A parameter-free fusion LASSO model solving method

Abstract

This invention discloses a parameter-free fusion LASSO model solution method. First, a linear signal model is established. Then, a fusion LASSO model is constructed by combining the total variation operator and the sparse norm. Next, the fusion LASSO model is redefined. By calculating the normalized weighting matrix and based on the covariance fitting criterion, parameter-free fusion LASSO solution models are constructed for both uniform and non-uniform noise conditions. Finally, the sparse reconstruction of the target is achieved by solving the parameter-free fusion LASSO model. This invention integrates the total variation operator and the sparse norm constraint, preserving target edge information while achieving sparse signal reconstruction. Compared to existing LASSO models, it exhibits better scale recovery capability. Furthermore, it constructs a parameter-free solution method based on the covariance fitting criterion, avoiding the complex regularization parameter selection problem during the solution process.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, specifically relating to a method for solving a parameter-free fusion LASSO model. Background Technology

[0002] In signal processing, fields such as radar angle super-resolution, speech signal processing, and comparative genomic hybridization can model signals as linear models. How to better recover signals from linear models is a current research hotspot in the field of signal processing. The Least Absolute Shrinkage and Selection Operator (LASSO) is a penalized least squares estimator widely used due to its excellent sparse regression constraint capabilities.

[0003] The paper "Funaki K. On Adaptive LASSO-based Sparse Time-Varying Complex ARSpeech Analysis 2021 12th International Symposium on Chinese Spoken Language Processing (ISCSLP). IEEE, 2021: 1-5" proposes a time-varying complex autoregressive method based on LASSO to estimate the formant frequencies in speech signals, addressing the linear prediction problem in speech signal processing. However, this method has unsatisfactory contour recovery capabilities. The paper "Luo J, Zhang Y, Yang J, et al. Online sparse DOA estimation based on sub-aperture recursive LASSO for TDM-MIMO radar. Remote Sensing, 2022, 14(9): 2133" applies the LASSO model to the sparse source localization method of time-division multiplexing MIMO radar system, achieving accurate signal arrival direction estimation and significant sidelobe suppression. However, this method requires the selection of regularization parameters. To address the problem of solving the LASSO model, the literature “Ye GB, Xie X. Split Bregman method for largescale fused Lasso. Computational Statistics & Data Analysis, 2011, 55(4): 1552-1569” proposes an iterative algorithm for large-scale fused LASSO models based on the split Bregman method. Test results show that this method can solve the large-scale fused LASSO problem quickly, but the selection of regularization parameters is still required, which is difficult in practical applications. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a parameterless fusion LASSO model solution method, which improves the contour preservation capability of existing LASSO models and solves the problem of difficulty in selecting regularization parameters.

[0005] The technical solution adopted in this invention is: a parameter-free fusion LASSO model solution method, the specific steps of which are as follows:

[0006] Step 1: Establish a linear signal model;

[0007] Establish a linear signal model:

[0008] y = Hs + e (1)

[0009] in, Represents the signal sample vector. Represents a dictionary matrix. Represents the signal vector to be recovered. Represents an additive noise vector. M represents the dimension of the vector space, M represents the dimension of the signal sample, and N represents the dimension of the signal to be recovered.

[0010] Step 2: Construct a fused LASSO model;

[0011] Given that the vector s is sparse, the LASSO optimization method is expressed as:

[0012]

[0013] in, represents the L2 norm distance between the model and the signal, ||s||1 represents adding a sparsity constraint to the vector s, and μ represents the parameter that controls the trade-off between the two terms.

[0014] By combining the total variation operator with sparse constraints, a fused LASSO optimization model is obtained, the expression of which is:

[0015]

[0016] Where D represents the total variation operator, specifically expressed as:

[0017]

[0018] Step 3: Redefine the fusion LASSO model;

[0019] set up It is an invertible matrix. By combining the total variation operator with sparse constraints, we can set:

[0020] B = HD -1 (5)

[0021] x=Ds (6)

[0022] Then equation (3) can be rewritten as:

[0023]

[0024] Step 4: Calculate the normalized weighted matrix;

[0025] Construct a normalized weighted matrix to remove the regularization parameter in equation (7), and let B = [b1, b2, ..., b M Then the weighted matrix can be expressed as:

[0026] W x=diag([w1...w M (8)

[0027] W σ =diag([w M+1 ...w M+N ]) (9)

[0028]

[0029] Where diag(·) represents converting a vector into a diagonal matrix, W x W represents the weighting matrix for the objective constraints. σ This represents the weighting matrix for noise constraints.

