# Bandlimited graph signal reconstruction by diffusion operator

- Lishan Yang
^{1}Email author, - Kangyong You
^{1}and - Wenbin Guo
^{1, 2}

**2016**:120

https://doi.org/10.1186/s13634-016-0421-4

© The Author(s) 2016

**Received: **27 April 2016

**Accepted: **4 November 2016

**Published: **11 November 2016

The Erratum to this article has been published in EURASIP Journal on Advances in Signal Processing 2017 2017:12

## Abstract

Signal processing on graphs extends signal processing concepts and methodologies from the classical signal processing theory to data indexed by general graphs. For a bandlimited graph signal, the unknown data associated with unsampled vertices can be reconstructed from the sampled data by exploiting the spatial relationship of graph signal. In this paper, we propose a generalized analytical framework of unsampled graph signal and introduce a concept of diffusion operator which consists of local-mean and global-bias diffusion operator. Then, a diffusion operator-based iterative algorithm is proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, the reconstructed residuals associated with the sampled vertices are diffused to all the unsampled vertices for accelerating the convergence. We then prove that the proposed reconstruction strategy converges to the original graph signal. The simulation results demonstrate the effectiveness of the proposed reconstruction strategy with various downsampling patterns, fluctuation of graph cut-off frequency, robustness on the classic graph structures, and noisy scenarios.

### Keywords

Diffusion operator Downsampling and reconstruction Bandlimited graph signal processing## 1 Introduction

Recent years have witnessed an enormous growth of interest in efficient paradigms and techniques for representation, analysis, and processing of large-scale datasets emerging in various fields and applications, such as sensor and transportation networks, social networks and economic networks, and energy networks [1, 2]. The irregular structure is the most important characteristic of those large-scale datasets, which limits the applicability of many approaches used for small-scale datasets. This big data problem motivates the emerging field of signal processing on graphs.

Signal processing on graphs extends the classical signal processing techniques and paradigms to the irregular domain [3–5]. Graphs are useful representation tools for representing large-scale datasets with geometric structures. The relational structure of large-scale dataset is represented with graph, in which data elements correspond to the vertices, the relationship between data elements is represented by the edge, and the strength of relationship is reflected in the edge weight. The graph signal can be regarded as a vector signal which contains the spatial relationship of the vertices. Due to the complicated relationship and large data volume, it is necessary to transform the original graph signal to small-scale modality. The downsampling can be treated as any decrease in dimension via an operator, and conversely, the interpolation can be treated as any increase in dimension via an operator. The main purpose of downsampling method is that the original graph signal may be reconstructed through its entries on only a subset of the vertices by exploiting the character of smoothness. Pesenson in [6] established a Paley-Wiener function based sampling theory on combinatorial graphs. He proposed a concept of uniqueness set for downsampling and gave a sufficient condition that the downsampling set needs to satisfy for unique reconstruction. For the reconstruction of sampled graph signal, the main methodology of current algorithms is to extend the Papoulis-Gerchberg algorithm [7, 8] from the classical regular domain to the graph irregular domain. S. K. Narang in [9] proposed an iterative least square reconstruction (ILSR) algorithm for reconstructing bandlimited graph signal from partially observed samples. ILSR adopts the method of projection onto convex sets (POCS) to iteratively project the sampled data onto the downsampling subspace and the low-pass filtering subspace. In [10, 11], X. Wang proposed a concept of local-set and two local-set based iterative reconstruction algorithms (IWR and IPR) for recovering the sampled bandlimited graph signal. The local-set is formed by partitioning the graph into several disjoint subgraphs. iterative propagating reconstruction (IPR) is also an iterative reconstruction algorithm based on the philosophy of POCS. Compared with ILSR, IPR propagates the reconstructed residual associated with the sampled vertex to the unsampled vertices in the respective local-sets. Given the benefit which is obtained from the propagating of reconstructed residual, IPR has faster convergence than ILSR. However, the graph signals associated with the unsampled vertices are influenced by all the sampled vertices in graph, which should not be limited in the local-set. Besides, since the differences among the reconstruction residuals associated with the unsampled vertices cannot be ignored, the even propagation may not be able to achieve the assignment.

