A Model-Driven Multi-UAV Spectrum Map Fast Fusion Method for Strongly Correlated Data Environments ^†

Abstract

Spectrum map fusion has emerged as an effective technique to enhance the accuracy of spectrum map construction. However, many existing fusion methods fail to address the strong correlation between spectrum data, resulting in sub-optimal performance. In this paper, we propose a new multi-unmanned aerial vehicle (UAV) spectrum map fusion method based on differential ridge regression. We first construct spectrum maps of UAVs by using differential features of spectrum data. Next, we present a spectrum map fusion model by leveraging the spatial distribution characteristic of spectrum data. To reduce the sensitivity of the fusion model to the strongly correlated data, a new map fusion regularization term is designed, which introduces $l_2$-norm to constrain the fusion regularization parameters and compress the ridge regression coefficient sizes. As a result, accurate spectrum maps can be constructed for the environments with highly correlated spectrum data. We then formulate a model-driven solution to the spectrum map fusion problem and derive its lower bound. By combining the propagation characteristics of the spectrum signal with the developed Lagrange duality, we can guarantee the convergence of map fusion processing while enhancing the convergence rate. Finally, we propose an accelerated maximally split alternating directions method of multipliers (AMS-ADMM) to reduce the computational complexity of spectrum map construction. Simulation results demonstrate that our proposed method can effectively eliminate external noise interference and outliers, and achieve an accuracy improvement of more than 27% compared to state-of-the-art fusion methods in spectrum map construction with low complexity.

1. Introduction

Spectrum maps provide convenience for spectrum management, interference detection, and planning [1,2,3]. Currently, constructing accurate spectrum maps has attracted a lot of attention. Spectrum map fusion, which can improve the accuracy of spectrum maps, has gradually been widely applied in spectrum data analyzing and processing [4]. Additionally, unmanned aerial vehicles (UAVs) are extensively applied in spectrum data sampling due to their convenience and flexibility [5].

Fusing multiple spectrum maps can effectively mitigate the noises and measurement errors induced by a single spectrum receiver. The authors of [6,7] proposed a weighted fusion method for cooperative spectrum sensing, where they adopted the weighted fusion method to fuse spectrum maps. In [8], the authors periodically used a combination of soft and hard decision-making methods for spectrum map fusion. In [9], the authors employed a compressed sensing approach to solve the problem of three-dimensional compressed spectrum map fusion, and an orthogonal matching tracking algorithm based on three-dimensional subspace was proposed to restore the spectral situation of three-dimensional compressed spectrum map fusion. The authors of [10,11] proposed a data-driven spectrum map fusion methods by learning from experience and a learning-based fusion method (LBFM) for radio occupancy map estimation, respectively. In [12], the authors used a compressed-sensing based method (CSBM) to complete the fusion of spectrum maps. However, spectrum maps constructed by these methods often suffer from performance degradation when spectrum data have strong correlation.

Most current spectrum maps are fused by constructing optimization problems solved by iterative algorithms. For instance, the authors of [13] employed a clustering algorithm and used iterative operations to complete spectrum map fusion. In [14], the authors proposed a Riemannian mean-based data fusion method and employed a gradient descent algorithm to calculate the Riemannian mean for fusing spectrum maps. The authors of [15] fused spectrum maps through the affine projection algorithm and employed delayed input vectors to assist in completing the algorithm iteration process. In [16], the authors exploited the strong directivity of directional antenna and used a channel propagation model to estimate the unknown spectrum data before spectrum map fusion. To demonstrate the effectiveness of their proposed method, they employed the ray-tracing technique to acquire the simulation data of the measurement area. The authors of [17] designed a path planning algorithm for UAVs to implement the three-dimensions (3D) spectrum occupancy measurement for the fusion of spectrum maps. However, most optimization problems of spectrum map fusion often cannot obtain the optimal solution and their computational complexity is very high.

In this paper, we propose a new multi-UAV spectrum map fusion method based on differential ridge regression, which can construct high-accuracy spectrum maps in the electromagnetic environment with strong-correlation data. We first construct a spectrum map fusion model by leveraging the spatial distribution characteristic of spectrum data. Then, we design a new map fusion regularization term to constrain the fusion regularization parameters and compress the ridge regression coefficient sizes. We also derive the lower bound of the spectrum map fusion problem and analyze its convergence. Finally, we propose an accelerated maximally split alternating directions method of multipliers (AMS-ADMM) algorithm to reduce the computational complexity of spectrum map construction. The key contributions are as follows:

• We construct a spectrum map fusion model based on differential ridge regression. By introducing the $l_2$-norm, a new regularization term is designed, which can constrain the map fusion parameters and compress regression coefficient sizes, effectively handling the correlation of spectrum data.
• We exploit the propagation characteristics of the spectrum signal and formulate a model-driven solution for the spectrum map fusion optimization problem. By developing the dual transformation, we derive the lower bound of the optimal value and ensure the convergence of the map fusion optimization problem.
• We design an AMS-ADMM algorithm to decompose the original fusion problem into several subproblems. The subproblems are coordinated to obtain the solution of the original problem, and, thus, the computational complexity is significantly reduced.

Part of this work was presented in [18]. In this paper, we analyze the convergence of the map fusion optimization problem and propose the new AMS-ADMM, which has lower computational complexity than the method used in [18]. Moreover, we consider a real electromagnetic propagation environment in this paper and leverage ray tracing (RT) to construct the ideal SEM for performance evaluation purposes. Multiple UAVs are also used for spectrum data sampling. In contrast, in [18], simulations were only performed in simple open environments.

The remaining part of this paper is arranged as follows. Section 2 presents the process of spectrum map modelling. We complete the construction of the spectrum map fusion model in Section 3. Section 4 describes the design of regularization terms. Simulation results are presented in Section 5 to demonstrate the superior performance of the proposed fusion method. Finally, the conclusion is provided in Section 6.

Notations: a, $\mathbf{a}$ and $\mathbf{A}$ stand for a scalar, a column vector, and a matrix, respectively; ${\mathbf{I}}_K$ represents a $K\times K$ identity matrix, and ${\mathbf{0}}_{M\times K}$ represents an $M\times K$ zero matrix; ${\mathbf{1}}_K$ denotes a vector of ones with a magnitude of K; ${\left[\mathbf{A}\right]}_{i,j}$ is the entry on the i-th row and j-th column of $\mathbf{A}$; ${\left[\mathbf{A}\right]}_{i,:}$ denotes the i-th row of $\mathbf{A}$; the inverse, transpose, and conjugate transpose of $\mathbf{A}$ are represented by ${\mathbf{A}}^{-1}$, ${\mathbf{A}}^T$, and ${\mathbf{A}}^H$, respectively; ${∥\mathbf{A}∥}_{\mathrm{F}}$ and $\mathrm{vec}(\mathbf{A})$ denote the Frobenius norm and vectorization of $\mathbf{A}$, respectively; ⊗, ⊕, and ⋄ denote the Kronecker product, Kronecker sum, and Khatri–Rao product, respectively; and expectation of a random variable is denoted by $\mathbb{E}\left\{·\right\}$.

2. Spectrum Map Modeling

We consider a typical urban area with buildings, vegetation, and vehicles, as shown in Figure 1. Multiple UAVs equipped with spectrum receivers are used to collect spectrum data from several radiation sources (RSs). We divide this area into $V_X\times V_Y$ grids. In this paper, we use the inverse distance weighting (IDW) to estimate the spectrum strength of an unknown grid and construct the spectrum map of the area [19]. This method is adaptable to different scenarios, and can distinguish the difference in spectrum strength at different distances by adjusting the weighted index, effectively constructing the spectrum maps for every UAV [20].

