cid:96) ,(cid:96) ) j,B . Since f is supported on ∪ v ∈ V B v , we can discard those of ψ ( (cid:96) ,(cid:96) ) j,B whose essen-tial support is not intersecting with any B v , v ∈ V . Then, f = (cid:104) f, ϕ (cid:105) ϕ + J − (cid:88) j =0 (cid:88) B ∈B j (cid:88) ≤ (cid:96) <(cid:96) ≤ c B , | supp ψ ∩ B v |(cid:54) =0 (cid:68) f, ψ ( (cid:96) ,(cid:96) ) j,B (cid:69) ψ ( (cid:96) ,(cid:96) ) j,B . That is, the restriction X ( {B j } Jj =0 | G ) of X ( {B j } Jj =0 ) on G is a tight frame for L ( G ). This completes the proof. (cid:4) In this section, we provide some examples to illustrate results in previoussections.4.1 Example 1: a tight frame on an undirected graphLet G = ( V, W ) = G be the graph in Fig. 5 (see also G x in Fig. 6). That is, W = , (20)where the rows and columns are ordered from 1 to 6 with respect to the vertices a, b, c, d, e, f in V .Applying clustering algorithms, e.g., the NHC algorithm in [8], to the graph G , we can obtain a coarse-grained chain G → := ( G , . . . , G ) of G as in Fig. 5.Each vertex in G j − is a cluster of vertices in G j . Based on such a coarse-grained chain, we can give a hierarchical sequence {I j } j =0 and build our tightframe system X ( {I j } j =0 ) = { ϕ } ∪ { Ψ j } j =0 as follows (see Fig. 5).1) The root node ( j = 0): G ←→ I := { I , = I = [0 , } . This is associatedwith ϕ = χ [0 , .
2) At level j = 1: G ←→ I := { I , , I , } . The graph G has two nodes[ a ] G = { a, b } (degree 3) and [ c ] G = { c, d, e, f } (degree 9). Accordingto their degrees and (16), we identify the nodes [ a ] G and [ c ] G with theintervals I , ([ a ]) = (cid:20) , (cid:19) = (cid:20) , (cid:19) and I , ([ c ]) = (cid:20) , (cid:21) = (cid:20) , (cid:21) , (21)respectively. The two subintervals I , and I , are the two children of I , .Hence, by (9), Ψ = { ψ } with ψ := 3 χ I , − χ I , √ . Note that (cid:107) ψ (cid:107) = 1 and (cid:107) ψ (cid:107) = 0.3) At level j = 2: G ←→ I := { I , , I , , I , } . The graph G has threenodes [ a ] G = { a, b } (degree 3), [ c ] G = { c, d, e } (degree 8), and [ f ] G = { f } (degree 1). Similarly, we identify them with the intervals I , ([ a ]) = (cid:20) , (cid:19) , I , ([ c ]) = (cid:20) , (cid:19) , I , ([ f ]) = (cid:20) , (cid:21) , (22)respectively. Only I , and I , are split from I , . Hence, by (9), Ψ = { ψ } with ψ := χ I , − χ I , √ . Note that (cid:107) ψ (cid:107) = 1 and (cid:107) ψ (cid:107) = 0.4) At level j = 3: G ←→ I := { I , , . . . , I , } . The graph G is the underlyinggraph with 6 vertices. According to their degrees, we identify the vertices a, b, c, d, e, f with the intervals I , ( a ) = (cid:20) , (cid:19) , I , ( b ) = (cid:20) , (cid:19) , I , ( c ) = (cid:20) , (cid:19) ,I , ( d ) = (cid:20) , (cid:19) , I , ( e ) = (cid:20) , (cid:19) , I , ( f ) = (cid:20) , (cid:21) , (23)respectively. Note that I , is split to I , and I , while I , is split to I , , I , , I , . Hence, by (9), Ψ = { ψ , ψ , ψ , ψ } with ψ := √ χ I , − χ I , ) , ψ := √
32 ( χ I , − χ I , ) ,ψ := √
32 ( χ I , − χ I , ) , ψ := (cid:114)
