Mobile network cooperative positioning method based on tensor completion

Abstract

The invention provides a mobile network cooperative positioning method based on tensor completion, and the method comprises the steps: obtaining a relative distance between unknown nodes in a wireless sensor network, and constructing an incomplete noisy Euclidean distance matrix at different moments; obtaining an Euclidean distance tensor according to the incomplete and noisy Euclidean distance matrix at different moments and the complete Euclidean distance matrix at different moments; the Euclidean distance tensor is complemented and denoised, and a denoised and complemented Euclidean distance matrix is obtained according to the denoised and complemented Euclidean distance tensor; performing multi-dimensional scale scaling on the de-noised and completed Euclidean distance matrix to obtain relative coordinates of an unknown node; the global position of the unknown node is obtained after the relative coordinates of the unknown node are subjected to coordinate registration through Print analysis, accurate recovery of the Euclidean distance matrix is achieved, and therefore the positioning accuracy of the unknown node is improved.

Description

Technical Field

[0001] The present invention relates to the field of signal processing technology, and more specifically to a mobile network collaborative positioning method based on tensor completion. Background Art

[0002] The Global Positioning System (GPS) has been favored by experts and scholars for its wide range of applications and high positioning accuracy. It provides relatively reliable positioning services outdoors. However, due to the complex indoor environment, signal propagation may encounter obstacles (pedestrians, walls, tables and chairs, etc.), causing signal reflection, refraction, or scattering, resulting in weakening of signal intensity. Therefore, GPS cannot achieve accurate positioning. Specifically, positioning accuracy is reduced in indoor environments with non-line-of-sight (NLOS).

[0003] Wireless sensor networks (WSNs) are an emerging distributed information collection technology that has been widely used in fields such as environmental monitoring, military surveillance, and smart cities. 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 collaborative localization technology plays a vital 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 relative distance information obtained from nodes, namely the Euclidean distance matrix (EDM), can contain errors and omissions. Improving positioning accuracy is currently a hot topic in indoor positioning research. Summary of the Invention

[0004] In order to solve the problem of poor positioning accuracy of unknown nodes in the existing NLoS environment, the present invention proposes a mobile network collaborative positioning method based on tensor completion.

[0005] In order to achieve the above technical effects, the technical solutions of the present invention are as follows:

[0006] A mobile network collaborative positioning 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 based on the incomplete and noisy Euclidean distance matrix at different times and the complete Euclidean distance matrix at different times;

[0009] The Euclidean distance tensor is completed and denoised, and the denoised and completed Euclidean distance matrix is ​​obtained according to the denoised and completed Euclidean distance tensor;

[0010] The denoised and completed Euclidean distance matrix is ​​scaled to obtain the relative coordinates of unknown nodes through multi-dimensional scaling;

[0011] The relative coordinates of the unknown nodes are aligned through Proctor analysis to obtain the global position of the unknown nodes.

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

[0013] The present invention proposes a mobile network collaborative positioning method based on tensor completion. The Euclidean distance matrix at the current moment is combined with the previously restored and completed Euclidean distance matrix to obtain a Euclidean distance tensor. The Euclidean distance tensor is completed and denoised. A denoised and completed Euclidean distance matrix is ​​obtained based on the denoised and completed Euclidean distance tensor. The relative coordinate information of the unknown node is obtained by multi-dimensional scaling. Finally, the known anchor point position information is used for coordinate alignment to obtain the precise unknown node position. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 This is a flow chart of a mobile network collaborative positioning method based on tensor completion according to an embodiment of the present invention.

[0015] Figure 2 This is a diagram showing the simulation positioning error results of different algorithms with different sparsity at a 5% outlier ratio according to an embodiment of the present invention.

[0016] Figure 3 This is a graph showing the cumulative distribution function simulation positioning error results of different algorithms under a 5% outlier ratio and 50% sparsity according to an embodiment of the present invention. DETAILED DESCRIPTION

[0017] Exemplary embodiments will be described in detail herein, examples of which are illustrated in the accompanying drawings. In the following description, when referring to the drawings, like numbers in different figures represent like or similar elements unless otherwise indicated. The embodiments described in the following exemplary embodiments are not intended to represent all possible embodiments consistent with the present invention. Rather, they are merely examples of apparatus and methods consistent with certain aspects of the present invention, as detailed in the appended claims.

[0018] The terms used in this invention are for the purpose of describing specific embodiments only and are not intended to limit the invention. The singular forms "a," "the," and "the" used in this invention and the appended claims are also intended to include 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 the present invention to describe various information, such information should not be limited to these terms. These terms are merely used to distinguish information of the same type from one another. For example, first information may also be referred to as second information, and similarly, second information may also be referred to as first information, without departing from the scope of the present invention. Depending on the context, the term "if" as used herein may be interpreted as "when," "when," or "in response to determining."