[0030] Step 5: Construct a parameter-free solution model;

[0031] Using the weighting matrix W from step four x and W σ Based on the covariance criterion, two parameter-free solution models are constructed for two different noise cases.

[0032] The first case refers to all noise variances σ k Equal uniform noise conditions. Assume all noise terms have equal variance, i.e., when... When, equation (7) is equivalent to:

[0033]

[0034] The second case refers to the noise variance σ k Non-uniform noise at different times. Assuming the noise term has different variances, equation (7) is equivalent to:

[0035]

[0036] Step Six: Model Transformation;

[0037] The model is transformed to solve for vector s, resulting in the final parametric-free fusion LASSO model solution method.

[0038] Under uniform noise conditions, substituting equations (5) and (6) into equation (11) yields:

[0039]

[0040] In the case of non-uniform noise, substituting equations (5) and (6) into equation (12) yields...

[0041]

[0042] Step 7: Solve the model;

[0043] The model obtained in step six is ​​a convex optimization problem, which is solved using the CVX toolbox in Matlab.

[0044] Under uniform noise, the solution is as follows:

[0045]

[0046] Under non-uniform noise conditions, the solution is as follows:

[0047]

[0048] in, This indicates the target reconstruction result, and CVX(·) indicates the solution process of the CVX toolbox.

[0049] Ultimately, sparse reconstruction of the target is achieved by solving the parameter-free fusion LASSO model.

[0050] The beneficial effects of this invention are as follows: First, a linear signal model is established. Then, a fused LASSO model is constructed by combining the total variation operator and the sparse norm. Next, the fused LASSO model is redefined. By calculating the normalized weighting matrix and based on the covariance fitting criterion, a parameter-free fused LASSO solution model is constructed for both uniform and non-uniform noise conditions. Finally, the target sparse reconstruction is achieved by solving the parameter-free fused LASSO model. This invention integrates the total variation operator and the sparse norm constraint, preserving target edge information while achieving sparse signal reconstruction. Compared to existing LASSO models, it has better scale recovery capabilities. Furthermore, it constructs a parameter-free solution method based on the covariance fitting criterion, avoiding the complex problem of selecting regularization parameters during the solution process. Attached Figure Description

[0051] Figure 1 This is a flowchart of a non-parametric fusion LASSO model solution method according to the present invention.

[0052] Figure 2 The solution results of the existing fusion LASSO model in this embodiment of the invention are shown in the figure.

[0053] Figure 3 The figure shows the solution results of the parameterless fusion LASSO model under uniform noise in an embodiment of the present invention.

[0054] Figure 4 The figure shows the solution results of the parameter-free fusion LASSO model under non-uniform noise in an embodiment of the present invention. Detailed Implementation

[0055] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0056] This embodiment uses measured data to demonstrate the effectiveness of the proposed method. All steps and conclusions of this invention are verified on the Matlab2020b simulation platform.

[0057] like Figure 1 The flowchart of a parameter-free fusion LASSO model solution method of the present invention is shown below, and the specific steps are as follows:

[0058] Step 1: Establish a linear signal model;

[0059] Many problems in signal processing, including radar super-resolution imaging, audio signal recovery, and signal orientation estimation, require the establishment of linear signal models for representation. In this embodiment, a bladder cancer dataset from comparative genomic hybridization is used as the sample vector y. To detect genomic copy number variations, the following linear signal model is constructed:

[0060] y = Hs + e (17)

[0061] in, Represents the signal sample vector. Represents a dictionary matrix. Represents the signal vector to be recovered. Represents an additive noise vector. Let represent the dimension of the vector space, M represent the dimension of the signal samples, and N represent the dimension of the signal to be recovered. By solving for s in this signal model, a large number of signal processing problems can be solved.