32 ( χ I , − χ I , ) . Note that (cid:107) ψ i (cid:107) = 0 for i = 2 , . . . , X ( {I j } j =0 ) = { ϕ , ψ , . . . , ψ } is a tight frame for L ( G ) =span { χ I ,k : k = 1 , . . . , } . daptive directional Haar tight framelets for digraph signal representations 19 G = ( V, W ) be determined by W = , where the rows and columns are ordered from 1 to 6 with respect to the vertices a, b, c, d, e, f in V , see Fig. 6. After applying the symmetrization processing asdiscussed in Section 3.2, we obtain two undirected graphs G x = ( V, W x ) and G y = ( V, W y ), where G x is the same graph considered as in Example 1 (see itsadjacency matrix in (20)) and G y is determined by W y : W y = . a b c d e f G a b c d e f G x a b c d e f G y Fig. 6
Symmetrization of G (bottom) gives a pair ( G x , G y ) of undirected graphs (top andmiddle). Applying clustering algorithms, e.g., the NHC algorithm in [8], to the undi-rected graphs G x , G y , we can obtain coarse-grained chains G x → := ( G x , . . . , G x ) and G y → := ( G y , . . . , G y ) of G x and G y , respectively. The coarse-grained chain G x and its associated interval collection {I xj } j =0 are already shown in Exam-ple 1 (or see Fig. 5). Now we describe the coarse-grained chain G y → and itsassociated interval collections {I yj } j =0 (see Fig. 7). Based on the hierarchicalinterval sequences {I xj , I yj } j =0 , we can build the hierarchical block sequence {B j = I xj ⊗ I yj } j =0 defined as in (17). See Fig. 8 (top) for the illustration ofblocks and the blocks in B j are labelled with a number from 0 to 49 in Fig. 8(bottom).1) The root node ( j = 0): G y ←→ I y := { I y , = I = [0 , } . Together with I x = { [0 , } , we have B = { I = [0 , } and ϕ = χ [0 , .2) At level j = 1: G y ←→ I y := { I y , , I y , } . The graph G y has two nodes[ a ] G y = { a, b, d } (degree 9) and [ c ] G y = { c, e, f } (degree 9). According totheir degrees, we identify the nodes [ a ] G y and [ c ] G y with the intervals I y , ([ a ]) = (cid:20) , (cid:19) = (cid:20) , (cid:19) and I y , ([ c ]) = (cid:20) , (cid:21) = (cid:20) , (cid:21) , respectively. From I x = { I x , , I x , } , see (21), B = I x ⊗ I y has four sub-blocks B , . . . , B in [0 , . Hence, by (9), Ψ = { ψ ( (cid:96) ,(cid:96) )0 , [0 , : 1 ≤ (cid:96) , (cid:96) ≤ } has 6 functions.3) At level j = 2: G y ←→ I y := { I y , , I y , , I y , } . The graph G y has threenodes [ a ] G y = { a, b, d } (degree 9), [ c ] G y = { c, e } (degree 6), and [ f ] G y = { f } (degree 3). Similarly, we identify them with the intervals I y , ([ a ]) = (cid:20) , (cid:19) , I y , ([ c ]) = (cid:20) , (cid:19) , I y , ([ f ]) = (cid:20) , (cid:21) , respectively. From I x in (22), B = I x ⊗ I y = { B , . . . , B } has ninesub-blocks split from B , . . . , B : B has c B = 1 sub-block B = B ; B has c B = 2 sub-blocks B and B ; B has c B = 2 sub-blocks B and B ; B has c B = 4 sub-blocks B , B , B , and B . Hence, by (9), Ψ = { ψ ( (cid:96) ,(cid:96) )1 ,B k : 1 ≤ (cid:96) , (cid:96) ≤ c B k , k = 1 , , , } has in total (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) = 8functions.4) At level j = 3: G y ←→ I y := { I y , , . . . , I y , } . The graph G y is the under-lying graph G y with 6 vertices. According