CN108121955A - A kind of smooth l based on adjusted mean approximation hyperbolic tangent function0Norm Method - Google Patents

Abstract

The invention discloses a smoothing _l0 norm method based on a modified approximate hyperbolic tangent function. The method adopts a modified approximate hyperbolic tangent function to approach the _l0 norm. The modified approximate hyperbolic tangent function has better Approximate performance, and then use Newton's method to solve the extreme value problem of the modified approximate hyperbolic tangent function, and then reconstruct the sparse signal with higher precision, which can significantly improve the sparse reconstruction performance of the SL0 algorithm.

Description

A Smooth l0 Norm Method Based on Modified Approximate Hyperbolic Tangent Function

technical field

The invention belongs to the technical field of sparse signal recovery, and in particular relates to a smoothing _l0 norm method based on a modified approximate hyperbolic tangent function.

Background technique

In the traditional signal sampling theory, in order to ensure the undistorted output of the signal, the required sampling rate is at least twice the signal bandwidth. Therefore, when processing signals with a large bandwidth, traditional signal sampling will cause great pressure on the hardware system. Compressed sensing (Compressing Sensing, CS) is a new type of signal sampling theory. It only needs very few sparse signal sampling values to complete the accurate reconstruction of the signal using the reconstruction algorithm. It has been widely used in radar imaging, signal processing and medical imaging. The sparse reconstruction problem is equivalent to the sparse solution problem of the underdetermined equation system y=Dx, where, is the measurement matrix, is the perception matrix, is a sparse vector. The solution model of the sparse problem can be expressed as

In the formula, ||·|| ₀ represents the l ₀ norm, that is, the number of non-zero elements of the vector. l The ₀ -norm minimization problem is an NP-hard problem, which makes formula (1) difficult to solve. The basic pursuit (Basic Pursuit, BP) algorithm is an effective algorithm to solve the above problems. This algorithm uses the l ₁ norm to replace the l ₀ norm, and the formula (1) is transformed into

Then formula (2) is solved by linear programming, so as to realize the reconstruction of sparse signals. Although the BP algorithm can effectively solve the sparse signal reconstruction problem, the algorithm has a large amount of calculation and cannot quickly reconstruct the sparse signal. In addition, there are many other sparse reconstruction algorithms, such as Iterative Re-weighted LeastSquares (IRLS) algorithm, Orthogonal Matching Pursuit (OMP) algorithm and Smoothed l ₀ norm (Smoothed l ₀ norm, SL0) algorithm, etc. Among them, the SL0 algorithm uses the Gaussian function to approximate the l ₀ norm, and realizes the reconstruction of the sparse signal through the steepest descent method and the gradient projection principle. Although the SL0 algorithm can quickly reconstruct the sparse signal, the Gaussian function used in the algorithm has poor approximation performance to the l ₀ norm, and when the steepest descent method is used to solve the extreme value problem of the function, a "sawtooth effect" will occur, thus The reconstruction accuracy of the SL0 algorithm is lower. Literature [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. ) proposed a Newton Smoothed l ₀ norm ₍ NSL0) algorithm, which uses an approximate hyperbolic tangent function to approximate the l ₀ norm, and uses the modified Newton method to solve the extreme value problem of the function, thereby improving The reconstruction performance of the SL0 algorithm is improved. Literature [2] (Feng JJ, Zhang G, Wen F Q. MIMO radarimaging based on smoothed l ₀ norm [J]. Mathematical Problems in Engineering, 2015, (8): 1-10.) uses another approximate hyperbolic The tangent function is used to approximate the l ₀ norm, and the extremum of the function is searched in the Newton direction, and then an approximate SL0 (Approximate Smoothed l ₀ norm, ASL0) algorithm is obtained, and then the algorithm is used for MIMO radar target imaging, effectively Improving its imaging performance. The core idea of the SL0 algorithm is to use the Gaussian function to approximate the l ₀ norm, thus transforming the l ₀ norm minimization problem into a smooth function extremum problem, avoiding the direct solution to the l ₀ norm minimization problem. Therefore, in order to further improve the reconstruction performance of the SL0 algorithm for sparse signals, it is very necessary to construct a function with better approximation performance to approximate the l ₀ norm and solve the sparse solution problem corresponding to this function.

