Dynamic Sensor Scheduling Based on Node Partitioning of Graphs

< Society logo(s) and publica- tion title will appear here. > Received XX Month, XXXX; revised XX Month, XXXX; accepted XX Month, XXXX; Date of publication XX Month, XXXX; date of current version XX Month, XXXX. Digital Object Identifier 10.1109/XXXX.2022.1234567 Dynamic Sensor Scheduling Based on Node P ar titioning of Graphs Ryosuke Ikura 1 , Juny a Hara 1 (Member , IEEE), Hiroshi Higashi 1 (Member , IEEE), and Y uichi T anaka 1 (Senior Member , IEEE) 1 Graduate School of Engineering, The University of Osaka, Osaka 565-0871, Japan Corresponding author : Ryosuk e Ikura (email: r.ikur a@sip.comm.eng.osaka-u.ac.jp). This work is suppor ted in par t by JSPS KAKENHI under Grant 23K26110 and 23K17461, and JST AdCORP under Grant JPMJKB2307. ABSTRA CT This paper proposes a dynamic sensor scheduling method for sensor networks. In sensor net- work applications, we often need multiple equally-informati ve node subsets that are acti vated sequentially to make a sensor network robust against concentrated battery consumption and sensor failures. In addition, quality of these subsets changes dynamically and thus we must adapt those changes. T o find those node subsets, we propose a graph node partitioning method based on sampling theory for graph signals. W e aim to minimize the average reconstruction error for signals obtained at all node subsets, in contrast to con ventional single subset selection. The graph node partitioning problem is formulated as a difference-of- con ve x (DC) optimization based on a subspace prior of graph signals, and is solved by the proximal DC algorithm. It guarantees conv ergence to a critical point. T o accommodate the online scenario where the signal subspace and optimal partitioning may change over time, we adaptively estimate the signal subspace from historical data and sequentially update the prior for our partitioning method. Numerical experiments on synthetic and real-world sensor network data demonstrate that the proposed method achieves lo wer av erage mean squared errors compared to alternative methods. INDEX TERMS Difference-of-con ve x optimization, graph signal processing, sampling theory , sensor network I. INTRODUCTION S ENSOR networks ha ve been used in v arious applications such as traf fic, infrastructure, and facility monitoring systems [1]–[3]. In practice, sensor networks often suffer from heavy power consumption and sensor failures during data collection and transmission [4]. T o mitigate such risks, controlling sensor activ ations ov er time is crucial for many sensor network applications. Sensor activ ation can be seen as a sensor selection prob- lem at each time instance. Its goal is to select a subset of K sensors from N candidates ( K < N ). The classical sensor placement problem often considers selecting static K sen- sors. Howe ver , this approach concentrates the sensing load on a fixed subset of nodes, which may shorten the lifespan of those sensors. As an alternative, we can group sensors into disjoint subsets and activ ate them sequentially [5]. This strat- egy is known as sensor scheduling [6]. An effecti ve sensor scheduling strategy must satisfy two essential requirements. 1) Accurate reconstruction: The whole signal can be accu- rately recovered from measurements obtained at activ e sensors at a gi ven time. 2) Load balancing: Sensing loads are balanced among all sensors ov er time to av oid concentrated energy consumption. T o satisfy these requirements, sensors must be partitioned into disjoint subsets, where each subset can accurately reconstruct the whole signal from its own measurements. W e consider this problem as graph node partitioning where sensor networks are mathematically represented as graphs. Nodes and edges in a graph correspond to sensors and their connectivity , respecti vely . Correspondingly , data collected through sensor networks can be modeled as graph signals — discrete signals with their domain as nodes [7]–[9]. Graph node partitioning relates to two other established approaches: sampling set selection and node clustering. While all three approaches in volve selecting or grouping node subsets, their objecti ves and applications differ fun- This work is licensed under a Creative Commons Attr ib ution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ VOLUME , 1 Author et al. : T ABLE 1. Comparison of graph node par titioning, sampling set selection, and clustering methods. Method Primary Objective Main Application Graph Node Partitioning Dividing nodes into multiple equally-informativ e node subsets. Sensor scheduling Sampling Set Selection Selecting one subset of nodes that accurately reconstruct the whole signal. Efficient sensing Node Clustering Grouping nodes sharing similar properties. Network analysis damentally . T able 1 summarizes the key distinctions among them. Sampling set selection of graph signals is widely studied in graph signal processing [10], [11]. In this approach, a designated number of nodes is selected so that the whole signal can be accurately reconstructed from the sampled measurements [12]. Howe ver , it only selects a single subset, and thus, for sensor scheduling, the activ ation load concen- trates on the selected node subset. Node clustering partitions nodes by assigning similar nodes to a group based on criteria such as cut minimization or submodularity maximization [13], [14]. While grouping similar nodes is beneficial for applications such as commu- nity detection, it is unsuitable for sensor scheduling. This is because nodes in the same cluster may have similar mea- surements and thus it is challenging to reconstruct signals in a dif ferent cluster . In summary , graph node partitioning is the strategy that satisfies the both requirements of high reconstruction accu- racy and load balancing for sensor scheduling. Existing graph node partitioning methods [5], [15] hav e the follo wing limitations. They typically rank nodes based on a predefined metric and sequentially assign them to subsets according to their ranking. Howe ver , these strategies are heuristic and lack theoretical guarantees. They also often rely on restrictiv e assumptions about the signal model, such as exact bandlimitedness. Unless the assumptions are satisfied, their performance of reconstruction is not theoretically guar - anteed. More importantly , these methods focus exclusi vely on static graph node partitioning: They may not be suitable for direct application to dynamic sensor scheduling , where signal statistics change ov er time. In this paper , we propose a graph node partitioning method that overcomes the limitations of existing approaches. The