to their degrees, we identify thevertices a, b, d, c, e, f with the intervals I y , ( a ) = (cid:20) , (cid:19) , I y , ( b ) = (cid:20) , (cid:19) , I y , ( d ) = (cid:20) , (cid:19) ,I y , ( c ) = (cid:20) , (cid:19) , I y , ( e ) = (cid:20) , (cid:19) , I y , ( f ) = (cid:20) , (cid:21) , (24)respectively. From I x in (23), B = I x ⊗ I y has in total 36 blocks B , . . . , B . Hence, by (9), Ψ = { ψ ( (cid:96) ,(cid:96) )2 ,B k : 1 ≤ (cid:96) , (cid:96) ≤ c B k , k = 5 , . . . , } containing (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) = 15 + 36 + 3 + 6 +15 + 1 + 1 + 3 = 80 functions. daptive directional Haar tight framelets for digraph signal representations 21 According to Theorem 3, the system X ( {B j } j =0 ) = { ϕ } ∪ { Ψ j } j =0 is a tightframe for L ( G x , G y ). a b d c e f G y [0 , ) [ , )[ , ) [ , ) [ , ) [ , a b d c e f G y [0 , ) [ , ) [ , a b d c e f G y [0 , ) [ , a b d c e f G y [0 , Fig. 7
A coarse-grained chain of G y . G y is the underlying graph G y . G yj − is from clustering of G yj for j = 1 , ,
3. Note that G y has one nodeonly. Here each box represents a node (or cluster) in the graph, the linesrepresent edges between vertices, and the arc on a same node indicatesa self-loop. G y can be identified as the root interval I y = [0 , G y as[0 , ) ∪ [ , G y as [0 , ) ∪ [ , ) ∪ [ , G y as [0 , ) ∪ [ , ) ∪ [ , ) ∪ [ , ) ∪ [ , ) ∪ [ , Next, we focus on the tight frame X ( {B j } j =0 | G ) for L ( G ) as in Corollary 5.Note that the vertices a, b, c, d, e, f in the digraph G are with respect to blocks B a = (cid:20) , (cid:19) × (cid:20) , (cid:19) , B b = (cid:20) , (cid:19) × (cid:20) , (cid:19) , B c = (cid:20) , (cid:19) × (cid:20) , (cid:19) ,B d = (cid:20) , (cid:19) × (cid:20) , (cid:19) , B e = (cid:20) , (cid:19) × (cid:20) , (cid:19) , B f = (cid:20) , (cid:21) × (cid:20) , (cid:21) . (25)The functions in X ( {B j } j =0 | G ) are as follows.a) Supported on B : { ϕ } ∪ { ψ ( (cid:96) ,(cid:96) )0 , [0 , : 1 ≤ (cid:96) < (cid:96) ≤ } . There are 7 = 1 + 6functions with support intersecting B v in (25).b) Supported on B : ψ ( (cid:96) ,(cid:96) )1 ,B k for k = 2 , ≤ (cid:96) < (cid:96) < c B k . There are6 = 1 + 5 functions with support intersecting B v in (25).c) Supported on B : ψ ( (cid:96) ,(cid:96) )2 ,B k for k = 5 , , ≤ (cid:96) < (cid:96) < c B k . There are26 = 9 + 8 + 9 functions with support intersecting B v in (25).We see that the number of functions in X ( {B j } j =0 | G ) (39) is significantly lessthan those in X ( {B j } j =0 ) (95). We remark that the 26 functions from B couldbe further reduced in view of the non-effective blocks in B . For example, the Fig. 8
Each big block is the unit square [0 , (totally 8). Top 4 unit squares (left toright): B , B , B , B from the coarse-grained chains G x → ( I xj , j = 0 , . . . ,
3) and G y → ( I yj , j = 0 , . . . ,