[0020] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0021] Example 1

[0022] This embodiment proposes a mobile network collaborative positioning method based on tensor completion, the flow chart of which is as follows: Figure 1 As shown, the following steps are included:

[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 based on the incomplete and noisy Euclidean distance matrix at different times and the complete Euclidean distance matrix at different times;

[0025] The Euclidean distance tensor is completed and denoised, and the denoised and completed Euclidean distance matrix is ​​obtained according to the denoised and completed Euclidean distance tensor;

[0026] The denoised and completed Euclidean distance matrix is ​​scaled to obtain the relative coordinates of unknown nodes through multi-dimensional scaling;

[0027] The relative coordinates of the unknown nodes are aligned through Proctor analysis to obtain the global position of the unknown nodes.

[0028] In this embodiment, the collaborative positioning of dynamic WSNs in an NLoS environment requires the restoration of the complete Euclidean distance matrix. Based on the precise Euclidean distance matrix, relative position information can be obtained through multidimensional scaling (MDS), and then accurate absolute position information can be obtained through coordinate registration. Accurate restoration of the Euclidean distance matrix enables high-precision positioning of unknown nodes.

[0029] Example 2

[0030] This example further explains the present invention based on Example 1.

[0031] How to define the tensor rank and nuclear norm is the key to tensor completion technology to restore incomplete tensors. First, we need to understand the tensor product, in which the Discrete Fourier Transform (DFT) plays a core role. Using the fast Fourier transform of the third dimension we can get Right now Similarly, by inverse fast Fourier transform from get Usually, we use represents a block diagonal matrix, where each value on the diagonal is a positive slice of the tensor, that is:

[0032]

[0033] In addition, the block circulant matrix of a tensor can also be expressed as:

[0034]

[0035] Use DFT to diagonalize the circulant matrix. The block circulant matrix is ​​similar. Its expression is:

[0036]

[0037] in, represents the Cramer-Rao product, represents the Fourier transform matrix, and Represents the unit tensor, where the first forward slice is the n1×n1 unit matrix and the remaining slices are all-zero matrices.

[0038] are orthogonal, as defined by:

[0039]

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

[0041]

[0042] Further deduction can obtain matrix multiplication similar to t product in the frequency domain, namely:

[0043]

[0044] (3) is calculated by (2), and then we can get:

[0045]

[0046] In addition, the forward slice of the tensor has the property expressed in (5), which can make the t-product operation more efficient:

[0047]

[0048] Similar to the matrix product, the tensor product also satisfies the associativity, that is

[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 is an orthogonal tensor Then it must satisfy in is the identity tensor.

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

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

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

[0054]

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

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

[0057]

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

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

[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 core norm; the tensor core norm expression is:

[0063]

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

[0065] Further, among them, is a tensor Obtained by t-SVD, is obtained through (7),

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

[0067] make is the measured EDM, Ω is the set of ranging information, is an orthogonal mapping of D, expressed as:

[0068]

[0069] Among them, 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, a Euclidean distance tensor is obtained based on an incomplete and noisy Euclidean distance matrix at different times and a complete Euclidean distance matrix at different times, T D Indicates the number of EDMs in the combination, and its expression is:

[0071]

[0072] Among them, D t represents the incomplete Euclidean distance matrix at time t, Represents the complete Euclidean distance matrix at each moment.

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

[0074]

[0075] Among them, ε represents the restored complement tensor, λ represents the regularization parameter, and ζ represents the sparse noise. represents an orthogonal mapping, A tensor representing noisy Euclidean distance observations.

[0076] Furthermore, to calculate the proximal operator of TNN, there is also a closed-form solution as the proximal operator of 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] Here, we define ρ + As the positive part of ρ, that is, ρ + =max(ρ,0) and the t-SVT operator is related to TNN, that is, it is only for The singular values ​​of the forward slice of A simple soft thresholding is performed, effectively reducing these values ​​to zero. The t-SVT problem can be formulated as:

[0084]

[0085] because The forward slice of satisfies property (5), and the soft threshold processing The same is also satisfied. Using properties (5) and (9), problem (18) can be expressed as:

[0086]

[0087] For this problem, property (5) is also applicable, making the t-SVT calculation more efficient.