[0062] In this embodiment, the data dimension N of the sample vector y is 2143, the dimension M of the signal to be recovered s is 2143, and the dictionary matrix H is an identity matrix with a dimension of 2143×2143.

[0063] Step 2: Construct a fused LASSO model;

[0064] Given that the vector s is sparse, the LASSO optimization method is an efficient way to solve this problem, which can be expressed as:

[0065]

[0066] in, represents the L2 norm distance between the model and the signal, ||s||1 represents adding a sparsity constraint to the vector s, and μ represents the parameter that controls the trade-off between the two terms.

[0067] To better preserve the target contour, this embodiment combines the total variation operator with sparse constraints to construct a fused LASSO optimization model, the expression of which is:

[0068]

[0069] Where D is the total variation operator, specifically expressed as:

[0070]

[0071] In this embodiment, the regularization parameter μ = 0.061.

[0072] Step 3: Redefine the fusion LASSO model;

[0073] set up It is an invertible matrix. In order to integrate the total variation operator with sparse constraints, let:

[0074] B = HD -1 (twenty one)

[0075] x=Ds (22)

[0076] Equation (19) can then be reformulated as:

[0077]

[0078] The vector x can be reconstructed using equation (23).

[0079] Step 4: Calculate the normalized weighted matrix;

[0080] By constructing a normalized weighted matrix, the regularization parameter in equation (23) can be removed, thus avoiding the complex parameter selection problem.

[0081] Let B = [b1, b2, ..., b M Then the weighted matrix can be expressed as:

[0082] W x =diag([w1...w M ]) (twenty four)

[0083] W σ =diag([w M+1 …w M+N (25)

[0084]

[0085] Where diag(·) represents converting a vector into a diagonal matrix, W x W represents the weighting matrix for the objective constraints. σ This represents the weighting matrix for noise constraints.

[0086] Step 5: Construct a parameter-free solution model;

[0087] Using the weighting matrix W from step fourx and W σ Based on the covariance criterion, two parameter-free solution models are constructed for two different noise cases.

[0088] The first case refers to all noise variances σ k Equal uniform noise conditions. Assume all noise terms have equal variance, i.e., when... When, equation (23) is equivalent to:

[0089]

[0090] The second case refers to the noise variance σ k Non-uniform noise at different times. Assuming the noise term has different variances, equation (23) is equivalent to:

[0091]

[0092] Step Six: Model Transformation;

[0093] To solve for vector s, the model needs to be transformed to obtain the final parameterless fusion LASSO model solution method.

[0094] Under uniform noise conditions, substituting equations (21) and (22) into equation (27) yields:

[0095]

[0096] In the case of non-uniform noise, substituting equations (21) and (22) into equation (28) yields:

[0097]

[0098] Among them, the weighting matrix W in equations (29) and (30) x With HD -1 Yes, there is a connection. Through this design, the method of this invention can effectively preserve the target contour while achieving sparse reconstruction of the target, and also avoids the problem of complex parameter selection.

[0099] Step 7: Solve the model;

[0100] The model obtained in step six is ​​a convex optimization problem, which is solved using the CVX toolbox in Matlab.

[0101] Under uniform noise, the solution is as follows:

[0102]

[0103] Under non-uniform noise conditions, the solution is as follows:

[0104]

[0105] in, This indicates the target reconstruction result, and CVX(·) indicates the solution process of the CVX toolbox.

[0106] Ultimately, sparse reconstruction of the target is achieved by solving the parameter-free fusion LASSO model.

[0107] To demonstrate the effectiveness of the method of this invention, it was applied to the detection of gene copy number variations in comparative genomic hybridization (CGL). The experiment was conducted on a 64-bit Matlab 2020b platform. In tumor cells, mutations typically lead to the deletion or insertion of a large DNA fragment into the chromosome; this phenomenon is called copy number variation (CGL). Accurate detection of CGL in CGL can greatly advance cancer research. Existing fusion-based LASSO signal approximators are effective tools for detecting CGL in CGL. The proposed parameter-free fusion-based LASSO model was then verified as an effective method for solving the problem of detecting CGL in CGL.