core idea is to reformulate graph node partitioning as a multiple sampling subsets selection [5], which naturally extends from a single sampling subset selection, to identify multiple disjoint node subsets, each capable of accurate signal reconstruction. In addition, our formulation is free from the bandlimitedness assumption by using generalized sampling of graph signals based on a subspace prior [12], [16], [17]. The single subset selection [12] minimizes the reconstruc- tion error for one subset. W e extend this to graph node partitioning by minimizing the average reconstruction error across all subsets. This problem is encoded as a difference- of-con ve x (DC) optimization whose objectiv e function is the difference of two or more con vex functions [18]. This is solved by the proximal DC algorithm (PDCA) [19], [20], which guarantees con ver gence to a critical point. T o address online scenarios where the signal subspace is unkno wn and time-varying, we also propose a weighted dictionary learning scheme as an extension of the existing dictionary learning for graph signals [21]. The e xisting method requires pre-training with the availability on the whole data to estimate an initial signal subspace. Ho wev er , it is impractical for some applications. In contrast, our extension enables robust subspace estimation without relying on the pre-training. Experiments using both synthetic and real-world sensor network data demonstrate that the proposed method outper- forms existing partitioning methods in terms of the mean squared error (MSE) of the reconstructed signals. The remainder of this paper is org anized as follows. Section II revie ws existing graph node partitioning methods that are related to this work. In Section III, we introduce mathematical preliminaries for our method. W e propose our graph node partitioning method in Section IV. Signal recon- struction experiments for synthetic and real-world signals are presented in Section V. Section VI concludes the paper . Notation: Bold lowercase and uppercase letters denote vectors and matrices, respectiv ely . W e denote the i th col- umn and ( i, j ) element of a matrix X by [ X ] i and [ X ] ij , respectiv ely . Similarly , [ x ] i is an i th element of vector x . The operator diag( v ) denotes the diagonal matrix with the elements of vector v on its diagonal, whereas Diag ( V ) extracts the diagonal elements of a square matrix V as a vector . The ℓ 2 -norm and Frobenius norm are denoted by ∥ · ∥ and ∥ · ∥ F , respectively . Calligraphic letters represent sets of indices; for a set B , its complement is denoted by B c . The superscripts ⊤ and † denote the transpose and Moore-Penrose pseudo-in verse, respectiv ely . The operator ∇ represents the gradient. In addition, ⊙ represents the element-wise product. II. RELA TED W ORK In this section, we briefly revie w two e xisting graph node partitioning methods for sensor scheduling: Selection on rele vance (SRel) and that based on minimum Frobenius norm (SFrob) [5]. Both SRel and SFrob share a common two-step strategy: (1) ranking all nodes based on a predefined importance criterion, and then (2) sequentially assigning these ranked 2 VOLUME , < Society logo(s) and publication title will appear here. > nodes to partitioned subsets in a c yclic manner . The primary distinction lies in the definition of the importance of nodes. W e describe them below . SRel: It ranks nodes based on the topology of the graph. This approach first performs node clustering by maxi- mizing modularity [22], [23]. Then, within each cluster , nodes are sorted based on the eigen vector centrality and sorted nodes are assigned to sampling subsets according to the rank. Ho wev er , since this approach relies solely on structural features and disregards the characteristics of signals, its partitioning may be suboptimal. SFrob: In contrast, it considers signal properties in the ranking algorithm. This approach extends the sampling theory for bandlimited graph signals [15] to graph node partitioning. Specifically , it ranks nodes according to their indi vidual contribution to reducing the recon- struction error and assigns them to partitioned subsets follo wing the rank. While this approach incorporates signal properties, its applicability is limited because of the bandlimited assumption. In summary , e xisting approaches either ignore signal char - acteristics or rely on restrictiv e assumptions on signal sub- spaces. Additionally , both are designed for static settings and do not adapt when signal characteristics change over time. Thus, they are unsuitable for many real-world applications. III. PRELIMINARIES In this section, we introduce a sampling frame work for graph signals. First, we present the graph signal sampling theory under a subspace prior . Second, we describe the sampling set selection criterion based on the sampling framew ork . W e first introduce the graph basics. W e consider a weighted undirected graph G = ( V , E ) , where V ( |V | = N ) and E denote the set of nodes and edges, respectiv ely . The adjacency matrix of G is denoted by W , where its ( i, j ) element is the weight of the edge between the i th and j th nodes if they are connected, and 0 otherwise. The degree matrix D is defined as D = diag ( d 0 , d 1 , · · · , d N − 1 ) , where d n = P m W nm . W e use graph Laplacian L = D − W as a graph variation operator [9]. The graph signal x ∈ R N is defined as a mapping from the node set to the set of real numbers, i.e., x : V − → R . The graph Fourier transform (GFT) of x is defined as ˆ x = U ⊤ x where the GFT matrix U is obtained by the eigen-decomposition of the graph Laplacian L = UΛU ⊤ with the eigenv alue matrix Λ = diag ( λ 0 , λ 1 , . . . λ N − 1 ) . W e refer to λ i as the i th graph fr equency . A. Graph Signal Sampling Theory Under Subspace Prior Subspace prior is a fundamental model in graph signal sampling theory , in which graph signals are assumed to lie in a known subspace [12]. This includes the well-kno wn bandlimited model as a special case. A graph signal subspace is defined as follows [12]: A : = { x | x = A d for d ∈ R M } , (1) where A ∈ R N × M , M ≤ N , is a generation transform depending on a graph, and d ∈ R M is a vector composed of expansion coefficients. Here, we define a node domain sampling operator as follows: Definition 1 (Node domain sampling [12]): Let I MV ∈ { 0 , 1 } K × N be the submatrix of the identity matrix inde xed by M ⊂ V ( |M| = K ) and V . The sampling operator is defined as follows: S ⊤ : = I MV G , (2) wher e G ∈ R N × N is an arbitrary linear graph filter . Thus, a sampled graph signal is given by y = S ⊤ x . In this paper, we consider the follo wing noisy