[0088] Based on t-SVT, the alternating direction method of multipliers (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 complete tensor and sparse the noise in the noisy Euclidean distance tensor; its Lagrangian function expression is:

[0090]

[0091] Among them, ε represents the restored complement tensor, λ represents the regularization parameter, and ζ represents the sparse noise. represents an orthogonal mapping, represents the noisy Euclidean distance observation tensor, μ represents the penalty parameter, l1, l2 represent the Lagrange multiplier, z represents the introduced auxiliary variable, and F represents the norm.

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

[0093]

[0094] Among them, ε represents the restored complement tensor, λ represents the regularization parameter, and ζ represents the sparse noise. represents an orthogonal mapping, represents the noisy Euclidean distance observation tensor, μ represents the penalty parameter, l1, l2 represent the Lagrange multiplier, represents the introduced auxiliary variable, and k represents the number of iterations.

[0095] In an optional embodiment, the auxiliary variable definition expression introduced is:

[0096]

[0097] Among them, Ω c It is defined as the complement of the index value Ω of the observed distance data, which represents the index set of missing data that needs to be completed, and k represents the number of iterations.

[0098] The SMACOF algorithm works by iteratively minimizing the difference between two distance matrices. In each iteration, the algorithm first computes the distance matrix for the lower-dimensional representation and compares it to the higher-dimensional distance matrix. The algorithm then optimizes the objective function using an optimization function to ensure that the distance matrix maintains monotonicity while updating the lower-dimensional representation. The algorithm then solves the minimization problem using methods such as gradient descent until convergence is achieved.

[0099] The problem to be solved can be expressed as: based on the EDM that has been fully restored, Solving for relative position This can be achieved by minimizing the following function; its expression is:

[0100]

[0101] in, Indicates the element in the recovered EDM, which is the relative distance between the i-th and j-th nodes. ij is the weight representing the measurement quality. Since it has been completed, the matrix is ​​a full 1 matrix. The stress function can be expanded as:

[0102]

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

[0104]

[0105] Among them, Γ ij Denotes the diagonal elements γ ii =γ jj =1,γ ij =γ ji = -1, and the rest of the 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] In addition, the third part can be changed to:

[0112]

[0113] In order to solve the non-convex optimization problem in (29), a simpler convex optimization function is found get:

[0114]

[0115] It can be further written as follows:

[0116]

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

[0118]

[0119] The solution to problem (28) is to solve The minimum value of The derivative value of is 0.

[0120] In an optional embodiment, the denoised and completed Euclidean distance matrix is ​​scaled by multidimensional scaling to obtain the relative coordinates of the unknown node, including using a SMACOF algorithm to iteratively minimize the difference between two Euclidean distance matrices to obtain the relative coordinates of the unknown node; the expression is:

[0121]

[0122] Among them, P represents the relative coordinates of the unknown node, represents the convex optimization function, V -1 represents the pseudo-inverse of V, R represents the auxiliary variable introduced by the Cauchy inequality, V is a symmetric matrix constructed by weights, and is defined as w ij Γ ij . .

[0123] The global coordinates and relative coordinates of the known m anchor points are 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 nodes are aligned by using Proctor analysis; the expression is:

[0125] P g =s p ·P r Q p +T p #(38)

[0126] Among them, P g represents the global coordinate, s p represents the scaling factor, P r Indicates relative coordinates, Q p represents the rotation matrix, T p 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] where Q p is an orthogonal matrix.

[0130] To simplify the rotation matrix Q p Calculation, move the center of matrix A and matrix B to the same point, usually the origin of the coordinate system. The matrices after translation are expressed as and in,

[0131]

[0132] The translation matrices can be expressed as and T a Represents the translation matrix of the anchor point's global coordinates to the origin, T B Represents the relative translation matrix of the global coordinates of the anchor point to the origin, Indicates the coordinates of the global coordinate center of the anchor point after translation to the origin. represents the global coordinates of the anchor point, m represents the number of anchor points, Indicates the coordinates of the anchor point after it is translated to the origin relative to the coordinate center. Represents the relative coordinates of the anchor point, j represents the index, and further elimination of the scaling effect is required, so normalization is required. The normalized matrix is ​​represented by and, simply:

[0133]

[0134] Among them, A0 represents the normalized global coordinate of the anchor point, and B0 represents the normalized relative coordinate of the anchor point.

[0135] After translation and normalization, in order to solve Q p Simplify (36) to:

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

[0137] According to the properties of the matrix trace, we can get:

[0138]

[0139] (45) in tr(B0 T B0) and tr(A0 T A0) and Q p It doesn’t matter. To get the minimum value, we only need to make 2tr(A0 T B0Q p ) takes the maximum value. T B0 is decomposed into singular values ​​to obtain and W p is a diagonal matrix. So A0 T B0 can be further expressed as:

[0140]

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

[0142]

[0143] Among them, σ i represents the singular value, h ii Indicates H p The main diagonal elements of . Since σ i is non-negative, (47) to take the maximum value, then h ii Take 1, H p is the identity matrix.