Related works on downsampling and reconstruction of graph signals include the methods proposed in [12, 13]. In [12], the authors proposed a sampling theory and reconstruction method on graphs for bandlimited graph signals. The sampling theory proposed in [12] focuses on the graph adjacency matrix and non-iterative reconstruction method. In [13], the authors proposed a sampling aggregation method for the graph signals, where the observations are aggregated to one vertex. Different from them, we focus on the iterative reconstruction method for bandlimited graphs in this paper.

The main contribution of this paper is that we present a generalized analytical framework of graph signals associated with the unsampled vertices to further improve the convergence rate of bandlimited graph signal reconstruction. We decompose the graph signals associated with the unsampled vertices into three components, i.e., the extrapolated component, the local-mean diffusion component, and the global-bias diffusion component. Based on this scheme, we propose an iterative diffusion operator-based reconstruction algorithm. The correspondence between the proposed algorithm and the current reconstruction algorithms (ILSR and IPR) is also analyzed, which will be helpful to future works on the reconstruction of bandlimited graph signal. Besides, the theoretical analysis for the proposed iterative reconstruction algorithm is also presented. Then, we demonstrate the performance of the proposed algorithm and the current algorithms with various downsampling patterns, fluctuation of graph cut-off frequency, robustness on the classic graph structures, and noisy scenarios. Finally, we adopt the temperature data of the USA and the electricity consumption data of Shandong province of China as the examples of real-world data to test the performance of reconstruction algorithms. The simulation results show that a better performance of the proposed algorithm can be achieved.

The rest of this paper is organized as follows. In Section 2, the previous works of downsampling and reconstruction for bandlimited graph signal are briefly reviewed. In Section 3, we propose the concept of diffusion operator and its corresponding iterative reconstruction algorithm. In Section 4, we analyze and prove the convergence of the proposed algorithm. In Section 5, we demonstrate the proposed algorithm by using the synthetic and real-world data on various graphs. In Section 6, conclusions are drawn.

## 2 The previous work for downsampling and reconstruction

*G*=(

*V*,

*E*) is a collection of vertices

*V*and edges

*E*, with

*V*={1,2,…,

*N*} representing the set of vertices of the graph and

*E*={

*w*

_{ i,j },∀

*i*,

*j*∈

*V*} representing the set of edges connecting vertex

*i*and

*j*with weight

*w*

_{ ij }, where

*w*

_{ ii }=0. The adjacency matrix of the graph is defined as

*A*(

*i*,

*j*)=

*w*

_{ ij }. The degree

*d*

_{ i }of a vertex

*i*is defined as the sum of the weights of the edges connected to the vertex

*i*. The degree matrix of the graph is a diagonal matrix defined as

*D*=diag{

*d*

_{1},

*d*

_{2},…,

*d*

_{ N }}. The Laplacian matrix of the graph is defined as

*L*=

*D*−

*A*. The normalized Laplacian matrix \(\mathcal {L}\) is a symmetric positive semi-definite matrix and can be decomposed as

where *U*
^{
T
} denotes the transpose of Laplacian eigenvector *U*, *Λ*=diag{*λ*
_{1},*λ*
_{2},…,*λ*
_{
N
}} is a diagonal matrix of real Laplacian eigenvalues ordering as *λ*
_{1}≤*λ*
_{2}≤⋯≤*λ*
_{
N
}, and its corresponding orthogonal set of Laplacian eigenvectors denoted as *U*={*u*
_{1},*u*
_{2},…,*u*
_{
N
}}, with *u*
_{
i
} is the *i*th column vector of Laplacian eigenvector matrix.

*f*is represented as a vector mapping

*f*:

*V*→

*R*

^{ N }, such that

*f*(

*i*) is the value of the signal on vertex

*i*. \({\hat f_{i}}=<{f},{u}_{i}>\) is the

*graph Fourier transform*(GFT) of

*f*. Similar with the Fourier transform in classical signal processing, graph Fourier transform performs the expansion of a graph signal into a Laplacian eigenvector basis of signals [14]. The eigenvectors and eigenvalues of the Laplacian matrix provide a spectral interpretation of the graph signal. For more concise comparison, the eigenvalues of the Laplacian matrix can be regarded as the graph frequency and form the spectrum of graph, and the Laplacian eigenvectors that correspond to a frequency

*λ*

_{ m }are called the graph frequency component corresponding to the

*m*th frequency.

