CN108121955A - A kind of smooth l based on adjusted mean approximation hyperbolic tangent function0Norm Method - Google Patents
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Abstract
The invention discloses a kind of smooth l based on adjusted mean approximation hyperbolic tangent function0Norm Method, this method approach l using a kind of adjusted mean approximation hyperbolic tangent function0Norm, the adjusted mean approximation hyperbolic tangent function has preferably approximation capability, then using the extreme-value problem of the Newton Algorithm adjusted mean approximation hyperbolic tangent function, and then sparse signal is gone out with higher accurate reconstruction, the sparse reconstruction property of SL0 algorithms can be significantly increased.
Description
Technical Field
The invention belongs to the technical field of sparse signal recovery, and particularly relates to a smoothing method based on a modified approximate hyperbolic tangent function 0 Norm method.
Background
In conventional signal sampling theory, to ensure distortion-free output of a signal, a sampling rate of at least twice the signal bandwidth is required. Therefore, conventional signal sampling can place a significant strain on hardware systems when processing signals of a large bandwidth. Compressed Sensing (CS) is a new type of signal samplingTheoretically, the method can complete accurate reconstruction of signals by using a reconstruction algorithm only by using few sparse signal sampling values, and is widely applied to the fields of radar imaging, signal processing, medical imaging and the like. The sparse reconstruction problem is equivalent to a sparse solution problem with an underdetermined equation set y = Dx, where,in order to measure the matrix of the measurements,is a matrix of the perception that is,is a sparse vector. The solution model for the sparse problem can be expressed as
In the formula, | · the luminance | | 0 Is represented by 0 Norm, i.e. the number of non-zero elements of the vector. l 0 The norm minimization problem is an NP-hard problem, resulting in equation (1) being difficult to solve. The Base Pursuit (BP) algorithm, which utilizes l, is an effective algorithm to solve the above-mentioned problems 1 Norm instead of l 0 Norm from which (1) is converted to
And then solving the equation (2) through linear programming, thereby realizing the reconstruction of the sparse signal. Although the BP algorithm can effectively solve the problem of sparse signal reconstruction, the calculation amount of the BP algorithm is large, and sparse signals cannot be rapidly reconstructed. In addition, there are many other sparse reconstruction algorithms, such as Iterative weighted Least Squares (IRLS) algorithm, orthogonal Matching Pursuit (OMP) algorithm, and smoothing l 0 Norm (smoothened l) 0 norm, SL 0), etc. WhereinThe SL0 algorithm approximates l with a Gaussian function 0 And performing norm reconstruction on the sparse signal by a steepest descent method and a gradient projection principle. Although the SL0 algorithm can reconstruct sparse signals quickly, the Gaussian function pair l adopted by the algorithm 0 The approximation performance of the norm is poor, and a sawtooth effect is generated when the extreme value problem of the function is solved by using the steepest descent method, so that the reconstruction accuracy of the SL0 algorithm is low. Document [1]](Zhang Y,Yu J,Bai H,et al.Improved Sparse Signal Reconstruction based on Approximate Hyperbolic Tangent Function with Smoothed l 0 Norm[J]International Journal of Science,2017,4 (2): 201-210.) proposes Newton smoothing 0 Norm (Newton smoothened l) 0 norm, NSL 0) algorithm that approximates l using an approximate hyperbolic tangent function 0 Norm, and solving the extreme value problem of the function by using a modified Newton method, thereby improving the reconstruction performance of the SL0 algorithm. Document [2]](Feng J J,Zhang G,Wen F Q.MIMO radar imaging based on smoothed l 0 norm[J]Chemical schemes in Engineering 2015, (8): 1-10.) another approximate hyperbolic tangent function is used to approximate l 0 Norm, and searching the extreme value of the function in Newton direction to obtain an Approximate SL0 (Approximate smoothened l) 0 norm, ASL 0), and then the MIMO radar target imaging is carried out by using the algorithm, so that the imaging performance is effectively improved. The core idea of the SL0 algorithm is to approximate l by adopting a Gaussian function 0 Norm such that l 0 The norm minimization problem is converted into the extreme value problem of a smooth function, and the problem of l is avoided 0 Direct solution of the norm minimization problem. Therefore, in order to further improve the reconstruction performance of the SL0 algorithm on sparse signals, a function with better approximation performance is constructed to approximate l 0 It is very necessary to norm and solve the sparse solution problem corresponding to the function.
