A mobile network cooperative positioning method based on tensor completion

Abstract

The application provides a mobile network cooperative positioning method based on tensor completion, acquires relative distances between unknown nodes in a wireless sensor network, constructs incomplete and noisy Euclidean distance matrices at different times, obtains Euclidean distance tensors according to the incomplete and noisy Euclidean distance matrices at different times and complete Euclidean distance matrices at different times, completes and denoises the Euclidean distance tensors, obtains denoised and completed Euclidean distance matrices according to the denoised and completed Euclidean distance tensors, obtains relative coordinates of the unknown nodes through multi-dimensional scaling of the denoised and completed Euclidean distance matrices, and obtains global positions of the unknown nodes through coordinate registration of the relative coordinates of the unknown nodes by means of Procrustes analysis, so that the Euclidean distance matrices are accurately recovered, and the positioning accuracy of the unknown nodes is improved.

Description

Technical Field

[0001] This invention relates to the field of signal processing technology, and more specifically to a mobile network cooperative localization method based on tensor completion. Background Technology

[0002] The Global Positioning System (GPS) has gained popularity among experts and scholars due to its wide range of applications and high positioning accuracy, providing relatively reliable positioning services outdoors. However, indoor environments are more complex, and signals encounter obstacles (pedestrians, walls, tables, chairs, etc.) during propagation, causing reflection, refraction, or scattering, resulting in weakened signal strength. Therefore, GPS cannot achieve precise positioning in indoor environments with non-line-of-sight (NLOS) conditions, where positioning accuracy decreases.

[0003] Wireless sensor networks (WSNs) are an emerging distributed information acquisition technology that has been widely applied in environmental monitoring, military monitoring, smart cities, and other fields. In these applications, the location information of sensor nodes is often critical data, directly affecting the accuracy of data processing and the operational efficiency of the network. Therefore, node cooperative localization technology plays an important role in WSNs. However, in many practical application scenarios, such as indoor or urban environments, there are numerous obstacles. In NLoS environments, due to the multipath effect and intensity attenuation of signal propagation, the obtained relative node ranging information, i.e., the Euclidean distance matrix (EDM), will contain errors and omissions. How to improve positioning accuracy is currently one of the hot issues in indoor positioning research. Summary of the Invention

[0004] To address the issue of poor positioning accuracy for unknown nodes in existing NLoS environments, this invention proposes a mobile network cooperative positioning method based on tensor completion.

[0005] To achieve the above-mentioned technical effects, the technical solution of the present invention is as follows:

[0006] A mobile network cooperative localization method based on tensor completion includes the following steps:

[0007] Obtain the relative distances between unknown nodes in a wireless sensor network and construct an incomplete and noisy Euclidean distance matrix at different times;

[0008] The Euclidean distance tensor is obtained by combining the incomplete and noisy Euclidean distance matrices at different times with the complete Euclidean distance matrices at different times.

[0009] The Euclidean distance tensor is completed and denoised, and the denoised and completed Euclidean distance matrix is ​​obtained from the denoised and completed Euclidean distance tensor.

[0010] The relative coordinates of unknown nodes are obtained by multi-dimensional scaling of the denoised and completed Euclidean distance matrix;

[0011] The global position of the unknown node is obtained by coordinate registration through Protodyakonov analysis of the relative coordinates of the unknown node.

[0012] Compared with the prior art, the beneficial effects of the technical solution of the present invention are:

[0013] This invention proposes a mobile network cooperative localization method based on tensor completion. The method combines the current Euclidean distance matrix with the previously recovered and completed Euclidean distance matrix to obtain an Euclidean distance tensor. This Euclidean distance tensor is then completed and denoised. Based on the denoised and completed Euclidean distance tensor, a denoised and completed Euclidean distance matrix is ​​obtained. Multi-dimensional scaling is used to obtain the relative coordinate information of unknown nodes. Finally, known anchor point location information is used for coordinate registration to obtain the accurate location of the unknown nodes. Attached Figure Description

[0014] Figure 1 This is a flowchart illustrating a mobile network cooperative localization method based on tensor completion, as shown in an embodiment of the present invention.