3) in Example 2. Horizontal axis is x while the vertical axis is y . Eachcolored sub-block represents B = I x × I y for some I x ∈ I xj and I y ∈ I yj . Note that the top-right square contains 36 sub-blocks from I x ⊗ I y with respect to V × V in G x = ( V, W x )and G y = ( V, W y ). Bottom 4 unit squares: sub-blocks in each of B j are selected so that | B ∩ B v | (cid:54) = 0 for some v ∈ V . White blocks are those discarded ones. This is with respect tothe system X ( {B j } j =0 | G ). Each block in the unit square is labelled with a number from 0to 49. The blocks B , B , B , B , B , B are the vertices a, b, c, d, e, f in the digraph,respectively. block B has 9 sub-blocks ( ψ ( (cid:96) ,(cid:96) )2 ,B in X ( {B j } j =0 | G ). But since the block B representing the vertex d is the only effective block, one function, say ψ (29 , ,B ,is indeed enough. The total number of functions in X ( {B j } j =0 | G ) could bereduced to 20 = 7(in B ) + 6(in B ) + 7(in B ). We conclude the paper by some remarks and potential future work.1) In [7,8], the systems used to represent graph (digraph) signals are orthonor-mal. They are obtained through the tensor product approach of two 1Dorthonormal systems coming from the GramSchmidt process. The tensorproduct orthonormal system is used for the digraph signal representation.In this paper, we do not use the Gram-Schmidt process but simply relyon the selection of two sub-blocks. The resulted systems are tight framesystems and could have more directionality than the tensor product ap-proach orthonormal systems. Note that when each block has at most twochildren sub-blocks, our resulted system is also orthonormal. However, suchan orthonormal system is different to those in [7,8]. daptive directional Haar tight framelets for digraph signal representations 23
2) Our construction of the system X ( {B j } j ∈ N ) is not necessarily restrictedto compact subsets K ⊆ R d . It is possible to extend the construction ofthe system on any σ -finite measurable subsets of R d . More generally, ourconstruction could be extended to abstract measurable spaces as well ascompact Riemannian manifolds.3) Since it is Haar-type, the framelet ψ ∈ Ψ j,B has only vanishing momentof order 1: (cid:82) K ψ ( x ) dx = 0. To increase the sparse representation ability,one could consider the construction of similar tight framelet systems withhigher vanishing moments.4) The number of elements in X ( {B j } Jj =0 | G ) could be significantly reducedby discarding non-effective framelet functions as pointed out in the end ofExample 2.5) Since there is a natural nested structure of the block sequences, fast algo-rithms could be developed for framelet transforms. We provide proofs of Lemma 1 and Lemma 2 here.
Proof (Proof of Lemma 1)
We prove that A (cid:62) A = I m by showing that thecolumns of A are orthogonal. Indeed, the 2-norm of the (cid:96) -th column of A is n (cid:88) i =0 | a i,(cid:96) | = | a ,(cid:96) | + (cid:88) ≤ i (cid:96) b i = 1 , ≤ (cid:96) ≤ m. Moreover, the dot product of the (cid:96) -th column and the (cid:96) -th column of A with (cid:96) < (cid:96) is n (cid:88) i =0 a i,(cid:96) a i,(cid:96) = a ,(cid:96) a ,(cid:96) + (cid:88) ≤ i (cid:96) a ( (cid:96) ,i ) ,(cid:96) a ( (cid:96) ,i ) ,(cid:96) + (cid:88) i <(cid:96) a ( i ,(cid:96) ) ,(cid:96) a ( i ,(cid:96) ) ,(cid:96) + (cid:88) i >(cid:96) a ( (cid:96) ,i ) ,(cid:96) a ( (cid:96) ,i ) ,(cid:96) + (cid:88) i <(cid:96) ,i (cid:54) = (cid:96) a ( i ,(cid:96) ) ,(cid:96) a ( i ,(cid:96) ) ,(cid:96) = (cid:112) b (cid:96) b (cid:96) + (cid:88) i >(cid:96) a ( (cid:96) ,i ) ,(cid:96) a ( (cid:96) ,i ) ,(cid:96) = (cid:112) b (cid:96) b (cid:96) + (cid:88) i >(cid:96) (cid:112) b i ( − (cid:112) b (cid:96) δ i ,(cid:96) )= (cid:112) b (cid:96) b (cid:96) − (cid:112) b (cid:96) (cid:112) b (cid:96) = 0 . Similarly, (cid:80) ni =0 a i,(cid:96) a i,(cid:96) = 0 for (cid:96) > (cid:96) . Hence, A (cid:62) A = I m is an identitymatrix. (cid:4) Proof (Proof of Lemma 2)
To prove that Ψ B is tight, by linearity, it sufficesto show that ψ ( (cid:96) ,(cid:96) ) = (cid:88) ≤ (cid:96) (cid:48) <(cid:96) (cid:48) ≤ m (cid:68) ψ ( (cid:96) ,(cid:96) ) , ψ ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) (cid:69) ψ ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) , ≤ (cid:96) < (cid:96) ≤ m. In fact, from (8), we have ψ ( (cid:96) ,(cid:96) ) = (cid:112) b (cid:96) γ (cid:96) − (cid:112) b (cid:96) γ (cid:96) = (cid:80) m(cid:96) =1 a ( (cid:96) ,(cid:96) ) ,(cid:96) γ (cid:96) ,where ( a ( (cid:96) ,(cid:96) ) ,(cid:96) ) m(cid:96) =1 is the row of the matrix A in Lemma 1. By A (cid:62) A = I andthe orthogonality of { γ (cid:96) : 1 ≤ (cid:96) ≤ m } , we have (cid:88) ≤ (cid:96) (cid:48) <(cid:96) (cid:48) ≤ m (cid:68) ψ ( (cid:96) ,(cid:96) ) , ψ ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) (cid:69) ψ ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) = (cid:88) ≤ (cid:96) (cid:48) <(cid:96) (cid:48) ≤ m ( (cid:88) (cid:96) a ( (cid:96) ,(cid:96) ) ,(cid:96) a ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) ,(cid:96) (cid:88) ˜ (cid:96) a ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) , ˜ (cid:96) γ ˜ (cid:96) )= (cid:88) (cid:96) a ( (cid:96) ,(cid:96) ) ,(cid:96) (cid:88) ˜ (cid:96) (cid:88) ≤ (cid:96) (cid:48) <(cid:96) (cid:48) ≤ m a ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) ,(cid:96) a ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) , ˜ (cid:96) γ ˜ (cid:96) = (cid:88) (cid:96) a ( (cid:96) ,(cid:96) ) ,(cid:96) (cid:88) ˜ (cid:96) ( δ (cid:96), ˜ (cid:96) − (cid:112) b (cid:96) b ˜ (cid:96) ) γ ˜ (cid:96) = (cid:88) (cid:96) a ( (cid:96) ,(cid:96) ) ,(cid:96) ( γ (cid:96) − (cid:88) ˜ (cid:96) (cid:112) b (cid:96) b ˜ (cid:96) γ ˜ (cid:96) )=( (cid:112) b (cid:96) γ (cid:96) − (cid:88) ˜ (cid:96) (cid:112) b (cid:96) b (cid:96) b ˜ (cid:96) γ ˜ (cid:96) ) − ( (cid:112) b (cid:96) γ (cid:96) − (cid:88) ˜ (cid:96) (cid:112) b (cid:96) b (cid:96) b ˜ (cid:96) γ ˜ (cid:96) )= (cid:112) b (cid:96) γ (cid:96) − (cid:112) b (cid:96) γ (cid:96) = ψ ( (cid:96) ,(cid:96) ) , where the 3 rd equation follows from that (cid:112) b (cid:96) b ˜ (cid:96) + (cid:80) ≤ (cid:96) (cid:48) <(cid:96) (cid:48) ≤ m a ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) ,(cid:96) a ( (cid:96) (cid:48) ,(cid:96) (cid:48) ) , ˜ (cid:96) is the dot product of the (cid:96) -th column and ˜ (cid:96) -th column of A , which equals to δ (cid:96), ˜ (cid:96) . Therefore, Ψ B is a tight frame for W B . (cid:4) References
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