measure- ment model y = S ⊤ x + η , (3) where η ∼ N ( 0 , σ 2 I ) is additive white Gaussian noise with standard deviation σ . Under the subspace prior in (1), the best possible recovery is obtained by solving the following minimax problem [24]. ˜ x = argmin ˜ x ∈A max S ⊤ x = y ∥ ˜ x − x ∥ 2 = A ( S ⊤ A ) † y . (4) Under the noiseless case, perfect recovery , i.e., x = ˜ x is achiev ed when S ⊤ A is inv ertible. The condition is so-called dir ect sum condition [12]. B. Sampling Set Selection According to (4), the e xpected v alue of MSE is upper bounded by the follo wing relationship [24]: E [ ∥ ˜ x − x ∥ 2 ] = tr ( xx ⊤ − E xx ⊤ E ⊤ ) + tr ( A ( S ⊤ A ) † Γ η ( A ⊤ S ) † A ⊤ ) ≤ tr ( Γ η ) tr ( A ⊤ A ) tr (( S ⊤ AA ⊤ S ) − 1 ) , (5) where E = A ( S ⊤ A ) † S ⊤ and Γ η = E [ η η ⊤ ] . Hereafter, we suppose that S ⊤ AA ⊤ S is in vertible for simplicity 1 . T o minimize (5), the optimal sampling set M can be obtained by solving the follo wing problem. M ∗ = argmin M⊂V tr (( S ⊤ AA ⊤ S ) − 1 ) . (6) This problem is a combinatorial optimization and NP-hard. Therefore, existing sampling set selection methods typically perform a greedy selection [25], [26], which yields subopti- mal solutions in general. Note that ev en if M ∗ in (6) is the global optimum, the remaining subset M c = V \M ∗ is generally not an equally- informativ e subset compared to M ∗ . Therefore, sampling set selection cannot be applied to graph node partitioning straightforwardly . 1 The same formulation can be easily derived even if not invertible. VOLUME , 3 Author et al. : Signal subspace Graph node set Graph signal Reconstructed signal Reconstructed signal Node partitioning Sampling operator Sampling operator Reconstruction filter Reconstruction filter Node partitioning Signal sampling Signal reconstruction FIGURE 1. Overview of the proposed method. For simplicity , the selection with two subsets is illustrated. IV . GRAPH NODE P ARTITIONING AND ONLINE SENSOR SCHEDULING In this section, we first present the static version of the proposed graph node partitioning based on graph signal sampling theory . Next, we extend it to the online sensor scheduling problem where the optimal partitioning can vary ov er time. Finally , we formulate a dictionary learning prob- lem for estimating a time-varying signal subspace from the observed data. A. Static Graph Node P artitioning Fig. 1 illustrates the overvie w of the proposed static graph node partitioning. For simplicity , we describe the case of bipartitioning with noiseless observations. It consists of three stages: Graph node partitioning, signal sampling, and signal reconstruction. First, with the give n signal subspace A in (1), the node set V is divided into subsets M k . Based on this partitioning, the sampling operators S k are determined to sample the graph signal x . Finally , the full graph signal is reconstructed from the sampled measurements S ⊤ k x in (3) using the reconstruction filter A ( S ⊤ k A ) † in (4). Our primary contribution is the design of ef ficient graph node partitioning for both static and time-varying signal subspaces. Accordingly , we employ existing methods [12], [16] for the signal sampling and reconstruction. W e describe the graph node partitioning in the following. 1) Problem Formulation Here, we assume that the signal subspace A is given and it is specified by generation transform A . W e focus on a bipartitioning scenario M 1 , M 2 ⊂ V where all subsets are non-ov erlapping and the number of nodes in each subset is equal, i.e., |M 1 | = |M 2 | = N / 2 . 2 Note that, it can be applied to the 2 k partitioning by hierarchically cascading the bipartitioning to the resulting subsets. The sampling operators for M 1 and M 2 are defined as S ⊤ 1 = I M 1 V and S ⊤ 2 = I M 2 V , respectiv ely , where we set G = I in (2) for simplicity . According to (3), two sampled signals are expressed as y 1 = S ⊤ 1 x + η 1 and 2 For odd N , one subset has ⌈ N/ 2 ⌉ nodes and the other one has ⌊ N / 2 ⌋ nodes. y 2 = S ⊤ 2 x + η 2 . Similar to (4), minimax recovery solutions of each observation are obtained as ˜ x 1 = A ( S ⊤ 1 A ) † y 1 and ˜ x 2 = A ( S ⊤ 2 A ) † y 2 . Our purpose is to minimize the avera ge reconstruction errors across all subsets. Based on (5), the a verage recon- struction error is upper bounded as 1 2  E [ ∥ ˜ x 1 − x ∥ 2 + E [ ∥ ˜ x 2 − x ∥ 2 ]  ≤ C  tr (( S ⊤ 1 AA ⊤ S 1 ) − 1 ) + tr (( S ⊤ 2 AA ⊤ S 2 ) − 1 )  , (7) where C = 1 2 tr ( Γ η ) tr ( A ⊤ A ) is a constant. Based on (6) and (7), the following optimization problem can be considered for graph node partitioning: ( M ∗ 1 , M ∗ 2 ) = argmin M 1 , M 2 ⊂V tr (( S ⊤ 1 AA ⊤ S 1 ) − 1 ) + tr (( S ⊤ 2 AA ⊤ S 2 ) − 1 ) s.t. M 1 ∩ M 2 = ∅ , |M 1 | = |M 2 | = N 2 . (8) Similar to the single subset selection, it is combinatorial and NP-hard. Furthermore, it in volv es matrix in version that requires huge computational burden. For tractability , we first approximate the calculation of the matrix inv erses with the second-order Neumann series approximation [27]. The details are shown in Appendix. As a result, (8) is approximated as ( M ∗ 1 , M ∗ 2 ) = argmin M 1 , M 2 ⊂V tr (( S ⊤ 1 AA ⊤ S 1 ) 2 + ( S ⊤ 2 AA ⊤ S 2 ) 2 ) s.t. M 1 ∩ M 2 = ∅ , |M 1 | = |M 2 | = N 2 . (9) Let m k ∈ { 0 , 1 } N ( k ∈ { 1 , 2 } ) be the indicator vectors, whose i th element [ m k ] i is defined as [ m k ] i = ( 1 if i ∈ M k , 0 otherwise. (10) Then, we further re write (9) with the c yclic property of trace and m k as follo ws: m ∗ 1 = argmin m 1 ∈{ 0 , 1 } N tr (( A ⊤ diag ( m 1 ) A ) 2 ) + tr (( A ⊤ diag ( 1 − m 1 ) A ) 2 ) s.t. m ⊤ 1 ( 1 − m 1 ) = 0 , 1 ⊤ m 1 = N 2 , (11) 4 VOLUME , < Society logo(s) and publication title will appear here. > where we use the relationship m 1 + m 2 = 1 . Note that (11) is still combinatorial due to the binary m . Therefore, we introduce a con vex relaxation of (11) by considering a continuous m relaxed ∈ [0 , 1] N instead of m . Finally , the problem to be solved is represented as follows. m ∗ relaxed = argmin m relaxed ∈ [0 , 1] N tr (( A ⊤ diag ( m relaxed ) A ) 2 ) + tr (( A ⊤ diag ( 1 − m relaxed ) A ) 2 ) s.t. m ⊤ relaxed ( 1 − m relaxed ) = 0 , 1 ⊤ m relaxed = N 2 . (12) Although the relaxed variables reside in the con vex set [0 , 1] N , (12) is non-conv ex due to the constraint m ⊤ relaxed ( 1 − m relaxed ) = 0 . Howe ver , as this constraint function is a DC function, the solution presented below leads to a critical point. 