According to the IDW, the weights of the unknown grid are determined by the distance between the unknown grid and the sampled grid. Specifically, the weight decreases as the grid distance increases. Therefore, the spectrum strength value $P_{s,i}$ of the unknown grid can be expressed as

$$Z_{s,0}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}{\omega }_i{Z}_{s,i}}{{\displaystyle \sum _{i=1}^n}{\omega }_i}},$$

(1)

where

$${\omega }_i={\displaystyle \frac{1}{d_{s,i}^p}},$$

(2)

$Z_{s,i}$ denotes the spectrum strength of the known sample grid $s_i(i=1,2,\dots ,N)$, and $d_{s,i}$ denotes the distance between the unknown grid and the known sampling grid. According to (2), weight allocation is mainly dependent on the weighting index p, which controls the decay rate of weight distance. Generally, the spectrum strength value of the unknown grid estimated by this method gradually approaches the strength value of the nearest known grid as the weighting index increases. Therefore, IDW can distinguish the spectrum strength differences effectively, and the spectrum fusion model construction is also smooth enough for further analysis. Figure 1 illustrates the process of the inverse distance weighting method for estimating unknown grid weights. In (2), $d_{s,i}$ can be expressed as

$$d_{s,i}=\sqrt{{(x_0-x_i)}^2+{(y_0-y_i)}^2},$$

(3)

where $(x_i,y_i)$ is the center coordinate of the i-th grid and $(x_0,y_0)$ is center coordinate of the grid to be measured. By applying the above method, we can estimate the spectrum strength of the unknown grid and construct the spectrum maps of UAVs.

3. Spectrum Map Fusion Model

In this section, we analyze the propagation characteristics of signals according to the path loss model to design the spectrum map fusion model [21].

When the signal frequency is in MHz and the distance is in km, assuming the antenna gain of both the transmitting and receiving ends is 1, the path loss model of urban scenarios can be expressed as

$$P_t(\ell )=P_t({\ell }_0)+10\vartheta {log}_{10}({\displaystyle \frac{\ell }{{\ell }_0}})+X_{\sigma },$$

(4)

where $P_t({\ell }_0)$ is the path loss at the reference distance; $\vartheta$ is the path loss exponent; and $X_{\sigma }$ is the shadow fading. In the urban scenarios, considering the attenuation of RSs with the increase of distance during propagation in space, the received signal strength is [22]

$$P={\displaystyle \frac{P_{\mathrm{t}}{G}_{\mathrm{t}}{G}_{\mathrm{r}}{\lambda }^2}{{(4\pi d)}^2}},$$

(5)

where $P_{\mathrm{t}}$ is the transmission power of the RS; $G_{\mathrm{t}}$ and $G_{\mathrm{r}}$ are the gains of transmitter and the receiver antennas, respectively; $\lambda$ is the signal wavelength; and d is the linear distance between the radiation source and UAVs [23].

From (5), we can see that the signal strength decreases with the attenuation of distance and/or frequency. Based on the path loss model (5), we can see that the distribution of spectrum strength near the RSs exhibits aggregation characteristics [24]. Therefore, we can further complete the construction of the spectrum map model.

The spectrum strength of adjacent frequency bands often have strong correlations due to external interference factors, which can cause abnormal regression coefficients or numerical distortions in the fusion model, as well as data redundancy, thus masking key useful information [25]. To demonstrate the effectivity of our method in addressing strong correlations, we employ the Cramer’s V method to measure the correlation between the spectrum data, as given by

$$\varpi =\sqrt{{\displaystyle \frac{{{\chi }_{\tau }}^2}{\iota (\kappa -1)}}}$$

(6)

where the value of $\varpi$ is between 0 and 1; ${\chi }_{\tau }$ is the Chi square test statistic of different scenarios; $\iota$, which can be denoted by $\iota =V_X\times V_Y$, is the number of spectral strength collected by UAVs; and $\kappa =\min \{V_X,V_Y\}$. According to [25], the data has strong correlation when V is greater than 0.5, and the data has weak correlation when V is less than 0.3.

To demonstrate the use of differential ridge regression to handle strong correlations between spectrum data, we plot Figure 2. Figure 2a shows the energy distribution of the raw spectrum data. Then, the raw spectrum data is differential processed in Figure 2b,c, enhancing local mutation features while suppressing stationary background noises and reducing data redundancy [16]. Multi-directional differential combination also improves the applicability of map fusion methods in various environments [26]. The differentiated data often still exhibits feature correlation between adjacent regions, while $l_2$ regularization based on ridge regression can effectively compresses regression coefficients to reduce the impact of data correlation on the stability of the fusion model. As is shown in Figure 2d, the potential noise components can be suppressed and useful signal components are preserved by adjusting the regularization parameter based on ridge regression. After reducing data redundancy through differential processing, the regularization of ridge regression can efficiently suppress local residual interference, improving the accuracy of spectrum map construction.

Based on Figure 2, we design a differential ridge regression regularization term to improve the accuracy of spectrum map fusion. Define ${\mathbf{Q}}_g\in {\mathbb{R}}^{V_X\times V_Y}$ as the g-th spectrum strength data to be fused, where $V_X$ and $V_Y$ denote the numbers of rows and columns of the spectrum strength data matrix, respectively. Then, ${\mathbf{a}}_g$ $(g=1,2,\cdots ,G)$ is defined as the g-th column vector of the fused spectrum strength data and ${\mathbf{q}}_g$ as the vectorization of ${\mathbf{Q}}_g$. Subsequently, the spectrum strength matrix can be given as

$$\mathbf{A}=[{\mathbf{a}}_{\mathbf{1}}^{\mathbf{T}},{\mathbf{a}}_{\mathbf{2}}^{\mathbf{T}},\cdots ,{\mathbf{a}}_{\mathbf{G}}^{\mathbf{T}}].$$

(7)

Fusing the spectrum maps of different UAVs can effectively suppress abnormal data in the constructed spectrum map [27]. Spectrum maps are supplemented to address potential anomalies and interference. We employ $l_2$-norm to measure the difference, as given by

$${∥\mathbf{A}∥}_2=\sum _m^M\varphi ({∥{\mathbf{A}}_m∥}_2),$$

(8)

where ${\mathbf{A}}_m$ is the m-th row vector of $\mathbf{A}$, $M=V_X{V}_Y$, and $\varphi (·)$ is an indicative function, which can be expressed as

$$\varphi ({∥{\mathbf{A}}_m∥}_2)=\left\{\begin{array}{c}1,\mathrm{if}{∥{\mathbf{A}}_{\mathbf{m}}∥}_2\succ 0\hfill \\ 0,\mathrm{else}\hfill \end{array}.\right.$$

(9)

We construct the loss function as 

$$\underset{{\mathbf{a}}_g}{min}\sum _g^G{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2.$$

(10)

We proceed to impose constraints on (10), as given by

$${∥\mathbf{A}∥}_2\le \sigma ,$$

(11)

where $\sigma$ is the tolerance factor. Therefore, we have the optimization problem as

$$\begin{array}{c}{\displaystyle \underset{{\mathbf{a}}_g}{\min }}{\displaystyle \sum _g^G}{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2,\hfill \\ s.t.{∥\mathbf{A}∥}_2\le \mu .\hfill \end{array}$$

(12)

To ensure the rationality of the optimization problem calculation results, we perform developed Lagrange dual transformation, which can be written as

$$L({\mathbf{a}}_g,\nu )=\sum _g^G{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2+\nu ({∥\mathbf{A}∥}_{2,1}-\mu ),$$

(13)

where $\nu$ is called dual variable. We perform dual transformation on (13), as given by

$$\begin{array}{cc}\hfill g({\mathbf{a}}_g,\nu )& =infL({\mathbf{a}}_g,\nu )\hfill \\ & =inf({\displaystyle \sum _g^G}{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2+\nu ({∥\mathbf{A}∥}_{2,1}-\mu )).\hfill \end{array}$$

(14)

Lemma 1.

The function obtained from the developed dual transformation of convex optimization problems based on our spectrum map fusion determines the lower bound of the optimal value of the original problem.

Proof. 