[0108] The data used in the experiment were bladder cancer genomic data, with a sample dimension of 2143. All solutions were obtained using the MATLAB CVX toolbox. Figure 2 The solution results of the existing fusion LASSO method are presented. The parameter μ = 0.061 is selected by using the generalized cross-validation method. The results show that the fusion LASSO method uses the l1 norm on the fusion penalty term, which enables the method to capture both piecewise flat modes and local mutation modes. At the same time, the control of the overall sparsity of the coefficient solution helps to filter out "cold" point regions. Figure 3 The results are obtained from the parametric fusion of the LASSO model under uniform noise, and are similar to those of existing methods. However, the method of this invention is more sensitive to local abrupt changes while ensuring local flatness. Figure 4 The results are obtained from the parametric-free fusion LASSO model under non-uniform noise. Under non-uniform noise, the method of this invention is insensitive to local mutations and exhibits stronger performance against off-point data. More importantly, the parametric-free fusion LASSO model proposed in this invention avoids the selection of hyperparameters, which greatly simplifies the computation process. Therefore, the method of this invention can effectively solve the problem of detecting copy number variations in comparative genomic hybridization techniques and avoids the complex problem of hyperparameter selection.

[0109] In summary, the method of this invention integrates the total variation operator and the sparse norm constraint, which can preserve the target edge information while realizing sparse reconstruction of the signal. Compared with the existing LASSO model, it has better scale recovery capability. At the same time, it constructs a parameterless solution method based on the covariance fitting criterion, avoiding the complex regularization parameter selection problem in the solution process.

[0110] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Various modifications and variations can be made to the invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the invention should be included within the scope of the claims of the invention.

Claims

1. A method for solving a parameter-free fusion LASSO model, the specific steps of which are as follows: Step 1: Establish a linear signal model; Establish a linear signal model: y = Hs + e (1) in, Represents the signal sample vector. Represents a dictionary matrix. Represents the signal vector to be recovered. Represents an additive noise vector. M represents the dimension of the vector space, and N represents the dimension of the signal sample. Step 2: Construct a fused LASSO model; Given that the vector s is sparse, the LASSO optimization method is expressed as: in, represents the L2 norm distance between the model and the signal, ||s||1 represents adding a sparse constraint to the vector s, and μ represents the parameter that controls the trade-off between the two terms; By combining the total variation operator with sparse constraints, a fused LASSO optimization model is obtained, the expression of which is: Where D represents the total variation operator, specifically expressed as: Step 3: Redefine the fusion LASSO model; set up It is an invertible matrix. By combining the total variation operator with sparse constraints, we can set: B=HD -1 (5) x=Ds (6) Then equation (3) can be rewritten as: Step 4: Calculate the normalized weighted matrix; Construct a normalized weighted matrix to remove the regularization parameter in equation (7), and let B = [b1, b2, ..., b M Then the weighted matrix can be expressed as: W x =diag([w1...w M ]) (8) IN σ =diag([in M+1 ...In M+N ]) (9) Where diag(·) represents converting a vector into a diagonal matrix, W x W represents the weighting matrix for the objective constraints. σ This represents the weighting matrix for noise constraints; Step 5: Construct a parameter-free solution model; Using the weighting matrix W from step four x and W σ Based on the covariance criterion, two parameter-free solution models are constructed for two different noise cases. The first case refers to all noise variances σ k Equal uniform noise conditions; assume all noise terms have equal variance, i.e., when When, equation (7) is equivalent to: The second case refers to the noise variance σ k For non-uniform noise cases at different times; assuming that the noise term has different variances, equation (7) is equivalent to: Step Six: Model Transformation; The model is transformed to solve for vector s, resulting in the final parametric-free fusion LASSO model solution method. Under uniform noise conditions, substituting equations (5) and (6) into equation (11) yields: In the case of non-uniform noise, substituting equations (5) and (6) into equation (12) yields... Step 7: Solve the model; The model obtained in step six is ​​a convex optimization problem, which is solved using the CVX toolbox in Matlab. Under uniform noise, the solution is as follows: Under non-uniform noise conditions, the solution is as follows: in, The result represents the target reconstruction, and CVX(·) represents the solution process of the CVX toolbox. Ultimately, sparse reconstruction of the target is achieved by solving the parameter-free fusion LASSO model.

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