2) Solver W e reformulate (12) into the applicable form to PDCA [20] as follo ws: m ∗ relaxed = argmin m relaxed f ( m relaxed ) + g ( m relaxed ) − h ( m relaxed ) , (13) where f ( m relaxed ) and h ( m relaxed ) are differentiable con vex, and g ( m relaxed ) is non-differentiable b ut con ve x. In addition, g is proximable, i.e., whose proximity operator , which is defined as prox γ g ( m relaxed ) : = argmin y g ( y )+ 1 2 γ ∥ m relaxed − y ∥ 2 2 , (14) can be solved efficiently with high precision [28]. W e define functions in (13) as follows. f ( m relaxed ) = tr (( A ⊤ diag ( m relaxed ) A ) 2 ) + tr (( A ⊤ diag ( 1 − m relaxed ) A ) 2 ) , g ( m relaxed ) = ι C card ( m relaxed ) + ι C box ( m relaxed ) , h ( m relaxed ) = β ( 1 ⊤ ( m relaxed ⊙ m relaxed ) − 1 ⊤ m relaxed ) , (15) where β is the parameter . In addition, the indicator function is defined as ι C ( m relaxed ) = ( 0 if m relaxed ∈ C + ∞ otherwise , (16) where C is a conv ex set. W e can con vert the hard constraint 1 ⊤ m relaxed = N 2 in (12) into the objectiv e function in (15) by C card = { m relaxed | 1 ⊤ m relaxed = N 2 } with (16). W e also constrain m relaxed ∈ [0 , 1] N as C box = { m relaxed | m relaxed ∈ [0 , 1] N } . W ith an appropriate choice of β , (13) becomes identical to (12) [29]. Algorithm 1 sho ws the detailed steps for solving (13). W e use the follo wing operators in the algorithm. ∇ f ( m relaxed ) = 2 Diag ( AA ⊤ (2 diag ( m relaxed ) − I ) AA ⊤ ) , ∇ h ( m relaxed ) = β (2 m relaxed − 1 ) . (17) Algorithm 1 Static graph node partitioning Require: m (0) relaxed ∈ [0 , 1] N , Lipschitz constant L > 0 , signal subspace A . 1: Set step size γ ← 1 /L . 2: k ← 0 3: while conv ergence criterion is not met do 4: Compute a gradient u ( k ) ← ∇ h ( m ( k ) relaxed ) . 5: Update v ariable: m ( k +1) relaxed ← prox γ g ( m ( k ) relaxed − γ ( ∇ f ( m ( k ) relaxed ) − u ( k ) )) 6: k ← k + 1 7: end while 8: Binarize the continuous vector by thresholding: [ m ] i = ( 1 if [ m ( k +1) relaxed ] i > 1 2 0 otherwise. (18) 9: Output: m ∈ { 0 , 1 } N The computation of prox γ g in (14) is a con vex optimiza- tion since g ( m relaxed ) consists of multiple conv ex functions. Therefore, it can be solved via an existing con vex solver . Specifically , we use alternating direction method of multipli- ers (ADMM) [30]. Since the resulting m relaxed ∈ [0 , 1] N is a real-valued vector , we binarize it by thresholding at the end of the algorithm to obtain the binary vector m ∈ { 0 , 1 } N . B. Online Graph Node P artitioning W e extend the static graph node partitioning to the online scenario. Since the signal subspace may be time-v arying in this setting, the optimal graph node partitioning also varies at each time instance. W e use the subscript t to specify the time instance of variables. For example, the signal subspace at t is defined by A t . W e describe the case of M partitioning ( M = 2 k , k = 1 , 2 , . . . ) in the following. Here, we assume that { A t } is giv en. In our online graph node partitioning, the following process is performed at e very M time instances. First, V is partitioned into M disjoint subsets based on the current signal subspace A t . Specifically , by recursively ex- ecuting the Algorithm 1, we obtain partitioned node subsets {M i } M i =1 satisfying M 1 ⊕ . . . ⊕ M M = V . Second, during the subsequent M time instances, we perform signal sampling on the nodes associated with the specific subset selected at each time step, i.e., y t = S ⊤ t x t + η t . (19) By defining the index l = (( t − 1) ( mod M ) + 1) , the sampling operator S t corresponds to the subset M l . In addition, x t represents the original signal at t . Finally , as shown in (4), the whole signal is reconstructed using the corresponding signal subspaces A t , i.e., VOLUME , 5 Author et al. : ˜ x t = A t ( S ⊤ t A t ) † y t . (20) C. Online Dictionary Learning In the previous subsections, the signal subspace A or { A t } is assumed to be giv en. Howe ver , it is not explicitly provided in general. Therefore, we would like to adaptiv ely estimate { A t } from the previously reconstructed signals from t − D to t − 1 , which is written as ˜ X t − 1 = [ ˜ x t − D , . . . , ˜ x t − 1 ] . This problem can be referred to as subspace tracking. Since ˜ x τ in (20) contains both measurements at reliable nodes and potentially inaccurate reconstructed signals, we propose to differentiate between them by introducing a confidence matrix that weights the data fidelity term based on the sampling history . Let D t − 1 = [ d t − D , . . . , d t − 1 ] ∈ R N × D be the collection of estimated e xpansion coef ficients. Under the subspace prior in (1), we can e xpress ˜ X t − 1 ≈ A t − 1 D t − 1 . W e formulate the follo wing subspace tracking problem as a dictionary learning problem: ( A ∗ t , D ∗ t ) = argmin A t , D t D − 1 X i =0 ∥ W i ([ ˜ X t − 1 ] i − A t [ D t ] i ) ∥ 2 F s.t. ∥ D t ∥ 1 ≤ K , (21) where the confidence matrix W i is defined as W i = diag ( w i ) . (22) The vector w i denotes a confidence vector whose i th el- ement is the confidence weight of the i th node. It could be determined by the sampling pattern at the corresponding time instance, such as high confidence at sampled signals and low confidence at unsampled ones. In (21), we impose a sparsity constraint on D t which is a similar setting to the well-studied dictionary learning problems [21], [31], [32]. The fundamental distinction between the proposed formu- lation and the existing online dictionary learning method for graph signals [21] lies in the introduction of the weighting matrix W i . The method in [21] assumes that A t changes smoothly over time and includes the regularization term ∥ A t − A t − 1 ∥ 2 F to ensure a stable learning. In contrast, our formulation utilizes W i to ensure that the subspace is learned primarily from measurements at reliable nodes. This mechanism prev ents error propagation from inaccurate reconstructions and stabilizes the learning process, thereby allo wing us to omit the temporal regularization term. W e decompose the optimization problem in (21) into two independent subproblems with respect to A t and D t , and solve them alternately , similar to the approach described in [21]. First, we optimize (21) with respect to D t by fixing A t . The problem is formulated as: D ∗ t = argmin D t ψ ( D t ) + ι C sparse ( D t ) , (23) where ψ ( D t ) = P D − 1 i =0 ∥ W i ([ ˜ X t − 1 ] i − A t [ D t ] i ) ∥ 2 F and ι C sparse is an indicator function in (16). Here, C sparse is defined as: C sparse = { D | ∥ D ∥ 1 ≤ K } . (24) Note that (23) is a con ve x optimization where the first term is differentiable and the second term is proximable. W e use proximal gradient descent [33] to solve it. The optimal solution is obtained by iterativ ely performing the follo wing update until con ver gence: D n +1 t = prox ι C sparse ( D n t − γ ∇ ψ ( D n t )) , (25) where the superscript n + 1 is the iteration number and γ is