Consider a common optimization problem, as given by

$$\begin{array}{c}\min f_0(x),\hfill \\ s.t.f_i(x)\le 0,i=1,2,\dots ,m,\hfill \\ h_i(x)=0,i=1,2,\dots ,p,\hfill \end{array}$$

(15)

where the domain of the independent variable is $x\in {\mathbb{R}}^n$. We abbreviate the domain as $dom(·)$. Assume that $D={\displaystyle \bigcap _{i=0}^m}dom(f_i)\cap {\displaystyle \bigcap _{i=0}^p}dom(h_i)$, which is a non empty set. $p^{*}$ is the optimal solution value. The Lagrange function of the optimization problem can be written as

$$L(x,\lambda ,\nu )=f_0(x)+\sum _{i=1}^m{\lambda }_i{f}_i(x)+\sum _{i=1}^p{\nu }_i{h}_i(x).$$

(16)

The domain of (16) is $dom(L)=D\times {\mathbb{R}}^m\times {\mathbb{R}}^p$. ${\lambda }_i$ is the Lagrange multiplier corresponding to the i-th inequality constraint $f_i(x)\le 0$. ${\nu }_i$ is the Lagrange multiplier corresponding to the i-th equality constraint $h_i(x)=0$. Vectors $\lambda$ and $\nu$ are called dual variables. Operate the dual transformation on (16), and we have

$$\begin{array}{c}g(\lambda ,\nu )=\underset{x\in \rho }{\inf }L(x,\lambda ,\nu )\hfill \\ =\underset{x\in \rho }{\inf }(f_0(x)+{\displaystyle \sum _{i=1}^m}{\lambda }_i{f}_i(x)+{\displaystyle \sum _{i=1}^p}{\nu }_i{h}_i(x).\hfill \end{array}$$

(17)

A random $\tilde{x}$ selected from the domain satisfies the constraint condition, and therefore we have

$$\sum _{i=1}^m{\lambda }_i{f}_i(x)+\sum _{i=1}^p{\nu }_i{h}_i(x)\le 0.$$

(18)

In (18), the first term on the left is non positive and the second term is zero. According to (18), we have

$$\begin{array}{c}L(\tilde{x},\lambda ,\nu )\hfill \\ =f_0(\tilde{x})+{\displaystyle \sum _{i=1}^m}{\lambda }_i{f}_i(\tilde{x})+{\displaystyle \sum _{i=1}^p}{\nu }_i{h}_i(\tilde{x})\le f_0(\tilde{x}).\hfill \end{array}$$

(19)

Therefore, (18) can be rewritten as

$$g(\lambda ,\nu )=\underset{x\in \rho }{inf}L(x,\lambda ,\nu )\le L(\tilde{x},\lambda ,\nu )\le f_0(\tilde{x}).$$

(20)

According to (20), $g(\lambda ,\nu )\le f_0(\tilde{x})$ holds for any feasible $\tilde{x}$, which also means $g(\lambda ,\nu )\le p^{*}$ holds. Thus, the dual function forms the lower bound of the optimal value of the original problem. The proof is complete.    □

4. Design of Regularization Term for Differential Ridge Regression

In this section, we use the differential characteristics of spectrum data to design a regularization term based on differential ridge regression and apply the regularization term to the fusion model, providing convergence solutions [28].

After analyzing the relationship between received signal strength and propagation distance, we calculate the first and second partial derivatives of received signal strength $P_r$ with respect to propagation distance d, as 

$${\displaystyle \frac{\partial (P_r)}{\partial d}}={\displaystyle \frac{\partial (P_t{G}_t{G}_r{\lambda }^2/{(4\pi d)}^2)}{\partial d}}=-2{\displaystyle \frac{P_t{G}_t{G}_r{\lambda }^2}{{(4\pi )}^2{d}^3}},$$

(21)

$${\displaystyle \frac{{\partial }^2(P_r)}{\partial d^2}}={\displaystyle \frac{\partial (-2P_t{G}_t{G}_r{\lambda }^2/{(4\pi )}^2{d}^3)}{\partial d}}=6{\displaystyle \frac{P_t{G}_t{G}_r{\lambda }^2}{{(4\pi )}^2{d}^4}}.$$

(22)

According to (21), the partial derivative of d with respect to $P_r$ is always negative, and the spectrum strength is inversely proportional to the third power of the propagation distance. Meanwhile, (22) indicates that the second partial derivative of d to $P_r$ is positive, implying that the change rate of spectrum strength $P_r$ with distance d continues to increase. Since a partial derivative is always negative, the change rate of spectrum strength with distance gradually approaches 0. Therefore, the rate of attenuation of spectrum strength slows down with the increasing distance. In other words, the data differences in the spectrum map exhibit variability and sparsity around the transmitter antenna [29]. Thus, we use ridge regression to address the sparsity of the spectrum map and compensate for the incompleteness of the collected spectrum strength data through regularization. In addition, the introduction of regularization in ridge regression helps improve the fitting and generalization ability of spectrum map fusion models.

On the other hand, we obtain the optimal value of the fusion model through computation. The proposed developed Lagrange duality can precisely combine mathematical models and optimization algorithms. We often get stuck in local optima during the process of solving the fusion model, while the developed Lagrange duality can ensure that the local optimum is the global optimum. By combining the differential derivation formula of spectrum strength with the original spectrum map, we can see that the spectrum strength value is much greater than zero, and the signal strength decays slowly with the increasing distance. Therefore, there is sparsity in the difference between adjacent signal strengths. To describe the difference in strength between adjacent spectra, we perform differential processing on the spectrum strength vector ${\mathbf{Q}}_g$, expressed as

$$\Delta {\mathbf{q}}_g=\left\{\begin{array}{c}{\mathbf{q}}_{g,m+1}-{\mathbf{q}}_{g,m},\mathrm{if}1\le m\le M-1,\hfill \\ {\mathbf{q}}_{g,1}-{\mathbf{q}}_{g,M},\mathrm{if}m=M.\hfill \end{array}\right.$$

(23)

Substituting (23) into (12), we have

$$\begin{array}{c}\underset{{\mathbf{a}}_g}{\min }{\displaystyle \sum _g^G}{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2,\hfill \\ s.t.{\displaystyle \sum _m^M}\sqrt{{\displaystyle \sum _g^G}{(\Delta {\mathbf{A}}_{m,g})}^2}\le {\phi }_1,\hfill \end{array}$$

(24)

where ${\phi }_1$ represents the tolerance factor. For the sake of computational convenience, we can instead solve the optimization problem to obtain the values of the independent variables corresponding to the optimal values, written as

$$\underset{{\mathbf{a}}_g}{arg\min }\sum _g^G{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2+\eta \sum _m^M{∥\Delta {\mathbf{A}}_m∥}_2.$$

(25)

In (25), $\eta$ is the regularization parameter of differential ridge regression for balancing the loss function and constraining the sparsity of the function.

Then, the Lagrange augmented function of the objective function can be written as

$$\begin{array}{c}{L}_{\rho }({\mathbf{q}}_g,{\mathbf{a}}_g,{\mathbf{A}}_m)={\displaystyle \sum _g^G}{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2+\eta {\displaystyle \sum _m^M}{∥\Delta {\mathbf{A}}_m∥}_2\hfill \\ +{\displaystyle \frac{\rho }{2}}{∥{\displaystyle \sum _m^M}\sqrt{{\displaystyle \sum _g^G}{(\Delta {\mathbf{A}}_{m,g})}^2}-{\phi }_1∥}_2.\hfill \end{array}$$

(26)

For the sake of computational convenience, we simplify the variables in (26) by reducing some of the parameters as follows

$$u=\sum _m^M\sqrt{\sum _g^G{(\Delta {\mathbf{A}}_{m,g})}^2}-{\phi }_1,$$

(27)

and then (26) can be simplified as

$$\begin{array}{c}{L}_{\rho }({\mathbf{q}}_g,{\mathbf{a}}_g,{\mathbf{A}}_m)={\displaystyle \sum _g^G}{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2\hfill \\ +\eta {\displaystyle \sum _m^M}{∥\Delta {\mathbf{A}}_m∥}_2+{\displaystyle \frac{\rho }{2}}{∥u∥}_2.\hfill \end{array}$$

(28)