### 2.1 Downsampling of a bandlimited graph signal

*w*), the graph signal is regarded as bandlimited in the range of [0,

*w*), with the

*w*is called the graph cut-off frequency [6, 15]. The space of bandlimited graph signal is often called Paley-Wiener space (PWS) and is denoted as

where *λ* denotes the Laplacian eigenvalue and \(\hat {f}(\lambda)\) denotes the graph frequency component corresponding to *λ*. We denote *S* as a downsampling set of the vertices of the graph, and *S*
^{
c
}=*V*∖*S* denotes its complementary set. The purpose of downsampling operation is to select valuable vertices to form the downsampling set *S*. The concept of *uniqueness set* is defined in [6], which provides a sufficient condition for exact reconstruction from the sampled graph signal. A subset of vertices *S*⊂*V* is a *uniqueness set* in PWS, if for any two signals *g*,*h*, the fact that they coincide on *S* implies they coincide on *V* : *g*(*S*)=*h*(*S*)⇒ *g*=*h*.

Currently, there are two solutions for finding an appropriate downsampling set. In [16], the author formulates a greedy heuristic algorithm to obtain an estimation of the optimal downsampling set. In [11], the author proposes a local-set based downsampling forming algorithm. The graph is divided into disjoint subgraphs and each subgraph selects one vertex as the downsampling vertex, which is called one-hop sampling method. The one-hop sampling method is a rather economical and efficient choice of downsampling set forming method when there is no restriction on the number of vertices in the downsampling set or no location-limited of downsampling vertices. In Fig. 1, we adopt the one-hop sampling method to form the downsampling set, where the sampled vertices are denoted as the red-star-vertices and the unsampled vertices are denoted as the blue-roundness-vertices.

### 2.2 Reconstruction of a bandlimited graph signal

where *f*
^{
c
} denotes the classical continuous signal, \({{P_{T}^{c}}}\) denotes the time domain downsampling operator, *I* denotes the identity operator, \({{P_{w}^{c}}}\) denotes the frequency domain cut-off operator, and \({{\mathcal {F}}_{c}}\) and \({\mathcal {F}}_{c}^{- 1}\) represent the classical Fourier transform and classical Fourier inverse transform, respectively. In each iteration, the Papoulis-Gerchberg algorithm replaces the downsampling part of the estimated reconstruction signal \({{f_{k}^{c}}}\) by the actual known segment and then combines the extrapolation segment to form the next iterative signal. In other words, at *k*th iteration, the solution \({f_{k}^{c}}\) is obtained from \(f_{k-1}^{c}\) and satisfies the two constraints of time domain downsampling and frequency domain bandlimited.

*S*to the actual given samples and then projects the graph signal onto the low-pass filtering subspace. Denote

*P*

_{ T }as the vertex domain downsampling operator and

*f*

_{ d }as the sampled graph signal, then the downsampling process can be represented as follows

*P*

_{ T }is a diagonal matrix

_{ S }is the set indicator vector, whose

*i*th entry is equal to one, if

*i*∈

*S*, or zero otherwise. The iterative process of ILSR can be written as follows

*k*th iterative reconstructed residual of ILSR,

*P*

_{ w }denotes the graph frequency cut-off operator,

*w*denotes the graph frequency domain cut-off frequency, and \({\mathcal {F}}\) and \({{\mathcal {F}}^{- 1}}\) denote the graph Fourier transform and inverse transform, respectively. At the first iteration, the initial reconstructed graph signal

*f*

_{0}is obtained by projecting the sampled graph signal

*f*

_{ d }onto the low-pass filtering subspace. In this paper, we define the difference between the graph signal

*f*and the sampled reconstructed graph signal

*f*

_{ k }as the reconstructed residual \({f_{k}^{s}}\), i.e., \({f_{k}^{s}} = {f} - {f_{k}}\). Moreover, the reconstructed graph signal