[0015] Figure 2 The figure shows the simulation positioning error results under different algorithms and sparsity at a 5% outlier ratio, as illustrated in the embodiment of the present invention.

[0016] Figure 3 The figure shows the simulation results of the cumulative distribution function of different algorithms for localization error under a 5% outlier ratio and 50% sparsity, as illustrated in the embodiment of the present invention. Detailed Implementation

[0017] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numerals in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the present invention. Rather, they are merely examples of apparatuses and methods consistent with some aspects of the invention as detailed in the appended claims.

[0018] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The singular forms “a,” “the,” and “the” used in this invention and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used herein refers to and includes any or all possible combinations of one or more of the associated listed items.

[0019] It should be understood that although the terms first, second, third, etc., may be used in this invention to describe various information, this information should not be limited to these terms. These terms are only used to distinguish information of the same type from one another. For example, first information may also be referred to as second information without departing from the scope of this invention, and similarly, second information may also be referred to as first information. Depending on the context, the word "if" as used herein may be interpreted as "when," "when," or "in response to a determination."

[0020] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0021] Example 1

[0022] This embodiment proposes a mobile network cooperative localization method based on tensor completion, the flowchart of which is shown below. Figure 1 As shown, it includes the following steps:

[0023] Obtain the relative distances between unknown nodes in a wireless sensor network and construct an incomplete and noisy Euclidean distance matrix at different times;

[0024] The Euclidean distance tensor is obtained by combining the incomplete and noisy Euclidean distance matrices at different times with the complete Euclidean distance matrices at different times.

[0025] The Euclidean distance tensor is completed and denoised, and the denoised and completed Euclidean distance matrix is ​​obtained from the denoised and completed Euclidean distance tensor.

[0026] The relative coordinates of unknown nodes are obtained by multi-dimensional scaling of the denoised and completed Euclidean distance matrix;

[0027] The global position of the unknown node is obtained by coordinate registration through Protodyakonov analysis of the relative coordinates of the unknown node.

[0028] In this embodiment, the cooperative localization problem of dynamic WSNs in an NLoS environment requires the recovery and completion of the Euclidean distance matrix. Based on the accurate Euclidean distance matrix, relative position information can be obtained through multidimensional scale (MDS), and then precise absolute position information can be obtained through coordinate registration. This achieves accurate recovery of the Euclidean distance matrix, thereby enabling high-precision localization of unknown nodes.

[0029] Example 2

[0030] This embodiment further explains the present invention based on Embodiment 1.

[0031] Defining the tensor rank and nuclear norm is crucial for tensor completion techniques to recover incomplete tensors. First, it's necessary to understand the tensor product, with the Discrete Fourier Transform (DFT) playing a central role. Let the tensor... Using the Fast Fourier Transform of the third dimension, we can obtain... Right now Similarly, from the inverse fast Fourier transform get Usually, we use Let represent a block diagonal matrix, where each value on the diagonal is a forward slice of a tensor, i.e.:

[0032]

[0033] Furthermore, the block cyclic matrix of a tensor can also be represented as:

[0034]

[0035] Diagonalizing a circulant matrix using the DFT works similarly for block circulant matrices, and its expression is as follows:

[0036]

[0037] in, Represents the Craméro product. Represents the Fourier transform matrix. and This represents the unit tensor, where the first positive slice is an n1×n1 identity matrix and the remaining slices are all-zero matrices.

[0038] They are orthogonal, hence the definition:

[0039]

[0040] Definition 1 (T-product): Let the tensor and t-product It can be represented as an n1×n4×n3 tensor, that is:

[0041]

[0042] Further derivation yields a matrix multiplication in the frequency domain similar to the t-product, namely:

[0043]

[0044] (3) Calculated from (2), we can then obtain:

[0045]

[0046] Furthermore, the forward slices of tensors have the property represented by (5), which can make the t-product operation more efficient:

[0047]

[0048] Similar to matrix products, tensor products also satisfy the associative law, i.e.

[0049] Definition 2 (Tensor Transpose): Let the tensor The tensor transpose can be obtained by transposing each forward slice and then reversing the order of forward slices 2 to n3.

[0050] Definition 3 (Orthogonal Tensor): If there exists an orthogonal tensor Then it must satisfy in It is a unit tensor.