We proceed to perform dual transformation on (28), which can be written as

$$\begin{array}{c}g({\mathbf{a}}_g,\nu )=inf{L}_{\rho }({\mathbf{q}}_g,{\mathbf{a}}_g,{\mathbf{A}}_m)\hfill \\ =inf{\displaystyle \sum _g^G}{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2\hfill \\ +\eta {\displaystyle \sum _m^M}{∥\Delta {\mathbf{A}}_m∥}_2+{\displaystyle \frac{\rho }{2}}{∥u∥}_2.\hfill \end{array}$$

(29)

Define k as the number of iterations and k as a natural number. We use $\alpha$ and $\gamma$ to constrain the calculation process. (In most cases, $\alpha =100$, $\gamma =0.01$ [30].) Repeat the following operation until convergence is reached.

$$\begin{array}{c}{\mathbf{a}}_g^{k+1}=\underset{{\mathbf{a}}_g}{arg\min }({\displaystyle \sum _g^G}{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2\hfill \\ +{\displaystyle \frac{\rho }{2}}{∥{\displaystyle \sum _m^M}{∥\Delta {\mathbf{A}}_m∥}_2+u^k∥}_2);\hfill \end{array}$$

(30)

$$\begin{array}{c}{\mathbf{A}}_m^{k+1}=\underset{{\mathbf{q}}_g}{arg\min }({\displaystyle \sum _g^G}{∥{\mathbf{q}}_g-{\mathbf{I}}_{\mathbf{M}}{\mathbf{a}}_g∥}_2^2\hfill \\ +{\displaystyle \frac{\rho }{2}}{∥{\displaystyle \sum _m^M}{∥\Delta {\mathbf{A}}_m∥}_2+u^k∥}_2);\hfill \end{array}$$

(31)

$$u^{k+1}=u^k+{\mathbf{a}}_g^{k+1}+{\mathbf{A}}_m^{k+1}.$$

(32)

After the iteration is completed, we input the final result of the Algorithm 1 into (29) and use the properties of Lagrange duality to determine the lower bound to verify whether the operation result meet the constraints, which ensures the rationality of the operation result.

Algorithm 1 Spectrum Map Fusing

Input: Spectrum maps matrix to be fused ${\mathbf{Q}}_g(g=1,2,\dots ,G),$ regularization parameter $\rho$, iteration counter $k=0$, tolerance factor ${\phi }_1$

Output: Spectrum map fusion result

Initialization: $\mathbf{A}=[{\mathbf{a}}_{\mathbf{1}}^{\mathbf{T}},{\mathbf{a}}_{\mathbf{2}}^{\mathbf{T}}\cdots {\mathbf{a}}_{\mathbf{3}}^{\mathbf{T}}],$ ${\mathbf{A}}_m=1,$$(m=1,2,\dots ,M)$

Update ${\mathbf{a}}_g$ using (30)

Update ${\mathbf{A}}_m$ using (31)

Update u using (32)

if $k\succ \alpha$ do

q = $q-\gamma$

restart the procedure

end if

Return the final values of ${\mathbf{a}}_g,{\mathbf{A}}_m,u$.

5. An Accelerated Maximally Split ADMM

In this section, we propose the accelerated split ADMM to solve the optimization problem.

5.1. ADMM for Solving the Fusion Problem

We note that in this paper, since we use UAVs to collect spectrum data, the data distribution is uniform, and we can obtain a convexity of the objective function. In the case of non-uniform distribution, we use traditional iterative methods for fusion directly. Then, we consider the general convex optimization problem

$$\underset{w}{\mathrm{minimize}f}(\mathbf{H}\mathit{w}-t)+g(\mathit{w})$$

(33)

where $\mathrm{H}\in {\mathbb{R}}^{M\times N}$ is the node data matrix, $w\in {\mathbb{R}}^N$ is the model weight vector, $t\in {\mathbb{R}}^M$ is the target output vector, $f(·)$ is a convex loss function, and $g(·)$ is a separable convex regularization function. Partition the node data matrix as $\mathbf{H}=[{\mathbf{H}}_1,{\mathbf{H}}_2,\dots ,{\mathbf{H}}_P]$ and the weight vector as $\begin{array}{cc}w& ={[{\mathit{w}}_1^T,{\mathit{w}}_2^T,\dots ,{\mathit{w}}_P^T]}^T\end{array}$ with ${\mathbf{H}}_p\in {\mathbf{R}}^{M\times n_p}$, ${\mathit{w}}_p\in {\mathbf{R}}^{n_p}$, and $n_1+n_2+\cdots +n_P=N$, and rewrite the regularization function as $g(\mathit{w})=g_1({\mathit{w}}_1)+g_2({\mathit{w}}_2)+\cdots +g_P({\mathit{w}}_P)$. After introducing $\mathrm{P}$ auxiliary variable vectors, $v_p={\mathbf{H}}_p{w}_p$ with $p=1,2,\dots ,P$, which are compacted as $\mathbf{V}=[v_1,v_2,\dots ,v_P]$, the convex optimization problem (32) is then transformed into

$$\begin{array}{c}\underset{\mathit{w},\mathbf{V}=[{\mathit{v}}_1,\dots ,{\mathit{v}}_P]}{\min }f\left({\displaystyle \sum _{p=1}^P}{\mathit{v}}_p-\mathit{t}\right)+{\displaystyle \sum _{p=1}^P}{g}_p({\mathit{w}}_p)\hfill \\ \mathrm{subject}\mathrm{to}{\mathbf{H}}_p{\mathbf{w}}_p={\mathit{v}}_p,p=1,2,\dots ,P\hfill \end{array}$$

(34)

Let k denote the iteration number; a variable with superscript k or $k+1$ represents the estimate of that variable in the kth or $(k+1)$th iteration. The ADMM for the P-partitioned model fitting problem (34) is described as

$$\begin{array}{c}{\mathit{w}}_p^{k+1}=arg{\displaystyle \underset{{\mathit{w}}_p}{\min }}\left\{g_p({\mathit{w}}_p)+{\displaystyle \frac{\rho }{2}}\left|\right|{\mathbf{H}}_p{\mathit{w}}_p-v_p^k+{\mathit{u}}_p^k{\left|\right|}_2^2\right\}\\ {\mathbf{V}}^{k+1}\left.=arg{\displaystyle \underset{v}{\min }}\left\{\begin{array}{c}f(v_1+v_2+\cdots +v_P-\mathit{t})\\ +{\displaystyle \frac{\rho }{2}}\sum _{p=1}^P\left|\right|{\mathbf{H}}_p{\mathit{w}}_p^{k+1}-v_p+{\mathit{u}}_p^k{\left|\right|}_2^2\end{array}\right.\right\}\\ u_p^{k+1}={\mathit{u}}_p^k+{\mathbf{H}}_p{\mathit{w}}_p^{k+1}-v_p^{k+1}\end{array}$$

(35)

where $\rho >0$ is a penalty parameter, and $u_1,u_2,\dots ,u_P$ are scaled Lagrangian multiplier vectors, and $p=1,2,\dots ,P$ in (35). We refer to the ADMM (35) as a P-partition ADMM.

Introduce the average auxiliary variable vector v, the average scaled multiplier vector u, and the average model output vector y as

$$\begin{array}{cc}& \mathit{v}=P^{-1}({\mathit{v}}_1+{\mathit{v}}_2+\cdots +{\mathit{v}}_P)\hfill \\ \hfill & \mathit{u}=P^{-1}({\mathit{u}}_1+{\mathit{u}}_2+\cdots +{\mathit{u}}_P)\hfill \\ \hfill & \mathit{y}=P^{-1}({\mathbf{H}}_1{\mathit{w}}_1+{\mathbf{H}}_2{\mathit{w}}_2+\cdots +{\mathbf{H}}_P{\mathit{w}}_P)=P^{-1}\mathbf{H}\mathit{w}.\hfill \end{array}$$

(36)

Problem (35) can be rewritten as

$$\underset{v,v_1,\dots ,v_P}{\mathrm{minimize}}\left\{f(\mathbf{Pv}-\mathit{t})+{\displaystyle \frac{\rho }{2}}\sum _{p=1}^P{∥{\mathbf{H}}_p{\mathit{w}}_p^{k+1}-{\mathbf{v}}_p+{\mathbf{u}}_p^k∥}_2^2\right\}$$

(37)

which is subject to the constraint (36).