Contents of the invention

Aiming at the problem that the smoothing functions such as Gaussian function and approximate hyperbolic tangent function adopted by the SL0 algorithm and its improved algorithm are unsatisfactory to the approximation effect of the _l0 norm, which leads to the problem of poor reconstruction performance of the algorithm, the present invention proposes a method based on modified approximation The smooth l ₀ norm method of hyperbolic tangent function adopts a modified approximate hyperbolic tangent function to achieve a better approximation to l ₀ norm, and solves the extremum problem of the smooth function by Newton's method, thereby effectively improving The sparse signal reconstruction performance of the SL0 algorithm is improved.

Realize above-mentioned technical purpose, reach above-mentioned technical effect, the present invention realizes through the following technical solutions:

A smooth _l0 norm method based on the modified approximate hyperbolic tangent function, comprising the following steps:

(1) a modified approximate hyperbolic tangent function designed to approach the _l0 norm;

(2) Establish a sparse problem model based on the modified approximate hyperbolic tangent function;

(3) Using the modified Newton method to solve the sparse problem represented by the modified function when the parameter σ decreases successively, σ is the shape parameter of the function, and the sparse signal is reconstructed.

Further, the expression of the modified approximate hyperbolic tangent function in the step (1) is:

Among them, x is a variable; σ is the shape parameter of the function.

Further, the step (2) is specifically:

Using the smooth function f _σ (x) to approximate the l ₀ norm, let:

in, is a sparse vector; x _i (i=1,2,...,N) is the i-th element in the sparse vector x.

can get:

Among them, ||·|| ₀ represents the l ₀ norm, then the sparse problem model based on the modified approximate hyperbolic tangent function is:

in, is the measurement matrix; is the perception matrix.

Further, the step (3) is specifically:

(3.1) Take σ as a set of descending sequences [σ ₁ >σ ₂ >,...>σ _J ], where σ _J is a positive number close to zero;

(3.2) Use the modified Newton method to sequentially analyze the time when σ=σ _i (1≤i≤J) Solve to make the vector x gradually approach the global optimal solution.

Further, the step (3.1) is specifically:

initialization:

(a) Set the initial value

(b) Select a suitable sequence [σ ₁ ,σ ₂ ,...,σ _J ], and σ _j+1 =ρσ _j , where ρ(0<ρ<1) is the attenuation factor,

Further, the step (3.2) is specifically:

Algorithm iteration:

forj=1,2,...,J

①Let σ=σ _j ,

②Search for the global minimum value of the function F _σ (x) successively in the modified Newton direction, and project the minimum value onto the feasible set;

for l=1,2,...,L

(a) The modified Newton direction is

(b)

(c) will projected onto the feasible set

③ order

output sparse vector solution

Compared with prior art, the beneficial effect of the present invention:

(1) The core idea of the SL0 algorithm is to use a smooth function to approximate the l ₀ norm, thereby transforming the l ₀ norm minimization problem into a sparse problem represented by a smooth function, and avoiding the NP-hard problem of l ₀ norm minimization direct solution of . A new smoothing function proposed by the present invention can achieve better approximation to the l ₀ norm, and its approximation degree is better than the existing Gaussian function and approximate hyperbolic tangent function. Therefore, the sparse problem represented by the smoothing function is the best It is close to the ideal situation, thus effectively improving the reconstruction performance of the SL0 algorithm for sparse signals.

(2) The smoothing function proposed by the present invention is simple and easy to derive, and the calculation of the modified Newton direction corresponding to the function is simple. Therefore, introducing the smoothing function proposed by the present invention into the SL0 algorithm can not only improve the sparse signal reconstruction performance, but also maintain maintain its high computational efficiency.

Description of drawings

Fig. 1 is the method implementation flowchart of an embodiment of the present invention;

Fig. 2 is the distribution figure of four kinds of functions when σ=0.1;

Figure 3 is a schematic diagram of the original sparse signal and the reconstructed signal obtained by using various algorithms;

Fig. 4 is a schematic diagram of the relationship between the reconstructed SNR and the change of the SNR of different algorithms;

Fig. 5 is a schematic diagram of the relationship between the reconstruction error and the signal-to-noise ratio of different algorithms;

Fig. 6 is a schematic diagram of the relationship between the reconstructed signal-to-noise ratio and the sparsity of each algorithm;

Fig. 7 is a schematic diagram of the relationship between the reconstruction error and the sparsity of each algorithm.

Detailed ways

In order to make the object, technical solution and advantages of the present invention more clear, the present invention will be further described in detail below in conjunction with the examples. It should be understood that the specific embodiments described here are only used to explain the present invention, not to limit the present invention.