Disclosure of Invention
For SL0 algorithm and improved algorithm thereof, smoothing function pairs l such as Gaussian function and approximate hyperbolic tangent function are adopted 0 The approximation of the norm is not ideal, resulting in a calculationThe invention provides a smoothing method based on a modified approximate hyperbolic tangent function 0 Norm method, using a modified approximate hyperbolic tangent function to pair l 0 The norm achieves better approximation, and the extreme value problem of the smooth function is solved through a Newton method, so that the sparse signal reconstruction performance of the SL0 algorithm is effectively improved.
The technical purpose is achieved, the technical effect is achieved, and the invention is realized through the following technical scheme:
smoothing method based on modified approximate hyperbolic tangent function 0 The norm method comprises the following steps:
(1) Designed to approximate l 0 The norm is corrected to approximate a hyperbolic tangent function;
(2) Establishing a sparse problem model based on a modified approximate hyperbolic tangent function;
(3) And solving the sparse problem represented by the correction function when the parameter sigma gradually decreases by using a correction Newton method, wherein the sigma is the shape parameter of the function, and reconstructing a sparse signal.
Further, the expression of the modified approximate hyperbolic tangent function in the step (1) is as follows:
wherein x is a variable; σ is the shape parameter of the function.
Further, the step (2) is specifically:
using a smoothing function f σ (x) To approximate l 0 Norm, order:
wherein,is a sparse vector; x is the number of i (i =1,2, …, N) is sparseThe ith element in vector x.
It is possible to obtain:
wherein | · | purple sweet 0 Is represented by 0 Norm, then the sparse problem model based on the modified approximate hyperbolic tangent function is:
wherein,is a measurement matrix;is the perceptual matrix.
Further, the step (3) is specifically:
(3.1) taking σ as a set of descending sequences [ σ ] 1 >σ 2 >,...>σ J ]Where σ is J A positive number close to zero;
(3.2) sequentially applying the modified Newton method to the number of σ = σ i (1. Ltoreq. I. Ltoreq.J)The solution is performed so that the vector x can gradually approach the global optimal solution.
Further, the step (3.1) is specifically:
initialization:
(a) Setting initial value
(b) Selecting a group of suitable sequences [ sigma ] 1 ,σ 2 ,...,σ J ]And σ j+1 ＝ρσ j Where ρ (0)<ρ&And (lt) 1) is an attenuation factor,
further, the step (3.2) is specifically:
iteration of the algorithm:
forj＝1,2,...,J
(1) let σ = σ j ，
(2) Searching function F successively in the modified Newton direction σ (x) And projecting the minimum onto the feasible set;
for l＝1,2,...,L
(a) Correcting Newton's direction to
(b)
(c) Will be provided withProjected onto a feasible set
(3) Order to
Output sparse vector solution
Compared with the prior art, the invention has the beneficial effects that:
(1) The core idea of the SL0 algorithm is to approximate l by adopting a smoothing function 0 Norm such that l 0 The norm minimization problem is converted into the sparse problem represented by a smooth function, and l is avoided 0 Direct solution of the norm-minimized NP-hard problem. The invention provides a new smoothing function pair 0 The norm achieves better approximation, and the approximation degree of the norm is superior to that of the existing Gaussian function and approximate hyperbolic tangent function, so that the sparse problem represented by the smooth function is closest to the ideal condition, and the reconstruction performance of the SL0 algorithm on sparse signals is effectively improved.
(2) The smoothing function provided by the invention is simple and easy to be derived, and the modified Newton direction corresponding to the function is simple to calculate, so that the smoothing function provided by the invention is introduced into the SL0 algorithm, the reconstruction performance of sparse signals can be improved, and the characteristic of high calculation efficiency of the smoothing function is still maintained.
Drawings
FIG. 1 is a flow chart of a method implementation of one embodiment of the present invention;
fig. 2 is a distribution plot of four functions at σ =0.1;
FIG. 3 is a schematic diagram of an original sparse signal and a reconstructed signal obtained using various algorithms;
FIG. 4 is a schematic diagram showing the relationship between the reconstructed SNR and the SNR variation of different algorithms;
FIG. 5 is a schematic diagram showing the relationship between reconstruction errors and signal-to-noise ratio variations of different algorithms;
FIG. 6 is a schematic diagram showing a variation relationship between a reconstructed signal-to-noise ratio and sparsity of each algorithm;
fig. 7 is a schematic diagram showing a change relationship between a reconstruction error and sparsity of each algorithm.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The following detailed description of the principles of the invention is provided in connection with the accompanying drawings.