Minimizing (37) over $v_1,v_2,\dots ,v_P$ with fixed v has the solution

$${\mathit{v}}_p-{\mathbf{H}}_p{\mathit{w}}_p^{k+1}-{\mathit{u}}_p^k=\mathit{v}-{\mathit{y}}^{k+1}-{\mathit{u}}^k.$$

(38)

Therefore, the V-update in (35) can be computed by solving

$${\mathit{v}}^{k+1}=arg\underset{\mathit{v}}{\min }\left\{f(P\mathit{v}-\mathit{t}))+{\displaystyle \frac{\rho }{2}}P{\parallel {\mathit{y}}^{k+1}-\mathit{v}+{\mathit{u}}^k\parallel }_2^2\right\}$$

(39)

Apply (38) for $v_1,v_2,\dots ,v_P$. The P-partition ADMM (35) for problem (34) is then simplified as

$$\begin{array}{cc}& {\mathit{w}}_p^{k+1}=arg\underset{{\mathit{w}}_p}{\min }\left\{g_p({\mathit{w}}_p)+{\displaystyle \frac{\rho }{2}}\right.∥{\mathbf{H}}_p({\mathit{w}}_p-{\mathit{w}}_p^k)\hfill \\ \hfill & -{{\mathit{v}}^k+{\mathit{y}}^k+{\mathit{u}}^k∥}_2^2\hfill \\ \hfill & y^{k+1}=P^{-1}\mathbf{H}{\mathit{w}}^{k+1}\hfill \\ \hfill & v^{k+1}=arg\underset{v}{\min }\left\{f(P\mathit{v}-\mathit{t}))+{\displaystyle \frac{\rho }{2}}P{∥{\mathit{y}}^{k+1}-\mathit{v}+{\mathit{u}}^k∥}_2^2\right\}\hfill \\ \hfill & u^{k+1}=u^k+y^{k+1}-v^{k+1}\hfill \end{array}$$

(40)

where $p=1,2,\dots ,P$ in (40).

5.2. An Accelerated Maximally Split ADMM for Fusion Problem Solving

As shown above, we take $P=N$ and $n_p=1,\forall p$. In this case, the block weight vector wp reduces to a weight scalar ${\mathit{w}}_{\mathit{p}}$ and the block data matrix $H_p$ reduces to a column data vector $h_p$. We assume that $h_n=0,\forall n=1,2,\dots ,N.$ The P-partition ADMM (40) for the general convex model fitting problem (34) then becomes

$$\begin{array}{cc}& w_n^{k+1}=arg\underset{w_n}{\min }\left\{g_n(w_n)+{\displaystyle \frac{\rho }{2}}\right.∥{\mathit{h}}_n\left(w_n-w_n^k\right)\hfill \\ \hfill & -{{\mathit{v}}^k+{\mathit{y}}^k+{\mathit{u}}^k∥}_2^2\hfill \\ \hfill & {\mathit{y}}^{k+1}=\overline{\mathbf{H}}{\mathit{w}}^{k+1}\hfill \\ \hfill & {\mathit{v}}^{k+1}=arg\underset{\mathit{v}}{\min }\left\{f\left(N(\mathit{v}-\overline{\mathit{t}})\right)+{\displaystyle \frac{\rho }{2}}N\parallel {\mathit{y}}^{k+1}-\mathit{v}+{\mathit{u}}^k{\parallel }_2^2\right\}\hfill \\ \hfill & u^{k+1}=u^k+y^{k+1}-v^{k+1}\hfill \end{array}$$

(41)

where $\overline{\mathbf{H}}=N^{-1}\mathbf{H},\overline{t}=N^{-1}t$, and $n=1,2,\dots ,N$ in (41).

It is obvious that the optimization over the weight vector w has been maximally split into univariate subproblems in (41).

If the loss function $f(v)$ is also decomposable with respect to the elements of v, the optimization problem (41) can also be split into univariate subproblems.

For the fusion problem, the loss function and regularization function of the model fitting problem (34) are expressed as

$$f(\mathit{v})={\displaystyle \frac{1}{2}}{\left|\right|\mathit{v}\left|\right|}_2^2,g_n(w_n)={\displaystyle \frac{1}{2}}{\gamma }_n^2{w}_n^2.$$

(42)

With (42), the $w_n$-minimization and the v-minimization in (41), respectively, can be rewritten as

$$\begin{array}{cc}& w_n^{k+1}=arg\underset{w_n}{\min }\left\{{\gamma }_n^2{w}_n^2+\rho {∥{\mathit{h}}_n\left(w_n-w_n^k\right)+{\mathit{y}}^k+{\mathit{u}}^k-{\mathit{v}}^k∥}_2^2\right\}\hfill \\ \hfill & v^{k+1}=arg\underset{v}{\min }\left\{{∥v-\overline{t}∥}_2^2+N^{-1}\rho {∥y^{k+1}-v+u^k∥}_2^2\right\}.\hfill \end{array}$$

(43)

After analytically solving (43) for $w_n^{k+1}$ and $v_n^{k+1}$, an accelerated maximally split for the fusion problem, which is denoted by the proposed solution, is obtained as

$$\begin{array}{cc}\hfill w_n^{k+1}& ={\displaystyle \frac{{∥{\overline{\mathit{h}}}_n∥}_2^2{w}_n^k-N^{-1}{\overline{\mathit{h}}}_n^T\left(\overline{\mathbf{H}}{\mathit{w}}^k+{\mathit{u}}^k-{\mathit{v}}^k\right)}{N^{-1}{\overline{\rho }}^{-1}{\overline{\gamma }}_n^2+{∥{\overline{\mathit{h}}}_n∥}_2^2}}\hfill \\ \hfill {\mathit{v}}^{k+1}& ={(1+\overline{\rho })}^{-1}\left[\overline{t}+\overline{\rho }\left(\overline{\mathbf{H}}{\mathit{w}}^{k+1}+{\mathit{u}}^k\right)\right]\hfill \\ \hfill u^{k+1}& =u^k+\overline{\mathbf{H}}{w}^{k+1}-v^{k+1}\hfill \end{array}$$

(44)

where ${\overline{\mathit{h}}}_n=N^{-1}{\mathit{h}}_n,\overline{\rho }=N^{-1}\rho ,$ and ${\overline{\gamma }}_n=N^{-1}{\gamma }_n$.

Then, the $w_n$-update reduces to

$$w_n^{k+1}=w_n^k-N^{-1}{∥{\overline{\mathit{h}}}_n∥}_2^{-2}{\overline{\mathit{h}}}_n^T\left(y^k+{\mathit{u}}^k-{\mathit{v}}^k\right),$$

(45)

The update for v and u remain the same as (44). In order to make the algorithm easier to solve the fusion problem, we have

$$f(·)={\displaystyle \frac{1}{2}}{∥\left[\begin{array}{c}\mathbf{H}\\ {\Gamma }^{1/2}\end{array}\right]\mathit{w}-\left[\begin{array}{c}t\\ {\mathbf{0}}_N\end{array}\right]∥}_2^2\mathrm{and}g(·)=0,$$

(46)

In the derivation of (46), $f(·)$ and $g(·)$ are taken as

$$f(\mathbf{H}\mathit{w}-\mathit{t})={\displaystyle \frac{1}{2}}\left|\right|\mathbf{H}\mathit{w}-{\mathit{t}\left|\right|}_2^2.$$

(47)

$$g(\mathit{w})={\displaystyle \frac{1}{2}}{\mathit{w}}^T\Gamma \mathit{w}$$

(48)

The decomposability of the cost function has been sufficiently utilized. Using (47) rather than (46) is significant, as it leads to a more efficient solution.