*f*

_{ k }is denoted by \({f_{k}^{L}}\) for ILSR and \({f_{k}^{P}}\) for IPR. In [10, 11], the author proposes a local-set based IPR algorithm. In each iteration, IPR adopts a local propagative operator to locally and evenly propagate the reconstructed residual. The iterative process of IPR is shown as follows

where \({f_{k}^{P}}\) denotes the *k*th iterative reconstructed graph signal of IPR, \({{\mathcal {Q}}_{w}}\left (\cdot \right)\) is a graph frequency domain cut-off operator, **δ**
_{
N(v)}=(*δ*
_{
N(v)}(1),*δ*
_{
N(v)}(2),…,*δ*
_{
N(v)}(*N*))^{
T
} with *δ*
_{
N(v)}(*m*)=1 only when *m*∈*N*(*v*). According to Eq. (7), IPR first propagates the reconstructed residual locally and evenly to the local-set that each sampled vertex belongs to and then projects the new signal onto the low-pass filtering subspace. Since IPR propagates the reconstructed residual in the local-set at each iteration, IPR converges faster than ILSR. However, IPR only focus on the propagating of reconstructed residual within the local-sets. In the next section, we propose a diffusion operator based iterative reconstruction strategy, which extends the reconstructed residual to more generalized diffusion.

Different from the POCS method, the sampling theory proposed in [12] recovers the sampled graph signal by employing an interpolation operator *Φ*=*U*
_{
w
}(*P*
_{
T
}
*U*
_{
w
})^{−1}, where *w* denotes the bandwidth of bandlimited graph signal, *U* denotes the eigenvector matrix of graph adjacency matrix *A*, *U*
_{
w
} denotes the first *w* columns of *U*, and *P*
_{
T
} denotes the sampling operator. In this paper, we follow the methodology of the POCS method.

## 3 The diffusion operator-based reconstruction strategy

In this section, we establish a generalized analysis framework of the graph signals associated with the unsampled vertices. The concept of local-mean and global-bias diffusion operator is firstly defined. Then, we propose an iterative diffusion operator-based reconstruction algorithm. Discussions on the current reconstruction algorithms are also included in this section.

### 3.1 The generalized analytical framework of unsampled graph signal

*f*

_{acu}denote the actual graph signals associated with the unsampled vertices,

*f*

_{rec}denote the extrapolated graph signal obtained by projecting onto the low-pass filtering subspace, and

*f*

_{res}denote the diffused reconstructed residuals obtained from the sampled vertices. Since ILSR directly projects the sampled signal onto the low-pass filtering subspace, it can be seen that

*f*

_{res}are set to zeros and

*f*

_{acu}is only obtained from

*f*

_{rec}. In IPR, the reconstructed residuals associated with sampled vertices are copied and assigned to the vertices in the corresponding local-set, and then the projection procedure is conducted. It can be seen that the vertices within the local-set have the same value of reconstructed residuals. It may not fit into the property of actual reconstructed residuals associated with the unsampled vertices. Besides, practically the graph signal associated with every vertex is either directly or indirectly influenced by all the vertices in a graph, which should not be limited in the local-set. Thus, for the purpose of accurate analysis, we decompose the diffused reconstructed residuals

*f*

_{res}into two components, and Eq. (8) can be rewritten as

where \(f_{\text {res}}^{\text {mean}}\) denote the local-mean component of the diffused reconstructed residuals, and \(f_{\text {res}}^{\text {bias}}\) denote the global-bias component of the diffused constructed residuals. The motivation of the diffused reconstructed residual decomposition is that we expect to establish three-layer analytical framework for the unsampled graph signals. The first layer of unsampled graph signal is obtained by projecting the sampled graph signal onto the low-pass filtering subspace, which is *f*
_{rec}. The second layer of unsampled graph signal is obtained from the reconstructed residual of adjacent sampled vertices, which is \(f_{\text {res}}^{\text {mean}}\). For the local-mean component, the reconstructed residuals associated with the sampled vertices are regarded as the mean of local region around the sampled vertices and are diffused from sampled vertices to their adjacent unsampled vertices. The third layer of unsampled graph signal is obtained by the reconstructed residual of all the sampled vertices, which is \(f_{\text {res}}^{\text {bias}}\). The main purpose of the global-bias component is that the graph signals associated with the unsampled vertices are influenced by all the sampled vertices in graph. The global-bias component is expected to further exploit the reconstructed residuals associated with the sampled vertices, which is beneficial to accelerate the convergence of reconstruction. Besides, the global-bias component is also used to form the differences among the graph signals associated with the unsampled vertices, since the local-mean component assigns the same value of reconstruction residuals from the sampled vertices to their adjacent vertices. The collaboration of the local-mean component and global-bias component is expected to approximate the actual reconstructed residuals associated with the unsampled vertices by further exploiting the smoothness of the bandlimited graph signals.