As shown in FIG. 1, the present invention provides a smoothing method based on a modified approximate hyperbolic tangent function 0 The norm method comprises the following concrete implementation steps:
step 1: designed to approximate l 0 The norm is corrected to approximate a hyperbolic tangent function;
the core idea of the SL0 algorithm is to approximate l by adopting a Gaussian function 0 Norm such that l 0 The norm minimization problem is converted into the extreme value problem of a smooth function, and the problem of l is avoided 0 Direct solution of the norm minimization problem.
The expression of the gaussian function employed in the SL0 algorithm is:
wherein x is a variable; σ is the shape parameter of the function.
Then
Order to
Wherein,is a sparse vector; x is the number of i (i =1,2, …, N) is the i-th element in the sparse vector x.
Can obtain
Equation (5) can be converted to:
wherein,is a measurement matrix.
In order to improve the smoothing function pair l in the SL0 algorithm 0 Approximation of norm [1]]And document [2]]Two different approximate hyperbolic tangent functions are respectively adopted to replace the standard Gaussian function, and the expressions of the two different hyperbolic tangent functions are respectively as follows:
therefore, finding a proper smooth function can effectively approximate l 0 Norm, is critical to the reconstruction performance of the SL0 algorithm. In the document [1]]And [2]]Based on the NSL0 algorithm and the ASL0 algorithm respectively provided, in order to further improve the smoothing function pair l 0 The approximation degree of norm, the invention designs a new smooth function (modified approximate hyperbolic tangent function) to approximate l 0 Norm to improve the approximation degree, wherein the modified approximate hyperbolic tangent function expression is as follows:
FIG. 2 shows a Gaussian function, document [1]]And document [2]]Two kinds of approximate hyperbolic tangent functions respectively adopted in the method and the modified approximate hyperbolic tangent function f provided by the invention σ (x) Function at σ =0.1And (5) distribution diagram. As can be seen from FIG. 2, the equation is similar to the Gaussian function, document [1]]And document [2]]Compared with the two approximate hyperbolic tangent functions, the modified approximate hyperbolic tangent function provided by the invention is in x E [ -0.5,0.5]The "steepness" of the inner is greater, indicating that the function is paired with l 0 The approximation degree of the norm is better.
Step 2, establishing a sparse problem solving model based on a modified approximate hyperbolic tangent function
Using a smoothing function f σ (x) To approximate l 0 Norm, let:
can obtain
Then the sparse problem model based on the modified approximate hyperbolic tangent function is:
and 3, step 3: solving a sparse problem represented by a correction function when a parameter sigma gradually decreases by using a modified Newton method
The shape parameter sigma controls F σ (x) To l 0 Approximation of norm, i.e. smaller sigma, F σ (x) The closer to l 0 Norm, but at the same time F σ (x) The more local extrema are present, then F σ (x) In the approximation of l 0 In the process of norm, the function is easy to fall into a local extreme value, and the solving difficulty of the global minimum of the function is greatly increased.
To solve this problem, the invention first takes σ as a set of descending sequences [ σ ] in series 1 >σ 2 >,...>σ J ]Where σ is J A positive number close to zero; then using the modified Newton methodFor σ = σ sequentially i And (i is more than or equal to 1 and less than or equal to J), solving the formula (13), so that the vector x can gradually approach to the global optimal solution.
The SL0 algorithm usually solves the global minimum of the gaussian function by using a steepest descent method, and although the steepest descent method has simple steps and a small iteration amount when calculating the function minimum each time, a "saw tooth effect" occurs in the process of searching the global minimum of the function, thereby adversely affecting the reconstruction accuracy of the SL0 algorithm. Aiming at the problem, the method solves the formula (13) by using a modified Newton method, namely the sparse problem based on the modified approximate hyperbolic tangent function, thereby improving the reconstruction accuracy of the sparse signal. For function F σ (x) The Newton direction is:
d＝-A 2 F σ (x)-1ΔF σ (x) (14)
in the formula:
wherein,
after calculating function F σ (x) After a Newton direction d, where the matrix Δ 2 F σ (x) Is a Hessen matrix which does not satisfy the positive definite condition and further cannot guarantee that the newton direction d is the descending direction. To ensure that d is the descending direction, the diagonal elements in equation (16) are modified to construct a new matrix H instead of Δ in Newton's direction d 2 F σ (x) And (4) matrix. The expression of the new matrix H is:
H＝Δ 2 (F σ (x))+ψ (18)
where psi is a diagonal matrix,for ease of calculation, the diagonal element ψ will be used herein i Is taken as
From this, the element on the ith diagonal in H is
From the above equation, the new matrix H after correction satisfies the positive definite condition, and the matrix H is used to replace Δ in the newton direction d 2 F σ (x i ) And (3) matrix, taking the ensured Newton direction d as a descending direction, namely the modified Newton direction is as follows:
solving a sparse problem represented by a correction function when the parameter sigma gradually decreases by using a correction Newton method, wherein in actual operation, the method specifically comprises the following steps of:
a1, initialization:
(a) Setting initial value
(b) Selecting a group of suitable sequences [ sigma ] 1 ,σ 2 ,...,σ J ]And σ j+1 ＝ρσ j Where ρ (0)<ρ&And (lt) 1) is an attenuation factor,
iteration of the A2 algorithm:
forj＝1,2,...,J
(1) let σ = σ j ，
(2) Searching function F successively in the modified Newton direction σ (x) And projecting the minimum onto the feasible set.