[0144] In an optional embodiment, the expressions of the rotation matrix, the scaling factor, and the translation matrix are respectively:

[0145]

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

[0147] Furthermore, keep the rotation matrix Q p If the scaling factor s0 is added to (44), the optimization problem shown in (44) can be obtained:

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

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

[0150]

[0151] (50) is a quadratic function with an upward opening about s0. To obtain the minimum, we need to:

[0152]

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

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

[0155]

[0156] Based on (53), the scaling factor s p and the translation matrix T p We can get:

[0157]

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

[0159]

[0160] Among them, T p represents the translation matrix, T A Represents the decomposed and translated matrix A, T B represents the decomposed and translated matrix B.

[0161] Furthermore, the absolute position information of unknown nodes is obtained, and high-precision positioning of dynamic node networks in NLoS environments is achieved.

[0162] For example, bold cursive English letters are used to represent function names in the present invention, for example Curly English letters represent tensors, for example Matrices are represented by capital letters, such as T; vectors and elements are represented by lowercase letters, such as t. The focus is on three-dimensional tensors. in represents a real number, Indicates an imaginary number. For the fast Fourier transform, the value after conversion from the time domain to the frequency domain is marked with an upper horizontal line. For example, Define (i, j, k) index value as T ijkOr T(i,j,k). T (i) Positioned as a tensor The i-th forward slice of for The conjugate complex number of is expressed as In addition, define For tensor tube; Defined as the l1 norm; is the F norm, where ||·|| represents the Euclidean norm, ‖T‖ * Represents the matrix nuclear norm of T. In a given mobile network scenario, there are N nodes, M anchor points with known location information, and N unknown nodes.

[0163] In this embodiment, the tensor completion and denoising technology is used to effectively overcome the problems of large noise in ranging signals and incomplete data in non-line-of-sight environments, thereby improving the robustness of positioning. The present invention introduces the Euclidean distance tensor in the time dimension, and uses the ranging information of the past and the current moment for collaborative optimization, which can dynamically track the movement trajectory of the node. Compared with the static positioning method, it can better adapt to changes in the mobile network environment. In terms of positioning technology, multi-dimensional scaling and Prof. Platts analysis are combined to first solve the relative position through distance-keeping constraints, and then align it to the global coordinate system through anchor nodes, avoiding the problem of direct nonlinear optimization easily falling into local optimality. The present invention provides a new technical solution for coping with NLoS environments in mobile network node positioning, which has high positioning accuracy even when most of the ranging data is missing. In dynamic WSNs in non-line-of-sight scenarios, to address the problem of completing and recovering incomplete noisy Euclidean distance matrices, the Euclidean distance matrix at that moment is combined with the previously recovered Euclidean distance matrix to obtain a Euclidean distance tensor. Based on the tensor tube rank, a tensor nuclear norm is proposed, transforming the tensor completion problem into an optimization problem of minimizing the tensor nuclear norm. High-precision completion is achieved using the alternating direction multiplication framework. Multidimensional scaling is then used to obtain relative position information. Finally, coordinate registration is performed using known anchor point locations to obtain the precise unknown node positions. Compared to traditional matrix completion methods, tensor completion utilizes more data information from dynamic scenarios, achieving higher-precision completion.

[0164] Example 3

[0165] This embodiment conducts a simulation test on the method proposed in the present invention, given a two-dimensional network space of 30M×30m, 95 unknown nodes n, 5 anchor points m, a ranging error outlier ratio 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 figure shows the simulation positioning error results of different algorithms under different sparsity with a 5% outlier ratio. The method can still estimate the node position with high precision in the presence of outliers and large sparsity.

[0168] like Figure 3 The figure shows the cumulative distribution function (CDF) simulation positioning error results of different algorithms under a 5% outlier ratio and 50% sparsity. The method can stably reduce noise in the presence of outliers and large sparsity, verifying its reliability and robustness.

[0169] Each embodiment of the present invention is described in a progressive manner. The same or similar parts between the embodiments can be referred to each other. Each embodiment focuses on the differences from other embodiments. In particular, for the device embodiment, since it is basically similar to the method embodiment, the description is relatively simple. For the relevant parts, refer to the partial description of the method embodiment. The device embodiment described above is merely exemplary. The modules described as separate components may or may not be physically separated. When implementing the scheme of the present invention, the functions of each module can be implemented in the same one or more software and / or hardware. It is also possible to select some or all of the modules according to actual needs to achieve the purpose of the scheme of this embodiment.