Besides, Eq. (9) can be regarded as the composition model of unsampled graph signal of ILSR, when \(f_{\text {res}}^{\text {mean}}\) and \(f_{\text {res}}^{\text {bias}}\) are set to zeros. Similarly, Eq. (9) can be regarded as the composition model of unsampled graph signal of IPR, when \(f_{\text {res}}^{\text {bias}}\) is set to zero. The *f*
_{rec} of IPR and ILSR are both obtained by directly projecting the sampled graph signal onto the low-pass filtering subspace. Based on the decomposition, in the following parts of this section, we will propose two diffusion operators (the local-mean diffusion operator and the global-bias diffusion operator) to achieve the local-mean diffusion and global-bias diffusion of reconstruction residuals associated with the sampled vertices.

### 3.2 The local-mean diffusion operator

In this part, the concept of local-mean diffusion operator is introduced. Then, discussions on the local-mean diffusion operator and local propagation are also included in this part.

According to the decomposition model of (9), the second layer of unsampled graph signal is the local-mean diffusion component. In this part, we proposed a local-mean diffusion operator, which is used to assign the reconstructed residual from the sampled vertices to their adjacent unsampled vertices, to achieve the local-mean diffusion operation. Denote by *ρ*(*v*,*S*) the fewest distance of an unsampled vertex *v* from the downsampling set *S*, i.e., the fewest number of edges in a shortest path from *v* to a vertex in the downsampling set *S*. Besides, we assume that the fewest distance of a sampled vertex from the downsampling set is zero.

###
**Definition 1**

*P*

_{ m }as

*N*-by-1 vector and denotes the local-mean component of reconstructed residual,

*B*

_{ i }=

*C*

*A*

^{ i },

*C*is a diagonal matrix with the diagonal element as the reciprocal of the degree of the corresponding vertex,

*A*denotes the adjacency matrix of graph,

*i*denotes the fewest number of edges in a shortest path from an unsampled vertex to a vertex in the downsampling set

*S*,

*r*denotes the maximal number of hop between sampled vertex and unsampled vertex on the graph, and

*δ*

_{ i }denotes

*δ*-function of

*ρ*with entries

In other words, *δ*
_{
i
} is a diagonal matrix with the diagonal element as the *ρ*(*u*,*S*) of corresponding vertex *u*.

Indeed, an underlying idea behind the local-mean diffusion operator is to incorporate the structure of the adjacency matrix into the local-mean diffusion operation of reconstructed residual. In definition 1, we employ the powers of the adjacency matrix *A* to sequentially diffuse the reconstructed residuals associated with the sampled vertices to the adjacency unsampled vertices. Besides, *C* is used to obtain the weighted value of diffused reconstructed residual.

In [11], the authors propose an operation *G* firstly to propagate the reconstructed residual in respective local-sets and then projects the combinatorial signal onto the low-pass filtering subspace. The first step of *G* provides a solution to propagate the reconstructed residuals associated with the sampled vertices with the help of local-sets. In this paper, we redefine the first propagating step of *G* as *G*
_{
p
}. It can be seen that *G*
_{
p
} can be regarded as a special case of local-mean diffusion operation based on local-set. Thus, it is easy to see that *G*
_{
p
} and *P*
_{
m
} achieve the same performance with the precondition of local-set, but *P*
_{
m
} can diffuse the reconstructed residuals in the more comprehensive scenarios.

### 3.3 The global-bias diffusion operator

The main motivation of the global-bias component is that the graph signals associated with the unsampled vertices are influenced by all the sampled vertices in graph, which should not be limited in the local region. Besides, the global-bias component is used to provide the difference of reconstructed residual of local region. In this part, we propose a global-bias diffusion operator, which is used to establish the relationship between one sampled vertex and all unsampled vertices, to achieve the global-bias diffusion operation.