the step size. The gradient ∇ ψ ( D t ) and the prox- imity operator pro x ι C sparse are computed as follo ws. W e calculate ∇ ψ ( D n t ) column-by-column. For the i th column ( i = 0 , . . . , D − 1 ), the gradient is giv en by ∇ [ D t ] i ψ ( D t ) = − 2 A ⊤ t W ⊤ i W i ([ ˜ X t − 1 ] i − A t [ D t ] i ) . (26) Follo wed by Moreau’ s decomposition [34], the proximity operator is computed via that of ℓ ∞ -norm, i.e., prox ι C sparse ( Y ) = Y − prox K ∥·∥ ∞ ( Y ) . (27) In addition, the second term in (27) is computed element- wise as [ prox K ∥·∥ ∞ ( Y )] ij = sign ([ Y ] ij ) min {| [ Y ] ij | , ζ i } , (28) where ζ i is the unique solution to the follo wing equation D − 1 X j =0 max { 0 , | [ Y ] ij | − ζ i } = K N . (29) Second, we update A t by solving (21) with fix ed D t , i.e., A ∗ t = argmin A t D − 1 X i =0 ∥ W i ([ ˜ X t − 1 ] i − A t [ D t ] i ) ∥ 2 F . (30) W e solve (30) by using gradient descent method [35]. The gradient of objecti ve function in (30) is given by ∇ A t D − 1 X i =0 ∥ W i ([ ˜ X t − 1 ] i − A t [ D t ] i ) ∥ 2 F = 2 D − 1 X i =0 W ⊤ i W i ( A t [ D t ] i − [ ˜ X t − 1 ] i [ D t ] ⊤ i ) . (31) W e repeat these two steps until con vergence. Algorithm 2 summarizes its algorithm. W e can simply integrate the signal subspace learning steps with the proposed graph node partitioning. After a signal reconstruction in (20), the reconstructed signal ˜ x t is appended to the buf fer of the reconstructed signals, i.e., ˜ X t = ( [ ˜ X t − 1 , ˜ x t ] if t ≤ D , [[ ˜ X t − 1 ] 2 , . . . , [ ˜ X t − 1 ] D , ˜ x t ] otherwise. (32) Then, we perform the dictionary learning (21) to estimate the signal subspace for the next time instance. The overall algorithm of the online sensor scheduling is summarized as Algorithm 3. 6 VOLUME , < Society logo(s) and publication title will appear here. > Algorithm 2 Dictionary learning with confidence matrix Require: ˜ X t − 1 , { W i } D − 1 i =0 , K , γ D , γ A 1: Initialize A t ← A t − 1 and D t ← 11 ⊤ . 2: while conv ergence criterion is not met do 3: Step 1: Update coefficients D t 4: n ← 0 , D ( n ) t ← D t . 5: while con ver gence criterion for D not met do 6: D ( n +1) t ← prox γ D ι C sparse  D ( n ) t − γ D ∇ ψ ( D ( n ) t )  7: n ← n + 1 8: end while 9: D t ← D ( n ) t . 10: Step 2: Update Dictionary A t 11: m ← 0 , A ( m ) t ← A t . 12: while con ver gence criterion for A not met do 13: G A ← 2 D − 1 P i =0 W ⊤ i W i ( A ( m ) t [ D t ] i − [ ˜ X t − 1 ] i [ D t ] ⊤ i ) 14: A ( m +1) t ← A ( m ) t − γ A G A 15: m ← m + 1 16: end while 17: A t ← A ( m ) t . 18: end while 19: Output: A t Algorithm 3 Online sensor scheduling Require: K > 0 , M = 2 k , ( k = 1 , 2 , . . . ) , A 0 = I 1: t ← 0 . 2: while the sensor system is activ ated do 3: Step 1: Graph Node P artitioning 4: Compute graph node partitioning {M k } M k =1 by solv- ing (12). 5: Step 2: Sequential Sampling & Reconstruction 6: f or k = 1 , . . . , M do 7: Acquire graph signals from the subset M k . 8: Reconstruct the full signal ˜ x t + k via (20). 9: Update the historical data buf fer to obtain ˜ X t + k − 1 according to (32). 10: Perform dictionary learning (21) using ˜ X t + k − 1 and update the signal subspace A t + k . 11: t ← t + 1 12: end for 13: end while In our implementation, we set the initial signal subspace as A 0 = I . On the other hand, man y alternati ve methods require it to be initialized more carefully [21], [31], [32]. T ypically , A 0 is determined via the singular value decomposition of a gi ven data. This difference stems from the fact that (21) can learn subspace stably o ver time with an y choice of A 0 , which is beneficial for online sensor scheduling under insufficient initial observ ations (i.e., cold start). V . EXPERIMENTS W e v alidate the effecti veness of the proposed method via re- construction experiments for synthetic and real-world graph signals. W e conduct three experiments: Static partitioning on synthetic graph signals, online partitioning on synthetic graph signals, and online partitioning on real-world data. A. Static P ar titioning on Synthetic Graph Signals First, in order to validate the effecti veness of our static graph node partitioning, we compare its performance to that of alternativ e graph node partitioning methods. For this experiment, we use Algorithm 1. 1) Graph and Signal Synthesis W e generate a random sensor graph using the following procedure: First, 256 nodes are randomly distributed in the 2-D space [0 , 1] × [0 , 1] . For each node, we connect edges to its k nearest neighbors and assign edge weights exp( − d 2 ) where d is the Euclidean distance between two nodes. T o simulate realistic sensor networks [36], k is randomly chosen between two to eight for each node. For graph signals, we consider two full-band signals: 1) Heat-diffusion (HD) graph signal: Its spectrum decays slowly as the graph frequency λ increases. According to (1), this signal is expressed as x heat = A heat d , (33) where A heat = U b H ( Λ ) U ⊤ with b H ( Λ ) = exp ( − α Λ ) . W e set α = 10 and d ∼ N ( 1 , I ) . 2) Piecewise smooth (PWS) graph signal: It forms clustered signals such that the signal has different av erage v alues in dif ferent clusters and also has smooth variations within the cluster . The PWS signal is there- fore represented as: x PWS = A PWS 1 d 1 + A PWS 2 d 2 , (34) where A PWS 1 = [ u 1 , . . . , u 32 ] and A PWS 2 = [ 1 T 1 , 1 T 2 , 1 T 3 ] ; The vector u i denotes the i th eigenv ec- tor of the graph Laplacian L , and 1 T is the indicator vector of cluster T , i.e., [ 1 T ] i = 1 if i ∈ T and [ 1 T ] i = 0 otherwise. W e obtain clusters {T i } i =1 , 2 , 3 by performing graph spectral clustering [13]. W e set d 1 ∼ N ( 1 , I ) and d 2 ∼ N ( 0 , 5 I ) . Here, the signal subspace is expressed as A PWS = [ A PWS 1 , A PWS 2 ] . 2) Setup W e partition the nodes into four subsets by cascading the proposed node bipartitioning twice. The cardinalities of all subsets are therefore |M i | = N/ 4 = 64 , i = 1 , . . . , 4 . Both noisy and noiseless cases are considered for sam- pling. For the noisy case, additiv e white Gaussian noise η ∼ N ( 0 , 10 − 3 I ) is added to the signals. In our algorithm, we experimentally set L in Algorithm 1 and β in (15) as ( L, β ) = (10 3 , 1) . The proposed method VOLUME , 7 Author et al. : T ABLE 2. Average reconstruction MSEs in decibels. Bold numbers denote the lowest MSE in each row . The columns labeled SS represent reconstruction with subspace prior in (4) . The columns labeled BL is the bandlimited reconstruction, where the numbers are the cutoff frequencies. Methods Prop. SRel SFrob SS SS BL SS BL B 10 32 100 256 10 32 100 256 HD Clean -26.2 -22.7 -17.1 -13.6 9.5 -1.3 -24.3 -17.1 -14.2 1.1 -1.3 Noisy -25.2 -22.3 -17.1 -12.6 32.6 -1.3 -23.6 -17.1 -13.7 28.0 -1.3 PWS Clean -297.2 -288.2 -14.8 3.7 -5.4 -1.2 -295.2 -15.2 -3.2 -5.3 -1.2 Noisy -33.4 -20.4 -14.9 14.7 3.9 -1.2 -31.9 -15.2 -3.3 -3.8 -1.2 is compared with two existing graph partitioning methods: SRel and SFrob [5], which are described in Section II. For all partitioning methods, the sampled signals are reconstructed according to (4) with given subspace A heat or A PWS to compare the sampling set qualities. In addition, for SRel and SFrob, we also perform their original reconstruction, i.e., those with the bandlimited as- sumption. Specifically , we use A = [ u 1 , . . . , u B ] for both HD and PWS graph signals. W e experimentally set four bandwidths B ∈ { 10 , 32 , 100 , 256 } . For all methods, 30 independent runs are performed and the av erage MSEs are compared. 