[0170] Obviously, the above embodiments of the present invention are merely examples for the purpose of clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Those skilled in the art will appreciate that other variations or modifications can be made based on the above description. It is not necessary and impossible to enumerate all embodiments here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the claims of the present invention.

Claims

1. A mobile network collaborative positioning method based on tensor completion, characterized in that: The following steps are involved: 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 based on the incomplete and noisy Euclidean distance matrix at different times and the complete Euclidean distance matrix at different times; The Euclidean distance tensor is completed and denoised, and the denoised and completed Euclidean distance matrix is ​​obtained according to the denoised and completed Euclidean distance tensor; The denoised and completed Euclidean distance matrix is ​​scaled to obtain the relative coordinates of unknown nodes through multi-dimensional scaling; The relative coordinates of the unknown nodes are aligned through Proctor analysis to obtain the global position of the unknown nodes.

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

3. The method for collaborative positioning in a mobile network based on tensor completion according to claim 1, characterized in that: The Euclidean distance tensor is obtained based on the incomplete and noisy Euclidean distance matrix at different times and the complete Euclidean distance matrix at different times; Its expression is: Among them, D t represents the incomplete Euclidean distance matrix at time t, Represents the complete Euclidean distance matrix at each moment, T D Indicates the number of EDMs in the combination.

4. The method for collaborative positioning in a mobile network based on tensor completion according to claim 1, wherein: Completing and denoising the Euclidean distance tensor includes recovering the completed tensor from the noisy Euclidean distance tensor and sparsely removing the noise; Its expression is: Among them, ε represents the restored complement tensor, λ represents the regularization parameter, and ζ represents the sparse noise. represents an orthogonal mapping, A tensor representing noisy Euclidean distance observations.

5. The method for collaborative positioning in a mobile network based on tensor completion according to claim 4, characterized in that: We exploit the alternating direction multiplier framework to recover the complete tensor and sparsify the noise in a noisy Euclidean distance tensor. Its Lagrangian function expression is: Among them, ε represents the restored complement tensor, λ represents the regularization parameter, and ζ represents the sparse noise. represents an orthogonal mapping, represents the noisy Euclidean distance observation tensor, μ represents the penalty parameter, ε1, l2 represents the Lagrange multiplier, represents the introduced auxiliary variable, and F represents the norm.

6. The method for collaborative positioning in a mobile network based on tensor completion according to claim 5, characterized in that: According to the framework of the alternating direction multiplier method, the restored completion tensor, sparse noise, introduced auxiliary variables, Lagrange multipliers and penalty parameters are updated; the expression is: Among them, ε represents the restored complement tensor, λ represents the regularization parameter, and ζ represents the sparse noise. represents an orthogonal mapping, represents the noisy Euclidean distance observation tensor, μ represents the penalty parameter, l1, l2 represent the Lagrange multiplier, represents the introduced auxiliary variable, and k represents the number of iterations.

7. The method for collaborative positioning in a mobile network based on tensor completion according to claim 6, characterized in that: The auxiliary variable definition expression introduced is: Among them, ε represents the restored complement tensor, ζ represents the sparse noise, represents an orthogonal mapping, Ω c Defined as the complement of the index value Ω of the observed distance data, it represents the index set of missing data that needs to be completed. represents the noisy Euclidean distance observation tensor, μ represents the penalty parameter, l1, l2 represent the Lagrange multiplier, represents the introduced auxiliary variable, and k represents the number of iterations.

8. The method for collaborative positioning in a mobile network based on tensor completion according to claim 1, wherein: The denoised and completed Euclidean distance matrix is ​​scaled by multi-dimensional scaling to obtain the relative coordinates of the unknown nodes, including using the SMACOF algorithm to iteratively minimize the difference between the two Euclidean distance matrices to obtain the relative coordinates of the unknown nodes; Its expression is: Among them, P represents the relative coordinates of the unknown node, represents the convex optimization function, V -1 Represents the pseudo-inverse of V, R represents the auxiliary variable in the optimization process, and V represents a symmetric matrix constructed by weights.

9. The method for collaborative positioning in a mobile network based on tensor completion according to claim 1, wherein: The relative coordinates of unknown nodes are aligned through Proctor analysis; Its expression is: P g =s p ·P r Q p +T p Among them, P g represents the global coordinate, s p represents the scaling factor, P r Indicates relative coordinates, Q p represents the rotation matrix, T p Represents the translation matrix.

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

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