*f*

^{ s }=[

*f*

^{ s }(1),0,…,0]

^{ T }, where

*f*

^{ s }(1) denotes the reconstructed residual resided on the first vertex and \({\hat f^{s}} = {\left [ {{u_{11}}{f^{s}}\left (1 \right),{u_{12}}{f^{s}}\left (1 \right),\ldots,{u_{1N}}{f^{s}}\left (1 \right)} \right ]^{T}}\) denotes graph frequency component. Since the graph signal is band-limited, we assume that there are

*m*graph frequency components within the bandwidth

*w*, that is, \({\hat f^{s}} = {\left [ {{u_{11}}{f^{s}}\left (1 \right),\ldots,{u_{1m}}{f^{s}}\left (1 \right),0,..,0} \right ]^{T}}\). Then, to transform \({\hat f^{s}}\) from the graph frequency domain to the vertex domain and denote \({\tilde f^{s}}\) as the new presentation, which can be written as follows

*S*. Thus, from the perspective of the single unsampled vertex, its estimated reconstructed residual is the sum of diffused reconstructed residuals of all the sampled vertices. Denote \({\bar f^{s}}\left (i \right)\) as the estimated reconstructed residual of the

*i*th unsampled vertex, which can be written as follows

The global-bias diffusion process can be understood from two aspects. From the perspective of sampled vertex, the reconstructed residual of the single sampled vertex is diffused to all the unsampled vertices. From the perspective of unsampled vertex, the estimated reconstructed residual of the single unsampled vertex is the sum of diffused reconstructed residuals of all the sampled vertices. Then, based on the discussion above, we define the global-bias diffusion operator to diffuse the reconstructed residual to all the unsampled vertices.

###
**Definition 2**

*P*

_{ b }as

where \(f_{\text {res}}^{\text {bias}}\) is an *N*-by-1 vector and denotes the global-bias diffusion component of estimated reconstructed residual, *U*
^{′} denotes the modified Laplacian eigenvector matrix which the rows corresponding to the sampled vertices are set to the zero sequence and the columns corresponding to out of the bandwidth *w* are set to the zero sequence, and *U*
^{″} denotes the modified Laplacian eigenvector matrix in which the rows corresponding to the unsampled vertices are set to the zero sequence and the columns corresponding to out of the bandwidth *w* are set to the zero sequence.

In each iteration, the global-bias diffusion operator *P*
_{
b
} diffuses the iterative reconstructed residual from the sampled vertices to all the unsampled vertices.

### 3.4 The diffusion operator-based reconstruction algorithm

In this part, we propose a diffusion operator based iterative reconstruction (IGDR) algorithm.

###
**Definition 3**

*P*

_{ d }as

where *P*
_{
m
} denotes the local-mean diffusion operator, and *P*
_{
b
} denotes the global-bias diffusion operator.

where \({f_{k}^{G}}\) denotes the *k*th iterative reconstructed signal of IGDR, \({\left ({{f_{d}} - {P_{T}}{f_{k}^{G}}} \right)}\) denote the reconstructed residual on the downsampling set, and \({\left ({I - {P_{T}}} \right){P_{d}}\left ({{f_{d}} - {P_{T}}{f_{k}^{G}}} \right)}\) denote the reconstructed residual is diffused from the downsampling set to the non-downsampling set by the diffused operator.

### 3.5 Discussion

Besides, it can be seen that ILSR can be regarded as a special case of IGDR, when the local-mean diffusion component and global-bias diffusion component are set to zeros. Similarly, IPR can be regarded as a special case of IGDR, when the local-mean diffusion component is designed by the local-set and the global-bias diffusion component is set to zero.

## 4 The analysis of iterative error and convergence

In this section, we present the theoretical analysis of convergence for the proposed iterative reconstruction algorithm.

###
**Proposition 1**

*f*can be reconstructed from its downsampling set

*S*by IGDR, for a given graph cut-off frequency

*w*, the reconstructed error bound of IGDR is

where *η*
^{
k
} denotes the *k*-th iterative error, \({\mathcal {F}}\) and \({{\mathcal {F}}^{- 1}}\) denote the graph Fourier transform and the inverse graph Fourier transform, respectively, *w* denote the graph cut-off frequency, *P*
_{
T
} denotes the downsampling operator which chooses the sampled vertices and pads the unsampled vertices with zeros, *P*
_{
w
} denote the graph frequency cut-off operator, *I* denotes the identity operator, and *P*
_{
d
} is diffusion operator. The notation *R*
_{max}, *J*
_{max}, *α*
_{1}, and *α*
_{2} are the parameters of graph signal and the downsampling set, and the detailed explanation can be seen in proof.

###
*Proof*

: The proof is postponed to Appendix. □

## 5 The simulation results

- 1.
Generate a random Gaussian signal on graph.