[0051] Definition 4 (F-diagonal tensor): If every slice of a tensor is a diagonal matrix, it is called an f-diagonal tensor.

[0052] Based on the tensor concept and t-product algorithm described above, tensor singular value decomposition (t-SVD) can be defined as follows:

[0053] Definition 5 (T-SVD): For any It can be decomposed into:

[0054]

[0055] in It is an orthogonal tensor, and It is an f-diagonal tensor.

[0056] The derivation is similar to (4), and we can obtain:

[0057]

[0058] Similarly, The forward slice also satisfies property (5).

[0059] Definition 6 (Tensor Rank): For Its tensor tube rank Defined as t-SVD factorization The number of non-zero tubes, i.e.:

[0060]

[0061] Definition 7 (Tensor nuclear norm): For The tensor nuclear norm is expressed as

[0062] In an optional embodiment, the Euclidean distance tensor is completed by introducing a tensor kernel norm; the tensor kernel norm is expressed as follows:

[0063]

[0064] in, Represents the tensor nuclear norm. Represents a tensor. Let r denote the diagonal tensor, r denote the rank of the tensor tube, i denote the index of the tensor tube, and n3 denote the tensor... The third dimension express The Fast Fourier Transform value of the singular value tensor after singular value decomposition, where j represents the index of the third dimension of the tensor. Represents a block diagonal matrix The matrix nuclear norm, This represents a block diagonal matrix.

[0065] Furthermore, among them, It is a tensor Obtained via t-SVD It was obtained through (7).

[0066] When defining the tensor nuclear norm, only the information from the first positive slice is used, which is also based on the characteristics of t-product and t-SVD.

[0067] make For the measured EDM, Ω is the set of distance measurement information. It is an orthogonal mapping of D, represented as:

[0068]

[0069] Where, d ij (1≤i,j≤m) represents the Euclidean distance between the i-th node and the j-th node.

[0070] In an optional embodiment, the Euclidean distance tensor T is obtained from the incomplete and noisy Euclidean distance matrices at different times and the complete Euclidean distance matrices at different times. D The number of EDMs in a combination is expressed as:

[0071]

[0072] Among them, D t Let represent the incomplete Euclidean distance matrix at time t. This represents the complete Euclidean distance matrix at each time step.

[0073] In an optional embodiment, completing and denoising the Euclidean distance tensor includes recovering the completed tensor from the noisy Euclidean distance tensor and sparsening the noise; its expression is:

[0074]

[0075] Where ε represents the recovered and completed tensor, λ represents the regularization parameter, and ζ represents sparse noise. Represents an orthogonal mapping. This represents the noisy Euclidean distance observation tensor.

[0076] Furthermore, when computing the proximal operators of TNNs, there is also a closed-form solution that serves as the proximal operator for the matrix nuclear norm, namely:

[0077]

[0078] ,in It can also perform t-SVD.

[0079] For any ρ > 0, we define t-SVT as:

[0080]

[0081] in,

[0082]

[0083] Wherein, ρ is defined + As the positive part of ρ, that is, ρ + =max(ρ,0) and the t-SVT operator is related to TNN, that is, it only applies to... Singular values ​​of forward slices A simple soft thresholding was performed, effectively reducing these values ​​to zero. The t-SVT problem can be formulated as:

[0084]

[0085] because The forward slice satisfies property (5), and the soft-thresholding process... This is also true. Using properties (5) and (9), problem (18) can be expressed as:

[0086]

[0087] Property (5) also applies to this problem, making t-SVT calculation more efficient.

[0088] Based on t-SVT, the Alternating Direction Multiplier Method (ADMM) framework is used to solve the problem (14).

[0089] In an optional embodiment, the alternating direction multiplier method framework is used to recover the completed tensor and sparse the noise in a noisy Euclidean distance tensor; its Lagrangian function expression is:

[0090]

[0091] Where ε represents the recovered and completed tensor, λ represents the regularization parameter, and ζ represents sparse noise. Represents an orthogonal mapping. Let f(x) represent the noisy Euclidean distance observation tensor, μ represent the penalty parameter, l1 and l2 represent the Lagrange multipliers, z represent the introduced auxiliary variable, and F represent the norm.