forl＝1,2,...,L
(a) Correcting Newton's direction to
(b)
(c) Will be provided withProjected onto a feasible set
(3) Order to
A3 output sparse vector solution
In order to verify the sparse reconstruction performance of the method, the invention designs several groups of SL0 algorithms, NSL0 algorithm proposed by document [1], ASL0 algorithm proposed by document [2] and comparison experiments of the algorithms. In the simulation experiment, the sparse source signal is randomly generated by a bernoulli-gaussian model, which is:
x i -p·N(0，δ on )+(1-p)·N(0，δ off )
where p is the probability of a large non-zero amount occurring in the source signal; n (0, delta) is white Gaussian additive noise, the mean value of the white Gaussian additive noise is zero, and the variance of the white Gaussian additive noise is delta; delta. For the preparation of a coating on And delta off Respectively, a larger non-zero coefficient and a smaller non-zero coefficient that constitute the source signal. Setting delta off ＜＜δ on And p < 1 to ensure the sparsity of the source signal.
In the simulation experiment, the test results of the test were,in the form of a randomly sampled matrix, the sampling matrix,in order to be a sparse matrix of the source signals,is a perceptual matrix. For y = Dx, the sparse source signal is reconstructed with the SL0 algorithm, NSL0 algorithm, ASL0 algorithm, and the inventive algorithm, respectively, knowing y and D. The parameter settings in the model for generating the sparse source signal are M =1000, n =400, δ, respectively on ＝1，δ off ＝10 -3 P =0.1; in the SL0 algorithm, NSL0 algorithm, ASL0 algorithm and the present invention algorithm, σ is set J =0.001, ρ =0.7, and the number of internal cycles L =5.
The signal-to-noise ratio is defined as:
in the formula, tr (-) indicates tracing the matrix.
The invention adopts the reconstruction signal-to-noise ratio and the reconstruction error to evaluate the reconstruction performance of each algorithm, and the reconstruction signal-to-noise ratio is defined as:
in the formula,for a sparse source signal x i The estimated solution of (2). The reconstruction error is defined as:
simulation experiment I: experiment of different methods for sparse signal reconstruction
Fig. 3 is a comparison of an original signal and a reconstructed signal obtained using methods. The source signal sparsity represents the number of non-zero elements in the signal matrix. In the simulation, the sparsity K =100 and the signal-to-noise ratio SNR =30dB are taken. The method adopts a modified approximate hyperbolic tangent function to approximate l 0 Norm, improved smoothing function pair l 0 The approximation degree of the norm, as can be seen from fig. 3, compared with the SL0 algorithm, the NSL0 algorithm, and the ASL0 algorithm, the sparse signal reconstructed by the method of the present invention is closest to the original source signal.
And (2) simulation experiment II: relationship between reconstruction performance and signal-to-noise ratio of each algorithm
Fig. 4 and 5 show the reconstructed signal-to-noise ratio and the variation of the reconstruction error and the signal-to-noise ratio of the four algorithms respectively. And (3) setting the signal-to-noise ratio variation range to be 20-40 dB and the signal sparsity K =100, and carrying out 200 times of simulation experiments. As can be seen from fig. 4 and 5, the approximate hyperbolic tangent function pair l is used in the NSL0 algorithm and the ASL0 algorithm 0 The approximation degree of the norm is superior to that of a Gaussian function, the extreme value problem of a smooth function is solved by using a modified Newton method, and a sawtooth effect generated in an iteration process by a steepest descent method is avoided, so that the reconstruction signal-to-noise ratio of an NSL0 algorithm and an ASL0 algorithm is higher than that of an SL0 algorithm, and the reconstruction error is lower than that of the SL0 algorithm. The method of the invention employs a modified approximate hyperbolic tangent function, which approximates to l 0 The norm degree is better than the smoothing function in the NSL0 algorithm and the ASL0 algorithm, and as can be seen from fig. 4 and 5, the reconstruction performance of the algorithm of the present invention is the best.