- 2.
Transform the graph signal into graph spectral domain by graph Fourier transformation and remove the frequency components higher than the given graph cut-off frequency.

- 3.
Transform the graph signal from the graph spectral domain to the vertex domain.

The Minnesota path graph is shown in Fig. 1. The proposed algorithm and the current algorithms are compared on the three classic graph structures for demonstrating the robustness. We also use the real-world data, which are the temperature (2014.1.1) of 94 cities of the USA and the electricity consumption data (2015) of Shandong province of China, to test the performance of the proposed reconstruction algorithm. Moreover, since IPR needs the help of local-set, we use one-hop sampling method to form the downsampling set for fair comparison. Besides, we define the concept of relative error. Let *x* denote a vector and \(\tilde {x}\) denote the estimated value of *x*, then the relative error is defined by \(\varepsilon = \left \| {\tilde x - x} \right \| \left / \left \| x \right \|\right.\).

### 5.1 The convergence performance

### 5.2 Graph cut-off frequency

### 5.3 Robustness with additive noise

### 5.4 Robustness with different downsampling sets

### 5.5 The performance on three classic graph structures

### 5.6 Sensitivity with imprecise knowledge of graph cut-off frequency

### 5.7 Real-world data

## 6 Conclusions

In this paper, the problem of the bandlimited graph signal reconstruction was studied. We established a generalized analytical framework of graph signals associated with the unsampled vertices. We defined a concept of diffusion operator, which consists of local-mean diffusion operator and global-bias diffusion operator. Employing the diffusion operator, we proposed an iterative algorithm to reconstruct the unknown data associated with the unsampled vertices from the observed samples. In each iteration, the reconstructed residuals associated with the sampled vertices are diffused to all the unsampled vertices. We also presented the analysis of iterative reconstructed error and convergence of the proposed algorithm. The simulation results are showed that the techniques presented in this paper perform beyond the current reconstruction algorithms. Moreover, the main purpose of this paper is to present a generalized model to accelerate the convergence rate of reconstruction algorithm. The design of the local-mean diffusion operator and global-bias diffusion operator according to the character of network topology and graph signals can be investigated in the future work.

## 7 Appendix

###
*Proof*

Key notation used in the proof

Symbol | Description |

| The graph frequency cut-off operator |

| The graph cut-off frequency |

| The downsampling set |

| The set of vertices of the graph |

| The set of edges connecting vertices |

\({\mathcal {F}}\) | The graph Fourier transform |

\({{\mathcal {F}}^{- 1}}\) | The inverse graph Fourier transform |

| The downsampling operator |

| The vertex |

| The diffusion operator |

| A bandlimited graph signal |

| The local-mean diffusion operator |

| The global-bias diffusion operator |

| The local-diffused unsampled vertices set of the sampled vertex |

| The degree of the vertex |

| The downsampling parameter |

| The cardinality of the downsampling vertices set |

| The cardinality of the graph vertices set |

| The cardinality of the eigenvalue less than the graph cut-off frequency |

| The maximal distance from sampled vertex |

| The cardinality of |

| The maximal value of the product of |

| The degree operator of |

| The maximal value |

*P*

_{ w }=diag{1

_{ w }}, where 1

_{ w }is the set indicator vector, whose

*i*th entry is equal to one, if

*i*∈[0,w), or zero otherwise (the range of normalized graph frequency is from 0 to 2). Thus, it can be seen that

*P*

_{ w }remove the high-frequency components ([

*w*,2]). Since only [0,

*w*) of the energy of graph signal can be preserved, \({{\mathcal {F}}^{- 1}}{P_{w}}{\mathcal {F}}\) is a contraction mapping. The downsampling operator is defined as a diagonal matrix

*P*

_{ T }=diag{1

_{ S }}, where 1

_{ S }is the set indicator vector, whose

*i*th entry is equal to one, if

*i*∈

*S*, or zero otherwise. Due to only the non-downsampling vertices are preserved, (