Then, we replace the fixed gain factor $N^{-1}$ with a tunable parameter $\overline{\alpha }$. This simple modification was shown to have caused an obvious increase in the convergence speed of the algorithm. Motivated by this modification, we replace the fixed scalar factor $N^{-1}$ in both the numerator and denominator of the $wn$-update in (44) by a tunable parameter $\overline{\alpha }$, which can be written as

$$\begin{array}{cc}& w_n^{k+1}={\displaystyle \frac{{∥{\overline{\mathit{h}}}_n∥}_2^2}{\overline{\alpha }{\overline{\rho }}^{-1}{\overline{\gamma }}_n^2+{∥{\overline{\mathit{h}}}_n∥}_2^2}}{w}_n^k-\overline{\alpha }{\displaystyle \frac{{\overline{\mathit{h}}}_n^T\left(\overline{\mathbf{H}}{\mathit{w}}^k+{\mathit{u}}^k-{\mathit{v}}^k\right)}{\overline{\alpha }{\overline{\rho }}^{-1}{\overline{\gamma }}_n^2+{∥{\overline{\mathit{h}}}_n∥}_2^2}}\hfill \\ \hfill & v^{k+1}={(1+\overline{\rho })}^{-1}\left[\overline{t}+\overline{\rho }\left(\overline{\mathbf{H}}{\mathit{w}}^{k+1}+{\mathit{u}}^k\right)\right]\hfill \\ \hfill & u^{k+1}=u^k+\overline{\mathbf{H}}{w}^{k+1}-v^{k+1}\hfill \end{array}$$

(49)

where $n=1,2,\dots ,N$ in (49).

After eliminating the multiplier vector u from (48), the fusion problem becomes

$$\begin{array}{cc}& w_n^{k+1}={\displaystyle \frac{{∥{\overline{\mathit{h}}}_n∥}_2^2{w}_n^k-\overline{\alpha }{\overline{\mathit{h}}}_n^T\left[\overline{\mathbf{H}}{\mathit{w}}^k-\left(1-{\overline{\rho }}^{-1}\right){\mathit{v}}^k-{\overline{\rho }}^{-1}\overline{\mathit{t}}\right]}{\overline{\alpha }{\overline{\rho }}^{-1}{\overline{\gamma }}_n^2+{∥{\overline{\mathit{h}}}_n∥}_2^2}}\hfill \\ \hfill & {\mathit{v}}^{k+1}={(1+\overline{\rho })}^{-1}\left[\overline{\rho }\overline{\mathbf{H}}{\mathit{w}}^{k+1}+{\mathit{v}}^k\right]\hfill \end{array}$$

(50)

where $n=1,2,\dots ,N$ in (50).

5.3. Convergence Analysis and Parameter Selection of the Accelerated Maximally Split ADMM

Based on the convergence analysis, optimal parameters of the algorithm for a particular regularization scheme are derived. We introduce the following equations to assist in analyzing the convergence of the fusion problem, as given by

$$\left\{\begin{array}{cc}\hfill {\overline{w}}^k& ={\overline{\mathbf{D}}}^{-1}{\mathit{w}}^k,{\overline{\mathit{v}}}^k=\overline{\mathbf{D}}{\overline{\mathbf{H}}}^T{\mathit{v}}^k\hfill \\ \hfill \mathrm{E}& =\mathrm{diag}\{{\overline{e}}_1,{\overline{e}}_2,\dots ,{\overline{e}}_N\}\hfill \\ \hfill {\overline{e}}_n& ={\left[\overline{\alpha }{\overline{\rho }}^{-1}{\overline{\gamma }}_n^2+\left|\right|{\overline{h}}_n{\left|\right|}_2^2\right]}^{-1}\left|\right|{\overline{h}}_n{\left|\right|}_2^2>0\hfill \\ \hfill \mathrm{D}& =\mathrm{diag}\{{\overline{d}}_1,{\overline{d}}_2,\dots ,{\overline{d}}_N\}\hfill \\ \hfill {\overline{d}}_n& =\sqrt{{\left[\overline{\alpha }{\overline{\rho }}^{-1}{\overline{\gamma }}_n^2+\left|\right|{\overline{h}}_n{\left|\right|}_2^2\right]}^{-1}\overline{\alpha }>0}\hfill \\ \hfill \overline{\mathrm{G}}& =\overline{\mathbf{D}}{\overline{\mathbf{H}}}^T\overline{\mathbf{H}}\overline{\mathbf{D}}\hfill \end{array}\right.$$

(51)

Thus, (49) can be rewritten as

$$\begin{array}{cc}& {\overline{\mathit{w}}}^{k+1}=\left(\overline{\mathbf{E}}-\overline{\mathbf{G}}\right){\overline{\mathit{w}}}^k+\left(1-{\overline{\rho }}^{-1}\right){\overline{\mathit{v}}}^k+{\overline{\rho }}^{-1}\overline{\mathbf{D}}{\overline{\mathbf{H}}}^T\overline{\mathit{t}}\hfill \\ \hfill & {\overline{\mathit{v}}}^{k+1}={(1+\overline{\rho })}^{-1}\left[\overline{\rho }\overline{\mathbf{G}}{\overline{\mathit{v}}}^{k+1}+{\overline{\mathit{v}}}^k\right].\hfill \end{array}$$

(52)

The fixed-point ${[{({\overline{\mathit{w}}}^{*})}^T,{({\overline{\mathit{v}}}^{*})}^T]}^T$ of the iterative algorithm (51) satisfies

$$\begin{array}{cc}& {\overline{\mathit{w}}}^{*}=\left(\overline{\mathbf{E}}-\overline{\mathbf{G}}\right){\overline{\mathit{w}}}^{*}+\left(1-{\overline{\rho }}^{-1}\right){\overline{\mathit{v}}}^{*}+{\overline{\rho }}^{-1}\overline{\mathbf{D}}{\mathbf{H}}^T\overline{\mathit{t}}\hfill \\ \hfill & {\overline{\mathit{v}}}^{*}={(1+\overline{\rho })}^{-1}\left[\overline{\rho }\overline{\mathbf{G}}{\overline{\mathit{w}}}^{*}+{\overline{\mathit{v}}}^{*}\right].\hfill \end{array}$$

(53)

Eliminating ${\overline{v}}^{*}$ from (52), and we can get

$$[\overline{\rho }\left(\mathbf{I}-\overline{\mathbf{E}}\right)+\overline{\mathbf{G}}]{\overline{\mathit{w}}}^{*}=\overline{\mathbf{D}}{\overline{\mathbf{H}}}^T\overline{\mathit{t}}.$$

(54)

Calculating the difference between $\mathrm{E}$ and ${\overline{d}}_n$ in (50), we obtain

$$[diag\left({\overline{\gamma }}_1^2,{\overline{\gamma }}_2^2,\dots ,{\overline{\gamma }}_N^2\right)+{\overline{\mathbf{H}}}^T\overline{\mathbf{H}}]\overline{\mathbf{D}}{\overline{\mathit{w}}}^{*}={\overline{\mathbf{H}}}^T\overline{\mathit{t}}.$$

(55)

From (54) we obtain ${\overline{\mathit{w}}}^{*}={\overline{\mathbf{D}}}^{-1}{\overline{\mathit{w}}}^{*}$ with $w^{*}$ in (50). Furthermore, inserting ${\overline{\mathit{w}}}^{*}$ into (52) gets us ${\overline{v}}^{*}=\overline{\mathbf{G}}{\overline{w}}^{*}=\overline{\mathbf{G}}{\overline{\mathbf{D}}}^{-1}{\overline{w}}^{*}$.