[0092] In an optional embodiment, the recovered complete tensor, sparse noise, introduced auxiliary variables, Lagrange multipliers, and penalty parameters are updated according to the alternating direction multiplier method framework; the expression is as follows:

[0093]

[0094] Where ε represents the recovered and completed tensor, λ represents the regularization parameter, and ζ represents sparse noise. Represents an orthogonal mapping. Let μ represent the noisy Euclidean distance observation tensor, l1 and l2 represent the Lagrange multipliers, and μ represent the penalty parameter. This represents the introduced auxiliary variable, and k represents the number of iterations.

[0095] In an optional embodiment, the introduced auxiliary variable is defined as follows:

[0096]

[0097] Among them, Ω c Defined as the complement of the index values ​​Ω of the observed distance data, representing the index set of the missing data that needs to be filled in, and k represents the number of iterations.

[0098] The SMACOF algorithm achieves its goal by iteratively minimizing the difference between two distance matrices. In each iteration, the algorithm first computes the distance matrix in the lower-dimensional representation and compares it with the higher-dimensional distance matrix. Then, the algorithm optimizes the objective function using an optimization function to ensure that the monotonicity of the distance matrix is ​​maintained when updating the lower-dimensional representation. Next, the algorithm uses methods such as gradient descent to solve the minimization problem until the convergence condition is met.

[0099] The problem that needs to be solved can be described as being based on the already completed and restored EDM. Solving for relative positions This can be achieved by minimizing the following function; its expression is:

[0100]

[0101] in, This represents the elements that have been recovered from the EDM, and is the relative distance between the i-th and j-th nodes. ij This represents the weights of the measured quality; since it has been padded, this matrix is ​​an all-one matrix. The stress function can be expanded as:

[0102]

[0103] The first part is a constant, and the second part is a squared term, which can be simplified to:

[0104]

[0105] Among them, Γ ij This indicates that the diagonal element γ ii =γ jj =1,γ ij =γ ji = -1, and all other elements are 0. tr(·) represents the trace of the matrix. (30) can be further expressed as:

[0106]

[0107] For the third part, using the Cauchy-Schwarz inequality, the following equation holds:

[0108]

[0109] in (33) The equality holds if and only if P = R. From (31), we can obtain:

[0110]

[0111] Furthermore, the third part can be changed as follows:

[0112]

[0113] To solve the non-convex optimization problem in (29), a simpler convex optimization function is needed. get:

[0114]

[0115] It can be further written in the following form:

[0116]

[0117] The elements in V and U(R) are defined as follows:

[0118]

[0119] The solution to problem (28) is obtained by solving... To find the minimum value, we only need to make The derivative value is 0.

[0120] In an optional embodiment, obtaining the relative coordinates of the unknown node from the denoised and completed Euclidean distance matrix through multi-dimensional scaling includes using the SMACOF algorithm to iteratively minimize the difference between two Euclidean distance matrices to obtain the relative coordinates of the unknown node; its expression is:

[0121]

[0122] Where P represents the relative coordinates of the unknown node. V represents a convex optimization function. -1 Let w denote the pseudo-inverse of V, R denote the auxiliary variable introduced by Cauchy's inequality, and V be a symmetric matrix constructed from weights. A symmetric matrix is ​​defined as w. ij Γ ij ...

[0123] Given m anchor points, their global coordinates and relative coordinates are as follows: and Coordinate registration uses the coordinates of m anchor nodes in two coordinate systems to obtain matrix B by translating, scaling, and rotating matrix A.

[0124] In an optional embodiment, the relative coordinates of the unknown node are registered using Protodyakonov analysis; the expression is:

[0125] P g =s p ·P r Q p +T p #(38)

[0126] Among them, P g Represents global coordinates, s p P represents the scaling factor. r Q represents relative coordinates. p Let T represent the rotation matrix. p This represents the translation matrix.

[0127] Furthermore, the coordinate registration problem is modeled as an optimization problem:

[0128] [s p Q p ,T p ] = argmin||s p BQ p +T p -A||#(39)

[0129] Q p It is an orthogonal matrix.