*I*−

*P*

_{ T }) is a contraction mapping. Besides, let

*f*denote a bandlimited graph signal and its graph cut-off frequency is

*w*. Notice that \({{\mathcal {F}}^{- 1}}{P_{w}}{\mathcal {F}}f = f\) and

*f*

_{ d }=

*P*

_{ T }

*f*, we have,

*P*

_{ d }consists of the local-mean diffusion operator

*P*

_{ m }and the global-bias diffusion operator

*P*

_{ b }. We have

*I*−

*P*

_{ m }

*P*

_{ T }∥. According to the analysis in Section 3.2, the local-mean diffusion operator

*P*

_{ m }assign the signal from sampled vertices to their adjacent unsampled vertices. Let

*f*denote a bandlimited graph signal,

*S*denote the downsampling set, and

*K*(

*u*) denote the local-diffused unsampled vertices set of the sampled vertex

*u*by the local-mean diffusion operator. Then, we have

*d*(

*v*) denote the degree of the vertex

*v*. It can be seen that the part

*a*of Eq. (18) is the downsampling part of ∥

*f*∥

^{2}, then

*α*

_{1}denote the proportion of quadratic sum of graph signals associated with sampled vertices in the quadratic sum of graph signals associated with all the vertices, i.e.,

*α*

_{1}is from 0 to 1. The value of

*α*

_{1}approximates to \(\frac {M}{N}\), when the graph signal

*f*is very smooth. For the part

*b*of Eq. (18), we have

*c*of Eq. (20). There is always a shortest path within

*K*(

*u*) from any local-diffused unsampled vertex

*v*∈

*K*(

*u*) to sampled vertex

*u*, which is denoted as (

*v*,

*v*

_{1},

*v*

_{2},…,

*v*

_{ r−1},

*u*). Then, we have

*H*

_{ K(u)}denote the maximal distance from sampled vertex

*u*to any local-diffused unsampled vertex within

*K*(

*u*). For each local-diffused unsampled vertex

*v*∈

*K*(

*u*), the path to the sampled vertex

*u*is not longer than

*H*

_{ K(u)}. Let

*X*

_{ K(u)}denote the cardinality of

*K*(

*u*). The maximal distance within

*K*(

*u*) is counted for no more than

*X*

_{ K(u)}times. Then, we have

*R*

_{max}denote the maximal value of the product of

*X*

_{ K(u)}and

*H*

_{ K(u)}in graph, i.e.,

*V*denotes the set of vertices of the graph and

*E*denotes the set of edges connecting vertices. Since

*f*is bandlimited, the components of \({\hat f}\) associated with the frequencies higher than the graph cut-off frequency

*w*are zero. Then, we analyze the character of the part

*d*of Eq. (20).

*J*(

*u*) is denoted as

*J*

_{max}denote the maximal value of

*J*(

*u*), i.e.,

*P*

_{ b }

*P*

_{ T }∥. According to the explanation in Section 3.3, the global-bias diffusion operation first projects the reconstructed residual associated with the downsampling set onto the low-pass filtering subspace, then preserves the non-downsampling set of new signal. Thus, we have \({P_{b}}{P_{T}} = \left ({I - {P_{T}}} \right){{\mathcal {F}}^{- 1}}{P_{w}}{\mathcal {F}}{P_{T}}\). Due to (

*I*−

*P*

_{ T }), \({{\mathcal {F}}^{- 1}}{P_{w}}{\mathcal {F}}\), and

*P*

_{ T }are contraction mapping, it can be seen that \(\left ({I - {P_{T}}} \right){{\mathcal {F}}^{- 1}}{P_{w}}{\mathcal {F}}{P_{T}}\) is contraction mapping, i.e.,

*α*

_{2}denote the proportion of quadratic sum of

*P*

_{ b }

*P*

_{ T }

*f*in the quadratic sum of graph signal

*f*. Therefore,

*α*

_{2}is from 0 to 1. Since

*P*

_{ T }and

*P*

_{ w }are the known parameter, the value of

*α*

_{2}can be obtained according to graph signal and downsampling set. Besides, the value of

*α*

_{2}approximates to \(\left ({\frac {{\left ({N - M} \right)}}{N}\frac {Y}{N}\frac {M}{N}} \right)\), when the graph signal

*f*is very smooth. Then, combining (17), (24) and (25), we have

*I*−

*P*

_{ d }

*P*

_{ T }) is a contraction mapping, if

and Proposition 1 is proved. □

## Notes

## Declarations

### Acknowledgements

The authors would like to thank the editor and the reviewers for constructive comments and suggestions that led to improvements in the manuscript.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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