With the fixed-point ${[{(w^{*})}^T,{(v^{*})}^T]}^T$, we rewrite the (44) as

$$\begin{array}{cc}& {\tilde{\mathit{w}}}^{k+1}=\left(\overline{\mathbf{E}}-\overline{\mathbf{G}}\right){\tilde{\mathit{w}}}^k+\left(1-{\overline{\rho }}^{-1}\right){\tilde{\mathit{v}}}^k\hfill \\ \hfill & {\tilde{\mathit{v}}}^{k+1}={\displaystyle \frac{\overline{\rho }}{1+\overline{\rho }}}\overline{\mathbf{G}}(\overline{\mathbf{E}}-\overline{\mathbf{G}}){\tilde{\mathit{w}}}^k+{\displaystyle \frac{1}{1+\overline{\rho }}}\left[\mathbf{I}-(1-\overline{\rho })\overline{\mathbf{G}}\right]{\tilde{\mathit{v}}}^k\hfill \end{array}$$

(56)

where

$${\tilde{\mathit{w}}}^k={\overline{\mathit{w}}}^k-{\overline{\mathit{w}}}^{*},{\tilde{\mathit{v}}}^k={\overline{\mathit{v}}}^k-{\overline{\mathit{v}}}^{*}.$$

(57)

The linear dynamical system can be written in the matrix-vector form, as

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\tilde{\mathit{w}}}^{k+1}\\ {\tilde{\mathit{v}}}^{k+1}\end{array}\right]& =\mathbf{Q}\left[\begin{array}{c}{\tilde{\mathit{w}}}^k\\ {\tilde{\mathit{v}}}^k\end{array}\right]\mathrm{with}\hfill \\ \hfill \mathbf{Q}& ={\displaystyle \frac{1}{1+\overline{\rho }}}\left[\begin{array}{cc}(1+\overline{\rho })(\overline{\mathbf{E}}-\overline{\mathbf{G}})& (\overline{\rho }-{\overline{\rho }}^{-1})\mathbf{I}\\ \overline{\rho }\overline{\mathbf{G}}(\overline{\mathbf{E}}-\overline{\mathbf{G}})& \mathbf{I}-(1-\overline{\rho })\overline{\mathbf{G}}\end{array}\right].\hfill \end{array}$$

(58)

Based on the above derivation, we are led to the conclusion that the fixed-point ${[{({\overline{\mathit{w}}}^{*})}^T,{({\overline{\mathit{v}}}^{*})}^T]}^T$ of (50) is

$$\left[\begin{array}{c}{\overline{\mathit{w}}}^{*}\\ {\overline{v}}^{*}\end{array}\right]=\left[\begin{array}{c}{\overline{\mathbf{D}}}^{-1}\\ \overline{\mathbf{G}}{\overline{\mathbf{D}}}^{-1}\end{array}\right]{\mathit{w}}^{*}$$

(59)

where

$$w^{*}={\left[\begin{array}{c}{\mathbf{H}}^T\mathbf{H}+\Gamma \end{array}\right]}^{-1}({\mathbf{H}}^{T}t)$$

(60)

We proceed to analyze the relationship between the convergence of fusion problem and the convergence ratio ${\mu }_{max}$, as given by

$$\underset{k\to \infty }{lim}{∥\left[\begin{array}{c}{\tilde{\mathit{w}}}^k\\ {\tilde{\mathit{v}}}^k\end{array}\right]∥}_{\mathbf{S}}^{-1}{∥\left[\begin{array}{c}{\tilde{\mathit{w}}}^{k+1}\\ {\tilde{\mathit{v}}}^{k+1}\end{array}\right]∥}_{\mathbf{S}}={\mu }_{\max }$$

(61)

The spectrum radius ${\mu }_{max}$ of matrix Q is small, where S is some positive definite matrix.

Next, we execute the optimal parameters and practical parameter selection. It is seen from the above analysis that the convergence ratio of (49) depends on the algorithm parameters $\overline{\alpha }$ and $\overline{\rho }$. Theoretically, the optimal values of $\overline{\alpha }$ and $\overline{\rho }$ for the fastest convergence can be determined by minimizing the spectrum radius ${\mu }_{max}$ of matrix Q over $\overline{\alpha }$ and $\overline{\rho }$. However, direct minimization of ${\mu }_{max}$ through eigenvalues of Q is very time consuming in general.

Fortunately, for the case with regularization factors, we obtain the succinct convergence condition (58). In addition, a concise analytic expression (59) for the spectrum radius ${\mu }_{max}$ of matrix Q in terms of the data matrix G, model parameter $\beta$, and algorithm parameters $\overline{\alpha }$ and $\overline{\rho }$ can also be determined. From (58) and (59), the optimal algorithm parameters ${\overline{\alpha }}^{*}$ and ${\overline{\rho }}^{*}$ for given data matrix and regularization factors can be quickly obtained by minimizing ${\mu }_{max}$ in (45) over $\overline{\alpha }$ and $\overline{\rho }$, subject to constraint (58).

We introduce

$$\tilde{\alpha }=v_N\overline{\alpha },\tilde{\beta }=v_N^{-1}\beta ,\chi =v_N^{-1}{v}_1.$$

(62)

With some algebra, it is known that ${\mu }_{max}$ is a function of $\overline{\alpha },\overline{\rho },\chi ,\mathrm{and}\overline{\tilde{\beta }}$, i.e., ${\mu }_{\max }(v_N,v_1,$ $\beta ,$ $\overline{\alpha },$ $\overline{\rho })={\tilde{\mu }}_{\max }(\chi ,\tilde{\beta },\overline{\alpha },\overline{\rho }).$

Then, for the fusion problem with ${\gamma }_n^2=\beta {\parallel {\mathit{h}}_n\parallel }_2^2$, the optimal parameters ${\overline{\alpha }}^{*}$ and ${\overline{\rho }}^{*}$ of (49) are determined by

$$\begin{array}{cc}\hfill \left(v_N{\overline{\alpha }}^{*},{\overline{\rho }}^{*}\right)& =\left({\tilde{\alpha }}^{*},{\overline{\rho }}^{*}\right)=arg\underset{(\tilde{\alpha },\overline{\rho })}{\min }{\tilde{\mu }}_{max}\left(\chi ,\tilde{\beta },\tilde{\alpha },\overline{\rho }\right)\hfill \\ \hfill \mathrm{subject}\mathrm{to}3\tilde{a}& <(2+\overline{\rho })\left(2+{\overline{\rho }}^{-1}\tilde{\alpha }\tilde{\beta }\right),\tilde{\alpha }>0,\overline{\rho }>0\hfill \end{array}$$

(63)

the smallest convergence ratio is

$${\mu }_{max}^{*}={\mu }_{max}\left(v_N,v_1,\beta ,{\overline{\alpha }}^{*},{\overline{\rho }}^{*}\right).$$

(64)

The optimal parameter ${\overline{\rho }}^{*}$ of (42), i.e., (49) with $\overline{\alpha }=N^{-1}$, is

$$\begin{array}{cc}\hfill {\overline{\rho }}^{*}& =arg\underset{\overline{\rho }}{\min }{\tilde{\mu }}_{max}\left(\chi ,\tilde{\beta },N^{-1}{v}_N,\overline{\rho }\right)\hfill \\ \hfill \mathrm{subject}\mathrm{to}3\widetilde{v_N}& <(2+\overline{\rho })\left(2N+{\overline{\rho }}^{-1}{v}_N\tilde{\beta }\right),\overline{\rho }>0\hfill \end{array}$$

(65)

and the smallest convergence ratio is

$${\mu }_{max}^{*}={\mu }_{max}\left(v_N,v_1,\beta ,N^{-1},{\overline{\rho }}^{*}\right).$$

(66)

By (62) and (64), the optimal algorithm parameters ${\overline{\alpha }}^{*}$ and ${\overline{\rho }}^{*}$ are functions of the fusion model parameters $\chi$ and $\tilde{\beta }$. We can compute the function values in advance on a 2-D dense grid of $(\chi ,\tilde{\beta })$, and then look up the optimal algorithm parameters very quickly for specific problems in application. For the fusion problem with general regularization factors, the algorithm parameters ${\overline{\alpha }}^{*}$ and ${\overline{\rho }}^{*}$ computed by (62) and (64) are not optimal. However, they are often good candidate parameters with which (49) converges fast, or they may be used as initial values in optimization of the algorithm parameters $\overline{\alpha }$ and $\overline{\rho }$.