[0130] To simplify the rotation matrix Q p The calculation involves shifting the centers of matrices A and B to the same point, typically the origin. The translated matrices are represented as follows: and in,

[0131]

[0132] The translation matrices can be expressed as follows: and T a T represents the translation matrix that moves the global coordinates of the anchor point to the origin. B This represents the relative translation matrix from the global coordinates of the anchor point to the origin. This represents the coordinates of the anchor point after its global coordinate center has been translated to the origin. This represents the global coordinates of the anchor points, and m represents the number of anchor points. This represents the coordinates of the anchor point after it has been translated relative to the coordinate center to the origin. Let represent the relative coordinates of the anchor point, and j represent the index. Further scaling is needed to eliminate the effects of scaling, so normalization is required. The normalized matrix is ​​represented by and , which can be achieved simply by:

[0133]

[0134] Where A0 represents the normalized global coordinates of the anchor point, and B0 represents the normalized relative coordinates of the anchor point.

[0135] After translation and normalization, in order to solve for Q p (36) can be simplified to:

[0136] Q p =argmin||B0Q p -A0||#(44)

[0137] Furthermore, based on the properties of the matrix trace, we can obtain:

[0138]

[0139] (45) tr(B0) T B0) and tr(A0) T A0) and Q p It doesn't matter; to get its minimum value, we only need to let 2tr(A0)... T B0Q p Take the maximum value for A0. T B0 is obtained by singular value decomposition. and W p It is a diagonal matrix. Therefore, A0 T B0 can be further represented as:

[0140]

[0141] in because Q p U p Both are orthogonal matrices, so H p It is also an orthogonal matrix. Therefore, (46) can be expressed as:

[0142]

[0143] Where, σ i H represents singular values. ii H represents p The main diagonal element. Because σ i Since it is non-negative, (47) to take the maximum value, then h ii Take 1, H p It is an identity matrix.

[0144] In an optional embodiment, the expressions for the rotation matrix, scaling factor, and translation matrix are as follows:

[0145]

[0146] Among them, Q p V represents the rotation matrix. p Represents an orthogonal matrix. U p Transpose of an orthogonal matrix;

[0147] Furthermore, maintain the rotation matrix Q p Without changing the problem, adding the scaling factor s0 to (44) yields the optimization problem shown in (44):

[0148] s0=argmin||s0B0Q p -A0||#(49)

[0149] Solving the rotation matrix Q p Similarly, based on (49), we get:

[0150]

[0151] (50) is a quadratic function with respect to s0 opening upwards. To obtain its minimum, we need to:

[0152]

[0153] Where s0 is The sum of singular values ​​after singular value decomposition.

[0154] From (49) and (51), we can approximately obtain s0B0Q. p ≈A0. Then, from (40)-(43), we get:

[0155]

[0156] Based on (53), the scaling factor s p Translation matrix T p We can obtain the following respectively:

[0157]

[0158] Among them, s p Let represent the scaling factor, and s0 represent the sum of singular values ​​after singular value decomposition. Let A represent the translated matrix. Represents the translated matrix B;

[0159]

[0160] Among them, T p Let T represent the translation matrix. A Let A and T be the translated matrices after decomposition. B Let B represent the matrix after decomposition and translation.

[0161] Furthermore, by obtaining the absolute location information of unknown nodes, high-precision positioning of dynamic node networks in the NLoS environment can be achieved.

[0162] For example, in this invention, function names are represented by bold, cursive English letters, such as... Cursive English letters represent tensors, for example Matrices are represented by uppercase letters, such as T; vectors and elements are represented by lowercase letters, such as t. The focus is on three-dimensional tensors. in Represent real numbers, This represents an imaginary number. For the Fast Fourier Transform, the value transformed from the time domain to the frequency domain is indicated by an overline. For example, Define the index value (i,j,k) as T ijkOr T(i,j,k). T (i) Positioned as a tensor The i-th forward slice, i.e. for The conjugate complex number is represented as In addition, the definition For tensor tubes; Defined as the l1 norm; Let |T| be the F-norm, where |T|·|| denotes the Euclidean norm. * Let T represent the matrix nuclear norm. In a given mobile network scenario, there are N nodes, M anchor points with known location information, and N unknown nodes.