3) Results The experimental results are summarized in T able 2. The proposed method exhibits the lowest MSEs for all cases. Even when all methods utilize the giv en subspace for signal reconstruction, the proposed method shows 2 – 5 dB smaller MSEs than those of SRel and SFrob . The gain becomes more significant when compared with the original recon- struction methods for SRel and SFrob with the bandlimited assumption. This is likely because only the proposed method incorporates the signal subspace into graph node partitioning, thereby enabling the selection of sampling subsets that are effecti ve to the signal model. W e also visualize the absolute errors between the original and the reconstructed signals in Fig. 2. It can be observed that the proposed method presents small reconstruction errors consistently regardless of the subsets. In contrast, SRel sometimes fails to reconstruct the signal as seen in Subset 2, presumably because it does not sample nodes in the upper- left area. Additionally , SFrob has large reconstruction errors in local regions without selected sensors, as in Subset 4. B. Online P ar titioning on Synthetic Graph Signals Second, we inv estigate the effecti veness of the subspace tracking on the performance of sensor scheduling. For this experiment, we use Algorithm 2 for sensor scheduling and Algorithm 1 for graph node partitioning. 1) Graph and Signal Synthesis In this experiment, we generate the same graph used in Section V-A. As graph signals, we generate time-v arying piece wise smooth graph signals based on (34). In this configuration, A PWS 1 and A PWS 2 in (34) are time-varying and are defined as A PWS 1 ( t ) = U exp ( − α ( t ) Λ ) U ⊤ , A PWS 2 ( t ) = [ 1 T 1 ( t ) , 1 T 2 ( t ) , 1 T 3 ( t ) ] , (35) where t denotes a time instance and α ( t ) = 2 + 1 8 t in which signals become smoother as time passes. W e utilize the same d 1 and d 2 as in the pre vious experiment. The clusters T 1 ( t ) , T 2 ( t ) , and T 3 ( t ) are initialized by using spectral clustering [13] at t = 0 . Subsequently , the cluster memberships of nodes near the boundaries are switched: Nodes located within two hops of the boundaries at t = 0 are randomly reassigned to one of the other two clusters at each time instance. Here, A t is e xpressed as A t = [ A PWS 1 ( t ) , A PWS 2 ( t )] . W e generate the graph signals for the duration of 64 . 2) Setup The set of nodes is partitioned into 16 subsets with equal size, i.e., |M i | = 16 , i = 1 , . . . , 16 . The additi ve white Gaussian noise η ∼ N ( 0 , 10 − 3 I ) is added to the signals. The hyperparameters are set as ( L, β ) = (10 3 , 1) . W e utilize the minimax recov ery (4) for signal reconstruction. Since there hav e been no online graph node partitioning methods, we use the following two methods as benchmarks: 1) Method 1 (Static partitioning + static signal subspace): In this method, graph node partitioning is fixed, i.e., the initial 16 partitions at t = 0 is used for all t . Furthermore, a signal subspace is also fixed to that defined by A 0 = [ A PWS 1 (0) , A PWS 2 (0)] for signal reconstruction. 2) Method 2 (Static partitioning + dynamic signal sub- space): In this method, graph node partitioning is also fixed like Method 1. F or signal reconstruction, the current signal subspace A t = [ A PWS 1 ( t ) , A PWS 2 ( t )] is utilized at each time instance. For all methods, 10 independent runs are performed and the av erage MSEs are compared. 8 VOLUME , < Society logo(s) and publication title will appear here. > FIGURE 2. Visualization of the absolute errors between original and reconstructed PWS graph signals. W e show the noisy case with reconstruction based on the subspace prior . From top to bottom: The proposed method, SRel, and SFrob. The leftmost column is the original signals (same for all methods). The other columns show the reconstructed signals from sampled subsets. The selected nodes are highlighted by red circ les. 0 10 20 30 40 50 60 T ime Step 10 8 6 4 2 0 MSE (dB) P r oposed Method 1: Static partitioning + static SS Method 2: Static partitioning + dynamic SS P r oposed Mean: -7.86 dB Method 1: Static partitioning + static SS Mean: -0.35 dB Method 2: Static partitioning + dynamic SS Mean: -7.35 dB FIGURE 3. MSE of reconstructed signals [dB]. The average MSE of each method is plotted as a horizontal dashed line . SS means signal subspace . V ertical dashed lines indicate the time instances when the proposed method updates the par titioning. 3) Results The results are shown in Fig. 3. As clearly observed, the proposed method and Method 2 are significantly better than Method 1, whose MSE immediately rises and stays high. This demonstrates that the proposed method successfully tracks the time-varying signal subspace, which is crucial for maintaining high reconstruction accuracy . Furthermore, the proposed method consistently outperforms Method 2. This indicates that not only considering the time-v arying signal subspace but also employing adaptive partitioning is important for accurate signal reconstruction. C. Online P ar titioning and Dictionar y Learning on Real-world Data Finally , we ev aluate the performance of the proposed online sensor scheduling method based on online graph node parti- tioning and dictionary learning. For this experiment, we use Algorithm 3 for sensor scheduling, Algorithm 2 for subspace learning, and Algorithm 1 for graph node partitioning. 