5.4. Complexity Analysis

The AMS-ADMM algorithm is used in this article to simplify the complexity of the differential ridge regression algorithm and improve its computational efficiency. The core complexity of spectrum construction comes from the complexity of the AMS-ADMM algorithm used in the spectrum map fusion algorithm. Since the spectrum strength vectors before and after fusion are denoted with ${\mathbf{q}}_g$ and ${\mathbf{a}}_g$, respectively, the computational complexity of the spectrum map fusion model is $O(M_{SMFM})=O(N_{{\mathbf{q}}_g}^2+N_{{\mathbf{q}}_g}{N}_{{\mathbf{a}}_g}+N_{{\mathbf{a}}_g}^2)$. The use of the proposed AMS-ADMM algorithm reduces the complexity to $O(M_{AMS})=O(N_{{\mathbf{q}}_g}^2+N_{{\mathbf{a}}_g}^2)$, while the algorithms using direct calculation have to take the spectrum matrices into account and have a complexity of $O(M_{DC})=O(N_{{\mathbf{q}}_g}^2+N_{{\mathbf{q}}_g}{N}_{{\mathbf{a}}_g}+N_{{\mathbf{A}}_m}+N_{{\mathbf{a}}_g}^2)$. Obviously, our algorithm effectively reduces computational complexity compared to algorithms using direct calculation.

Next, we compare the computational complexity between the proposed algorithm and other ADMM algorithms. Assuming the same input data, the main computation burden relies on evaluating and performing a sort operation for ${\mathit{v}}^k$. According to (41), the computational complexity of ADMM relies on an input signal strength vector whose complexity is $O(M_{ADMM})=O(N_{{\mathbf{q}}_g}^2+N_{{\mathbf{q}}_g}{N}_{{\mathbf{a}}_g}+N_{{\mathbf{a}}_g}^2)$. On the basis of [31], the computational complexity of the Tikhonov regularization method is $O(k{N}_1^2{N}_2^2{G}^2)$. According to [32], the computational complexity of the adaptative LASSO algorithm is $O(k{N}_1^2{N}_2^2{G}^2+N_1^2{G}^2)$. In comparison, the computational complexity of our AMS-ADMM algorithm is lower than these methods.

6. Simulation Results

In this section, we perform simulations to demonstrate the performance of the proposed spectrum map fusion method. Simulations are conducted in an area of 900 m × 900 m, which is divided into 160 × 160 grids and each grid has a side length of 5 m. We conduct spectrum map fusion in two typical scenarios with different data correlation and frequecies. In Scenario 1, the correlation of spectrum strength data is $\varpi =0.71>0.5$, so the spectrum strength data has strong correlation. In Scenario 2, the correlation of spectrum strength data is $\varpi =0.22<0.3$, indicating that the data has weak correlation. The signal power, signal center frequency, signal bandwidth, and signal positions of two typical scenarios are listed in

Table 1. Simulation parameter settings.

Scenarios	Radiation Source	Antenna Type	Signal Power	Signal Center Frequency	Signal Bandwidth	Position Index
Scenario 1	Radiation Source 1	Directional	10 dBm	2.45 GHz	100 MHz	(135, 45)
	Radiation Source 2	Omni	15 dBm	2.45 GHz	100 MHz	(50, 52)
	Radiation Source 3	Omni	10 dBm	2.45 GHz	100 MHz	(165, 162)
	Radiation Source 4	Omni	20 dBm	2.45 GHz	100 MHz	(92, 118)
	Radiation Source 5	Omni	15 dBm	2.45 GHz	100 MHz	(25, 160)
Scenario 2	Radiation Source 6	Omni	15 dBm	4.00 GHz	90 MHz	(29, 98)
	Radiation Source 7	Omni	20 dBm	4.00 GHz	90 MHz	(37, 165)
	Radiation Source 8	Omni	10 dBm	4.00 GHz	90 MHz	(110, 118)
	Radiation Source 9	Directional	15 dBm	4.00 GHz	90 MHz	(137, 52)
	Radiation Source 10	Omni	10 dBm	4.00 GHz	90 MHz	(181, 93)

.

Figure 3a–c show the spectrum map without noises, the spectrum map with SNR = 20 dB, and the fused spectrum map in Scenario 1, respectively. We can see that the method proposed in this paper effectively suppresses the noise interference and significantly improves the accuracy of the spectrum map by fusing the spectrum maps with strong data correlation. Due to the characteristic of differential ridge regression in preventing signal fluctuations, the fused spectrum map appears smooth. Comparing the spectrum maps in Figure 3b,c, fusing multiple spectrum maps by our method can reduce the construction root mean square error (RMSE) by over 30%. The proposed method introduces a differential regularization term to penalize sudden and discontinuous changes in spectrum strength during the fusion process of spectral maps, and thereby can enhance the smoothness and consistency of the fusion map while avoiding the noise overfitting in the spectrum strength data.

Figure 4a–c show the ideal spectrum map of Scenario 2, the spectrum map with SNR = 20 dB, and the fused spectrum map, respectively. By comparing Figure 3 with Figure 4, we can see that the method proposed in this paper can also suppress noise interference and improve the accuracy of the fused maps in an electromagnetic environment with weak correlation. This is because the proposed new regularization terms can constrain the map fusion parameters and compress regression coefficient sizes, effectively handling the correlation of spectrum data. We can also see that, although the frequencies, source locations, and source power in Scenario 2 are different from those in Scenario 1, our method can still achieve satisfactory performance in spectrum map construction. The reason is that we exploit the propagation characteristics of the spectrum signal to constrcut the spectrum map fusion model. This model suppress the local noise interferences in different electromagnetic environments.

We evaluate the performance of the fusion method numerically by calculating RMSE in two scenarios, as shown in Figure 5 and Figure 6. Figure 5a,b show the RMSEs of the constructed spectrum map versus the number of spectrum receivers. Comparing Figure 5a,b, we can see that the method proposed in this paper provided exceptional performance in spectrum map fusion in two different scenarios. This is because employing the proposed regularization term in differential ridge regression can improve the generalization ability of the fusion model. Figure 5 also shows that the spectrum map construction accuracy improves as the number of spectrum receivers increases, but the accuracy increases slowly when the number of spectrum receivers exceeds five. The reason is that the noises and measurement errors induced by a single receiver can be mitigated as the number of receivers increases.

We also compare the proposed method with the improved weighted average method (IWAM) [6], the group sparsity method (GSM) [33], the least absolute shrinkage and selection operator method (LASSOM), the learning-based fusion method (LBFM) [11], and the compressed-sensing-based method [12]. Figure 6a,b show the spectrum map construction RMSEs of five methods versus the SNRs. We can also see that our method outperform other methods in spectrum map construction. This is because the method proposed in this paper involves a differential ridge regression regularization term to suppress external interference, which can deal with complex constraint conditions through convex optimization derivation. The addition of regularization terms based on differential ridge regression improves the accuracy in fusing spectrum data with collinearity of features. Additionally, the proposed fusion method employ the developed Lagrange duality to solve the optimal value of the path loss function based on spectrum construction. By using our method, we can ensure the convergence of optimization problems and suppress the noises and abnormal interference in the process of spectrum map construction, thus improving the spectrum map accuracy.

7. Conclusions

This paper proposed a multi-UAV spectrum map fusion method based on differential ridge regression. Relying on the analysis of coefficient characteristics of spectrum data after differential processing, the fusion of spectrum maps was achieved by combining differential ridge regression with convex optimization derivation. A differential ridge regression regularization term was designed to suppress anomalous data in the spectrum map and handle the correlation of spectrum data, meanwhile providing a numerical solution. The simulation results showed that the method proposed in this paper achieved higher accuracy in fusing spectrum maps compared to existing spectrum map fusion methods. Further, our method can be employed to complete construction of spectrum maps in interference environments and UAV-assisted construction of high-precision spectrum maps. In future work, we plan to study the mathematical relationship between the operation costs and spectrum map construction accuracy, and estimate the optimal receiver quantity required for specific construction accuracy.