[0163] This embodiment effectively overcomes the problems of high noise and incomplete data in ranging signals under non-line-of-sight (NLOS) environments by using tensor completion and denoising techniques, thus improving positioning robustness. This invention introduces a time-dimensional Euclidean distance tensor and utilizes historical and current ranging information for collaborative optimization, enabling dynamic tracking of node movement trajectories. Compared to static positioning methods, it is more adaptable to changes in the mobile network environment. In terms of positioning technology, it combines multi-dimensional scaling and Protodyakonov analysis, first solving for relative positions through distance-preserving constraints, and then registering anchor nodes to the global coordinate system, avoiding the problem of getting trapped in local optima in direct nonlinear optimization. This invention provides a new technical solution for dealing with NLoS environments in mobile network node positioning, achieving high positioning accuracy even when most ranging data is missing. In dynamic WSNs operating outside of line-of-sight, this paper addresses the problem of incomplete and noisy Euclidean distance matrix completion and restoration. It combines the current Euclidean distance matrix with the previously restored Euclidean distance matrix to obtain a Euclidean distance tensor. Based on the tensor tube rank, a tensor kernel norm is proposed, transforming the tensor completion problem into an optimization problem of minimizing the tensor kernel norm. A high-precision completion and restoration is achieved using the alternating direction multiplier method framework. Multi-dimensional scaling is then employed to obtain relative position information. Finally, known anchor point position information is used for coordinate registration to obtain the precise location of unknown nodes. Compared to traditional matrix completion methods, tensor completion utilizes more data information from the dynamic scene, achieving higher accuracy.

[0164] Example 3

[0165] This embodiment simulates the method proposed in this invention. In a given two-dimensional network space of 30M×30m, there are 95 unknown nodes n, 5 anchor points m, a ranging error anomaly rate of 5%, a refresh rate of 1Hz, and a node speed of 0-2m / s.

[0166] The test results are as follows:

[0167] like Figure 2The results show the simulation positioning error results of different algorithms with different sparsity at a 5% outlier ratio. The method can still estimate the node position with high accuracy even when there are outliers and large sparsity.

[0168] like Figure 3 The results show the simulation positioning error results of the Cumulative Distribution Function (CDF) of different algorithms under a 5% outlier ratio and 50% sparsity. The proposed method can stably reduce noise even in the presence of outliers and with large sparsity, verifying its reliability and robustness.

[0169] The various embodiments in this invention are described in a progressive manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, the device embodiments are basically similar to the method embodiments, so the description is relatively simple; relevant parts can be referred to the descriptions in the method embodiments. The device embodiments described above are merely exemplary. The modules described as separate components may or may not be physically separate. When implementing the present invention, the functions of each module can be implemented in one or more software and / or hardware. Alternatively, some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs.

[0170] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the implementation of the present invention. Those skilled in the art can make other variations or modifications based on the above description. It is neither necessary nor possible to exhaustively describe all embodiments here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A mobile network cooperative localization method based on tensor completion, characterized in that, Includes the following steps: Obtain the relative distances between unknown nodes in a wireless sensor network and construct an incomplete and noisy Euclidean distance matrix at different times; The Euclidean distance tensor is obtained by combining the incomplete and noisy Euclidean distance matrices at different times with the complete Euclidean distance matrices at different times. The Euclidean distance tensor is completed and denoised, and the denoised and completed Euclidean distance matrix is ​​obtained from the denoised and completed Euclidean distance tensor. The relative coordinates of unknown nodes are obtained by multi-dimensional scaling of the denoised and completed Euclidean distance matrix; The global position of the unknown node is obtained by coordinate registration through Protodyakonov analysis of the relative coordinates of the unknown node.

2. The mobile network cooperative localization method based on tensor completion according to claim 1, characterized in that, The Euclidean distance tensor is completed by introducing a tensor nuclear norm; the expression for the tensor nuclear norm is: in, Represents the tensor nuclear norm. Represents a tensor. Let r denote the diagonal tensor, r denote the rank of the tensor tube, i denote the index of the tensor tube, and n3 denote the tensor... The third dimension express The Fast Fourier Transform value of the singular value tensor after singular value decomposition, where j represents the index of the third dimension of the tensor. Represents a block diagonal matrix The matrix nuclear norm, This represents a block diagonal matrix.