1) Setup W e use the global sea temperature dataset [37]. This dataset is composed of snapshots recorded every month from 2016 to 2021. From sensors all over the world, we randomly select 256 sensors corresponding to the regions of the Mediterranean Sea, the North Sea, Black Sea, and the Northwest Atlantic coast. Subsequently , we create a k -NN graph ( k = 8 ) based on geographical distances between nodes. W e visualize the created graph and the observed signals in Fig. 4. VOLUME , 9 Author et al. : 20°W 20°W 10°W 10°W 0° 0° 10°E 10°E 20°E 20°E 30°E 30°E 40°E 40°E 30°N 30°N 40°N 40°N 50°N 50°N 60°N 60°N 2 4 6 8 10 12 14 16 18 SST (°C) FIGURE 4. Visualization of a graph signal constructed from global sea surface temperature . W e partition the graph into eight subsets. For the proposed algorithm, parameters are e xperimentally set as ( L, β , D , K ) = (10 3 , 1 , 20 , 3 × 10 2 ) . The confidence matrix in (21), is defined as: W t = diag ( m t ) , (36) where m t ∈ { 0 , 1 } N is the sampling index in (10) for the t th instance. W e consider the noisy case where η in (3) is η ∼ N ( 0 , 5 × 10 − 1 I ) . The performance of the proposed method is compared with those of SRel and SFrob [5]. In this experiment, we employ a signal reconstruction method in (4) for all approaches to ev aluate only the quality of the partitioned sampling subsets. Furthermore, we conduct an ablation study comparing three dictionary learning configurations to validate the ef- fecti veness of the proposed method. Specifically , we ev aluate the av erage MSEs under the following configurations: 1) Configuration 1: It employs the confidence matrix de- fined in (36). It is identical to the proposed method. 2) Configuration 2: It assigns uniform weights across the entire reconstructed signal (i.e., W = I ). This setting corresponds to the approach used in con ventional dic- tionary learning methods [21]. 3) Configuration 3: In this configuration, online dictionary learning is performed with the observed signals rather than the reconstructed ones. Specifically , the subspace is learned using signals recon- structed via the least-squares reconstruction [24], i.e., ˜ x ∗ = argmin ˜ x ∥ S ⊤ ˜ x − y ∥ 2 2 = S ( S ⊤ S ) † y . (37) In our setting, this solution performs zero-padding, where non-sampled nodes are set to zero while the observed values are preserved exactly . W e also utilize the confidence matrix defined by (36). Hence, the dictio- nary learning is performed exclusi vely on the sampled v alues. Note that its performance is ev aluated based on signals obtained through the standard reconstruction method (4) with the learned signal subspace. T ABLE 3. MSE [dB] of reconstructed signals. Bold number s denote the lowest MSE. Method Config. 1 Config. 2 Config. 3 MSE [dB] − 16 . 52 − 1 . 52 − 16 . 31 The performance of these three methods is compared by calculating the av erage MSE over time. 2) Results W e visualize the MSEs of the reconstructed signals in Fig. 5. The proposed method exhibits the best performance among all methods. This is because the proposed method incorporates the signal model into its partitioning algorithm, enabling it to achiev e a more suitable online partitioning strategy for the learned signal subspace than the other methods. The absolute errors between the original and reconstructed signals are also visualized in Fig. 6. While SRel and SFrob tend to select nodes that are relativ ely uniformly distributed across the graph, the proposed method frequently concen- trates its selections in specific regions. This implies that the proposed method can partition nodes into subsets that are suitable for reconstructing full-band graph signals. The numerical result of the ablation study is shown in T able 3. As expected, Configuration 1 outperforms the Configuration 2. Configuration 2 could be biased by recon- struction errors in unobserved nodes. W e leave the search for the optimal W t beyond (36) for future work. Furthermore, Configuration 1 outperforms the Configuration 3. This is because Configuration 3 learns the subspace directly from noisy measurements, whereas Configuration 1 utilizes recov- ered signals for learning, thereby suppressing the influence of observ ation noise. VI. CONCLUSION This paper presents a graph node partitioning method based on graph signal sampling theory . It partitions nodes into multiple equally-informati ve subsets that minimize the av- erage reconstruction error . W e formulate the problem as a DC optimization. W e extend the graph node partitioning to the online scenario by introducing dictionary learning for time-varying signal subspace estimation. Experimental results on synthetic and real-world graphs and graph signals demonstrate that our proposed method outperforms existing graph partitioning methods. APPENDIX Here, we introduce the deriv ation of the approximation of (8) by (9). By applying the Neumann series approximation, each term in (8) is represented as tr (( S ⊤ i AA ⊤ S i ) − 1 ) = 1 α i ∞ X n =0 tr (( I − α i S ⊤ i AA ⊤ S i ) n ) , (38) 10 VOLUME , < Society logo(s) and publication title will appear here. > 1 5 9 13 17 21 25 29 33 37 T ime Step 25 20 15 10 5 0 Nor malized MSE (dB) P r oposed SF r ob SR el P r oposed (avg: -16.52 dB) SF r ob (avg: -11.85 dB) SR el (avg: -12.32 dB) FIGURE 5. MSE of reconstructed signals [dB]. The average MSE of each method is plotted as a horizontal dashed line . FIGURE 6. Comparison of the original signal and absolute reconstruction errors at t = 1 , 9 , 17 , 25 , 33 , all obtained using the same subset (Subset 1). The top row displays the original signal, followed by the absolute errors of the proposed method, SRel, and SFrob, respectively . Selected nodes are highlighted with red circles where i = 1 , 2 and α i is determined such that it satisfies ∥ S ⊤ i AA ⊤ S i ∥ op ≤ 1 /α in terms of the operator norm ∥ · ∥ op . W e then truncate the approximation up to the second order . 1 α i ∞ X n =0 tr (( I − α i S ⊤ i AA ⊤ S i ) n ) ≈ 1 α i 2 X n =0 tr (( I − α i S ⊤ i AA ⊤ S i ) n ) = 1 α i tr (3 I − 3 α i S ⊤ i AA ⊤ S i + ( α i S ⊤ i AA ⊤ S i ) 2 ) . (39) Since α i is sufficiently smaller than unity , the second-order approximation is justified. Follo wed by M 1 ⊕ M 2 = V , we can deriv e the following relationship: 2 X i =1 tr ( S ⊤ i AA ⊤ S i ) = 2 X i =1 tr ( I M i V A ⊤ A ⊤ I ⊤ M i V ) = tr ( AA ⊤ ) . (40) Hence, the second term in (39) is constant. By ignoring the constant terms, we can approximate (8) by (9). REFERENCES [1] S. Coleri, S. Y . Cheung, and P . V araiya, “Sensor netw orks for moni- toring traffic, ” in Allerton Conference on Communication, Control and Computing . Citeseer , 2004, pp. 32–40. [2] K. Aberer , M. Hauswirth, and A. Salehi, “Infrastructure for data processing in large-scale interconnected sensor networks, ” in 2007 International Confer ence on Mobile Data Management . IEEE, 2007, pp. 198–205. [3] M. G. Rodriguez, L. E. O. Uriarte, Y . Jia, K. Y oshii, R. Ross, and P . H. Beckman, “Wireless sensor network for data-center environmen- VOLUME , 11 Author et al. : tal monitoring, ” in 2011 Fifth International Conference on Sensing T