And (3) simulation experiment III: changing relation between reconstruction performance and sparsity of each algorithm
Fig. 6 and 7 are respectively a change relationship between a reconstruction signal-to-noise ratio and a reconstruction error of each algorithm and sparsity. The sparsity K is assumed to vary from 20 to 100, with a signal-to-noise ratio of 30dB. As can be seen from fig. 6 and 7, the reconstruction performance of the algorithm of the present invention is better than that of the SL0 algorithm, the NSL0 algorithm, and the ASL0 algorithm under different signal sparsity.
The foregoing shows and describes the general principles and features of the present invention, together with the advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (6)
1. Smoothing method based on modified approximate hyperbolic tangent function 0 The norm method is characterized by comprising the following steps:
(1) Designed to approximate l 0 The norm is corrected to approximate a hyperbolic tangent function;
(2) Establishing a sparse problem model based on a modified approximate hyperbolic tangent function;
(3) And solving the sparse problem represented by the correction function when the parameter sigma gradually decreases by using a correction Newton method, wherein the sigma is the shape parameter of the function, and reconstructing a sparse signal.
2. A smoothing method according to claim 1 based on a modified approximate hyperbolic tangent function 0 The norm method is characterized in that: the expression of the modified approximate hyperbolic tangent function in the step (1) is as follows:
wherein x is a variable; σ is the shape parameter of the function.
3. A smoothing method according to claim 2 based on a modified approximation hyperbolic tangent function 0 The norm method is characterized in that: the step (2) is specifically as follows:
using a smoothing function f σ (x) To approximate l 0 Norm, order:
wherein,is a sparse vector; x is the number of i (i =1,2, …, N) is the i-th element in the sparse vector x.
It is possible to obtain:
wherein | · | charging 0 Is represented by 0 Norm, then the sparse problem model based on the modified approximate hyperbolic tangent function is:
wherein,is a measurement matrix;is the perceptual matrix.
4. A smoothing method according to claim 3 based on a modified approximate hyperbolic tangent function 0 The norm method is characterized in that: the step (3) is specifically as follows:
(3.1) taking σ as a set of descending sequences [ σ ] 1 >σ 2 >,...>σ J ]Where σ is J A positive number close to zero;
(3.2) sequentially applying the modified Newton method to the number of σ = σ i (1. Ltoreq. I. Ltoreq.J)s.t.y = Dx, so that the vector x can gradually approach the global optimum solution.
5. A method according to claim 4, based on a modified approximation hyperbolic tangent function 0 The norm method is characterized in that: the step (3.1) is specifically as follows:
initialization:
(a) Setting initial value
(b) Selecting a group of suitable sequences [ sigma ] 1 ,σ 2 ,...,σ J ]And σ j+1 ＝ρσ j Where ρ (0)<ρ&And (lt) 1) is an attenuation factor,
6. a smoothing method according to claim 4 or 5 based on a modified approximate hyperbolic tangent function 0 The norm method is characterized in that: the step (3.2) is specifically as follows:
iteration of the algorithm:
forj＝1,2,...,J
(1) let σ = σ j ，
(2) Searching function F successively in the modified Newton direction σ (x) And projecting the minimum onto the feasible set;
for l＝1,2,...,L
(a) Correcting Newton direction to
(b)
(c) Will be provided withProjected onto a feasible set
(3) Order to
Output sparse vector solution
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN109462427A (en) * | 2018-10-12 | 2019-03-12 | 南京信息工程大学 | A kind of MIMO underwater acoustic channel estimation method optimizing smooth L0 norm based on improved ADAPTIVE MIXED |
| CN111862257A (en) * | 2020-07-17 | 2020-10-30 | 中国科学院光电技术研究所 | Compressed sensing image reconstruction method based on approximation of approximate L0 norm by arc tangent function |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN109462427A (en) * | 2018-10-12 | 2019-03-12 | 南京信息工程大学 | A kind of MIMO underwater acoustic channel estimation method optimizing smooth L0 norm based on improved ADAPTIVE MIXED |
| CN109462427B (en) * | 2018-10-12 | 2021-05-04 | 南京信息工程大学 | MIMO underwater acoustic channel estimation method |
| CN111862257A (en) * | 2020-07-17 | 2020-10-30 | 中国科学院光电技术研究所 | Compressed sensing image reconstruction method based on approximation of approximate L0 norm by arc tangent function |
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