3. The mobile network cooperative localization method based on tensor completion according to claim 1, characterized in that, The Euclidean distance tensor is obtained by combining the incomplete and noisy Euclidean distance matrices at different times with the complete Euclidean distance matrices at different times. Its expression is: Among them, D t Let represent the incomplete Euclidean distance matrix at time t. T represents the complete Euclidean distance matrix at each time step. D This indicates the number of EDMs in the combination.

4. The mobile network cooperative localization method based on tensor completion according to claim 1, characterized in that, The process of completing and denoising the Euclidean distance tensor includes recovering the completed tensor from the noisy Euclidean distance tensor and sparsening the noise. Its expression is: Where ε represents the recovered and completed tensor, λ represents the regularization parameter, and ζ represents sparse noise. Represents an orthogonal mapping. This represents the noisy Euclidean distance observation tensor.

5. A mobile network cooperative localization method based on tensor completion according to claim 4, characterized in that, We use the alternating direction multiplier method framework to recover the complete tensor and reduce noise in noisy Euclidean distance tensors; Its Lagrange function expression is: Where ε represents the recovered and completed tensor, λ represents the regularization parameter, and ζ represents sparse noise. Represents an orthogonal mapping. Let μ represent the noisy Euclidean distance observation tensor, μ represent the penalty parameter, and ε1, l2 represent the Lagrange multipliers. This indicates the introduced auxiliary variable, and F represents the norm.

6. The mobile network cooperative localization method based on tensor completion according to claim 5, characterized in that, The alternating direction multiplier method framework is used to update the restored and completed tensor, sparse noise, introduced auxiliary variables, Lagrange multipliers, and penalty parameters; its expression is: Where ε represents the recovered and completed tensor, λ represents the regularization parameter, and ζ represents sparse noise. Represents an orthogonal mapping. Let μ represent the noisy Euclidean distance observation tensor, l1 and l2 represent the Lagrange multipliers, and μ represent the penalty parameter. This represents the introduced auxiliary variable, and k represents the number of iterations.

7. A mobile network cooperative localization method based on tensor completion according to claim 6, characterized in that, The expression for the introduced auxiliary variable is defined as follows: Where ε represents the recovered and completed tensor, and ζ represents sparse noise. Ω represents an orthogonal mapping. c Defined as the complement of the index values ​​Ω of the observed distance data, representing the set of indices of the missing data that needs to be filled in. Let μ represent the noisy Euclidean distance observation tensor, l1 and l2 represent the Lagrange multipliers, and μ represent the penalty parameter. This represents the introduced auxiliary variable, and k represents the number of iterations.

8. The mobile network cooperative localization method based on tensor completion according to claim 1, characterized in that, The relative coordinates of unknown nodes are obtained by multi-dimensional scaling of the denoised and completed Euclidean distance matrix, including using the SMACOF algorithm to iteratively minimize the difference between two Euclidean distance matrices. Its expression is: Where P represents the relative coordinates of the unknown node. V represents a convex optimization function. -1 Let R denote the pseudo-inverse of V, R denote the auxiliary variables in the optimization process, and V denote a symmetric matrix constructed from the weights.

9. A mobile network cooperative localization method based on tensor completion according to claim 1, characterized in that, The relative coordinates of unknown nodes are registered using Protodyakonov analysis; Its expression is: P g =s p ·P r Q p +T p Among them, P g Represents global coordinates, s p P represents the scaling factor. r Q represents relative coordinates. p Let T represent the rotation matrix. p This represents the translation matrix.

10. A mobile network cooperative localization method based on tensor completion according to claim 9, characterized in that, The expressions for the rotation matrix, scaling factor, and translation matrix are as follows: Among them, Q p V represents the rotation matrix. p Represents an orthogonal matrix. U p Transpose of an orthogonal matrix; Among them, s p Let represent the scaling factor, and s0 represent the sum of singular values ​​after singular value decomposition. Let A represent the translated matrix. Represents the translated matrix B; Among them, T p Let T represent the translation matrix. A Let A and T represent the translated matrices after decomposition. B Let B represent the matrix after decomposition and translation.

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