echnolo gy . IEEE, 2011, pp. 533–537. [4] E. A. Evangelakos, D. Kandris, D. Rountos, G. Tselikis, and E. Anas- tasiadis, “Energy sustainability in wireless sensor networks: An ana- lytical survey , ” Journal of Low P ower Electronics and Applications , vol. 12, no. 4, 2022. [5] R. Chakraborty , J. Holm, T . B. Pedersen, and P . Popovski, “Finding representativ e sampling subsets in sensor graphs using time-series similarities, ” ACM T ransactions on Sensor Networks , vol. 19, no. 4, pp. 1–32, 2023. [6] V . Gupta, T . H. Chung, B. Hassibi, and R. M. Murray , “On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor co verage, ” A utomatica , v ol. 42, no. 2, pp. 251–260, 2006. [7] A. Ortega, P . Frossard, J. Ko va ˇ cevi ´ c, J. M. Moura, and P . V an- derghe ynst, “Graph signal processing: Overview , challenges, and ap- plications, ” Pr oceedings of the IEEE , vol. 106, no. 5, pp. 808–828, 2018. [8] A. Sandryhaila and J. M. Moura, “Discrete signal processing on graphs, ” IEEE T ransactions on Signal Pr ocessing , vol. 61, no. 7, pp. 1644–1656, 2013. [9] D. I. Shuman, S. K. Narang, P . Frossard, A. Ortega, and P . V an- derghe ynst, “The emerging field of signal processing on graphs: Ex- tending high-dimensional data analysis to networks and other irregular domains, ” IEEE Signal Pr ocessing Magazine , v ol. 30, no. 3, pp. 83– 98, 2013. [10] V . Isler, S. Kannan, and K. Daniilidis, “Sampling based sensor- network deployment, ” in 2004 IEEE/RSJ International Confer ence on Intelligent Robots and Systems (IR OS)(IEEE Cat. No. 04CH37566) , vol. 2. IEEE, 2004, pp. 1780–1785. [11] A. Sakiyama, Y . T anaka, T . T anaka, and A. Ortega, “Eigendecomposition-free sampling set selection for graph signals, ” IEEE T ransactions on Signal Processing , vol. 67, no. 10, pp. 2679–2692, 2019. [12] Y . T anaka, Y . C. Eldar, A. Ortega, and G. Cheung, “Sampling signals on graphs: From theory to applications, ” IEEE Signal Processing Magazine , vol. 37, no. 6, pp. 14–30, 2020. [13] U. V on Luxburg, “ A tutorial on spectral clustering, ” Statistics and Computing , v ol. 17, pp. 395–416, 2007. [14] T . Cour, F . Benezit, and J. Shi, “Spectral segmentation with multiscale graph decomposition, ” in 2005 IEEE Computer Society Conference on Computer V ision and P attern Reco gnition (CVPR’05) , vol. 2. IEEE, 2005, pp. 1124–1131. [15] M. Tsitsvero, S. Barbarossa, and P . Di Lorenzo, “Signals on graphs: Uncertainty principle and sampling, ” IEEE T ransactions on Signal Pr ocessing , v ol. 64, no. 18, pp. 4845–4860, 2016. [16] Y . T anaka and Y . C. Eldar, “Generalized sampling on graphs with subspace and smoothness priors, ” IEEE Tr ansactions on Signal Pr o- cessing , v ol. 68, pp. 2272–2286, 2020. [17] J. Hara and Y . T anaka, “Sampling set selection for graph signals under arbitrary signal priors, ” in ICASSP 2022-2022 IEEE International Confer ence on Acoustics, Speech and Signal Pr ocessing (ICASSP) . IEEE, 2022, pp. 5732–5736. [18] P . D. T ao and L. T . H. An, “ A dc optimization algorithm for solving the trust-region subproblem, ” SIAM Journal on Optimization , vol. 8, no. 2, pp. 476–505, 1998. [19] T . Okuno and Y . Ikebe, “ A new approach for solving mixed integer DC programs using a continuous relaxation with no integrality gap and smoothing techniques, ” Optimization , vol. 70, no. 1, pp. 55–74, 2021. [20] S. Ono and I. Y amada, “Hierarchical con vex optimization with primal- dual splitting, ” IEEE Tr ansactions on Signal Processing , v ol. 63, no. 2, pp. 373–388, 2014. [21] S. Nomura, J. Hara, H. Higashi, and Y . T anaka, “Dynamic sensor placement based on sampling theory for graph signals, ” IEEE Open Journal of Signal Processing , vol. 5, pp. 1042–1051, 2024. [22] D. A. Bader , H. Meyerhenke, P . Sanders, and D. W agner, Graph partitioning and graph clustering . American Mathematical Society Providence, RI, 2013, vol. 588. [23] U. Brandes, D. Delling, M. Gaertler , R. Gorke, M. Hoefer , Z. Nikoloski, and D. W agner , “On modularity clustering, ” IEEE T ransactions on Knowledge and Data Engineering , vol. 20, no. 2, pp. 172–188, 2007. [24] J. Hara, S. Ono, H. Higashi, and Y . T anaka, “Sensor placement problem on networks for sensors with multiple specifications, ” in Pr oc. Eur opean Signal Pr ocessing Confer ence (EUSIPCO) , 2024, pp. 2327– 2331. [25] F . W ang, Y . W ang, and G. Cheung, “ A-optimal sampling and robust reconstruction for graph signals via truncated neumann series, ” IEEE Signal Pr ocessing Letters , vol. 25, no. 5, pp. 680–684, 2018. [26] F . W ang, G. Cheung, and Y . W ang, “Low-comple xity graph sampling with noise and signal reconstruction via neumann series, ” IEEE T ransactions on Signal Pr ocessing , vol. 67, no. 21, pp. 5511–5526, 2019. [27] J. Neumann, “ Almost periodic functions in a group. 1, ” T ransactions of the American Mathematical Society , vol. 36, no. 3, pp. 445–492, 1934. [28] L. Condat, “ A primal–dual splitting method for con vex optimization in- volving lipschitzian, proximable and linear composite terms, ” Journal of Optimization Theory and Applications , vol. 158, no. 2, pp. 460–479, 2013. [29] R. T . Rockafellar , Con vex Analysis . Princeton University Press, 1997, vol. 28. [30] D. Gabay and B. Mercier , “ A dual algorithm for the solution of nonlinear variational problems via finite element approximation, ” Computers & Mathematics with Applications , vol. 2, no. 1, pp. 17–40, 1976. [31] M. Aharon, M. Elad, and A. Bruckstein, “K-svd: An algorithm for designing overcomplete dictionaries for sparse representation, ” IEEE T ransactions on Signal Pr ocessing , vol. 54, no. 11, pp. 4311–4322, 2006. [32] K. Engan, S. O. Aase, and J. H. Husoy , “Method of optimal directions for frame design, ” in 1999 IEEE International Conference on Acous- tics, Speech, and Signal Pr ocessing. Proceedings. ICASSP99 (Cat. No. 99CH36258) , v ol. 5. IEEE, 1999, pp. 2443–2446. [33] N. Parikh, S. Boyd et al. , “Proximal algorithms, ” F oundations and tr ends® in Optimization , vol. 1, no. 3, pp. 127–239, 2014. [34] P . L. Combettes and N. N. Reyes, “Moreau’ s decomposition in banach spaces, ” Mathematical Pr ogramming , vol. 139, no. 1, pp. 103–114, 2013. [35] S. Ruder, “ An overvie w of gradient descent optimization algorithms, ” arXiv pr eprint arXiv:1609.04747 , 2016. [36] F . Xue and P . R. Kumar , “The number of neighbors needed for connectivity of wireless networks, ” W ir eless Networks , v ol. 10, no. 2, pp. 169–181, 2004. [37] N. A. Rayner , D. E. Parker , E. Horton, C. K. Folland, L. V . Alexander , D. Rowell, E. C. Kent, and A. Kaplan, “Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century , ” Journal of Geophysical Researc h: Atmospher es , vol. 108, no. D14, 2003. 12 VOLUME ,