The application principle of the present invention will be described in detail below in conjunction with the accompanying drawings.

As shown in Figure 1, a kind of smooth ₁₀ norm method based on revising approximate hyperbolic tangent function that the present invention proposes concrete implementation steps are as follows:

Step 1: designing a modified approximate hyperbolic tangent function for approximating the _l0 norm;

The core idea of the SL0 algorithm is to use the Gaussian function to approximate the l ₀ norm, thus transforming the l ₀ norm minimization problem into a smooth function extremum problem, avoiding the direct solution to the l ₀ norm minimization problem.

The expression of the Gaussian function used in the SL0 algorithm is:

Among them, x is a variable; σ is the shape parameter of the function.

but

make

in, is a sparse vector; x _i (i=1,2,...,N) is the i-th element in the sparse vector x.

can get

Then formula (5) can be transformed into:

in, is the measurement matrix.

In order to improve the approximation degree of the smoothing function in the SL0 algorithm to the l ₀ norm, literature [1] and literature [2] respectively use two different approximate hyperbolic tangent functions to replace the standard Gaussian function. These two different hyperbolic tangent functions The tangent function expressions are:

It can be seen that finding a suitable smoothing function that can effectively approximate the l ₀ norm is crucial to the reconstruction performance of the SL0 algorithm. On the basis of the NSL0 algorithm and the ASL0 algorithm respectively proposed in literature [1] and [2], in order to further improve the approximation degree of the smoothing function to the l ₀ norm, the present invention designs a new smoothing function (modified approximate double Hyperbolic tangent function) to approximate the ₁₀ norm, thereby improving its degree of approximation, the modified approximate hyperbolic tangent function expression is:

Fig. 2 is Gaussian function, two kinds of approximate hyperbolic tangent functions adopted respectively in document [1] and document [2] and the modified approximate hyperbolic tangent function f _σ (x) that the present invention proposes at the function distribution of σ=0.1 place picture. As can be seen from Fig. 2, compared with two kinds of approximate hyperbolic tangent functions proposed in Gaussian function, literature [1] and literature [2], the modified approximate hyperbolic tangent function proposed by the present invention is at x∈[-0.5, 0.5], the "steepness" is greater, indicating that the function has a better approximation to the l ₀ norm.

Step 2: Build a sparse solution problem model based on the modified approximate hyperbolic tangent function

Using the smooth function f _σ (x) to approximate the l ₀ norm, let:

can get

Then the sparse problem model based on the modified approximate hyperbolic tangent function is:

Step 3: Use the modified Newton method to solve the sparse problem represented by the modified function when the parameter σ decreases successively

The shape parameter σ controls the approximation degree of F _σ (x) to the l ₀ norm, that is, the smaller σ is, the closer F _σ (x) is to the l ₀ norm, but at the same time, the local extremum of F _σ (x) The more , the easier it is for F _σ (x) to fall into a local extremum in the process of approaching the l ₀ norm, which greatly increases the difficulty of finding the global minimum of the function.

In order to solve this problem, the present invention first takes σ as a set of descending sequences [σ ₁ >σ ₂ >,...>σ _J ], where σ _J is a positive number close to zero; and then uses the modified Newton The method sequentially solves the formula (13) when σ=σ _i (1≤i≤J), so that the vector x can gradually approach the global optimal solution.

The SL0 algorithm usually uses the steepest descent method to solve the global minimum of the Gaussian function. Although the steps of the steepest descent method are simple, and the number of iterations is small each time the function minimum is calculated, the "sawtooth effect" will appear in the process of searching for the global minimum of the function. ”, thus adversely affecting the reconstruction accuracy of the SL0 algorithm. In view of this problem, the present invention uses the modified Newton method to solve the formula (13), that is, the sparse problem based on the modified approximate hyperbolic tangent function, thereby improving the reconstruction accuracy of the sparse signal. For the function F _σ (x), its Newton direction is:

d＝-A ² F _σ (x)-1ΔF _σ (x) (14)

In the formula:

in,

After calculating the Newton direction d of the function F _σ (x), the matrix Δ ² F _σ (x) is a Hessen matrix, which does not satisfy the positive definite condition, and thus cannot guarantee that the Newton direction d is the descending direction. In order to ensure that d is the descending direction, the diagonal elements in Eq. (16) are modified to construct a new matrix H to replace the Δ ² F _σ (x) matrix in Newton's direction d. The expression of the new matrix H is:

H＝Δ ² (F _σ (x))+ψ (18)

In the formula, ψ is a diagonal matrix. For the convenience of calculation, this paper takes its diagonal elements ψ _i as

From this, it can be obtained that the elements on the i-th diagonal in H are

It can be seen from the above formula that the new matrix H after modification satisfies the positive definite condition, and the matrix H is used to replace the Δ ² F _σ ( _xi ) matrix in the Newton direction d to ensure that the Newton direction d is the descending direction, that is, the improved modified Newton direction for:

Use the modified Newton method to solve the sparse problem represented by the modified function when the parameter σ decreases successively. In actual operation, the following steps are specifically included:

A1 initialization:

(a) Set the initial value

(b) Select a suitable sequence [σ ₁ ,σ ₂ ,...,σ _J ], and σ _j+1 =ρσ _j , where ρ(0<ρ<1) is the attenuation factor,

A2 algorithm iteration:

forj=1,2,...,J

①Let σ=σ _j ,

②Search for the global minimum value of the function F _σ (x) successively in the modified Newton direction, and project the minimum value onto the feasible set.

forl=1,2,...,L

(a) The modified Newton direction is

(b)

(c) will projected onto the feasible set

③ order

A3 output sparse vector solution

In order to verify the sparse reconstruction performance of the method of the present invention, several groups of SL0 algorithms, the NSL0 algorithm proposed in the literature [1], the ASL0 algorithm proposed in the literature [2] and the comparison experiments of the algorithm of the present invention are designed in the present invention. In the simulation experiment, the sparse source signal is randomly generated by the Bernoulli-Gaussian model, which is:

x _i -p·N(0, δ _on )+(1-p)·N(0, δ _off )

Among them, p is the probability of a large non-zero quantity appearing in the source signal; N(0, δ) is Gaussian additive white noise with a mean of zero and a variance of δ; δ _on and δ _off are the comparative Large nonzero coefficients and small nonzero coefficients. Set δ _off << δ _on , and p<<1 to ensure the sparsity of the source signal.

In the simulation experiment, is a random sampling matrix, is the sparse source-signal matrix, is the perception matrix. For y=Dx, when y and D are known, the sparse source signal is reconstructed by using the SL0 algorithm, the NSL0 algorithm, the ASL0 algorithm and the algorithm of the present invention respectively. The parameter settings in the model for generating sparse source signals are respectively M=1000, N=400, δ _on =1, δ _off =10 ⁻³ , p=0.1; in SL0 algorithm, NSL0 algorithm, ASL0 algorithm and the algorithm of the present invention , set σ _J =0.001, ρ=0.7, and the number of inner cycles L=5.

Define the signal-to-noise ratio as:

In the formula, tr(·) means to trace the matrix.

The present invention uses 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, is the estimated solution of the sparse source signal _xi . The reconstruction error is defined as:

Simulation experiment 1: Experiments on sparse signal reconstruction by different methods

Figure 3 is a comparison chart of the original signal and the reconstructed signal obtained by various methods. The source signal sparsity indicates the number of non-zero elements in the signal matrix. In this simulation, the sparseness K=100 and the signal-to-noise ratio SNR=30dB are taken. The inventive method has adopted a kind of revised approximate hyperbolic tangent function to approach _l0 norm, has improved the approximation degree of smooth function to _l0 norm, as can be seen from Fig. 3, compared with SL0 algorithm, NSL0 algorithm and ASL0 algorithm, The sparse signal reconstructed by the method of the present invention is closest to the original source signal.

Simulation experiment 2: the relationship between the reconstruction performance of each algorithm and the signal-to-noise ratio

Figure 4 and Figure 5 respectively show the reconstruction SNR and the relationship between reconstruction error and SNR of the four algorithms. Assuming that the signal-to-noise ratio ranges from 20 to 40dB, and the signal sparsity K is 100, 200 simulation experiments are carried out. From Figure 4 and Figure 5, it can be seen that the approximate hyperbolic tangent function used in the NSL0 algorithm and the ASL0 algorithm has a better approximation to the l ₀ norm than the Gaussian function, and the modified Newton method is used to solve the extremum of the smooth function The problem is to avoid the "sawtooth effect" produced by the steepest descent method in the iterative process, so the reconstruction signal-to-noise ratio of the NSL0 algorithm and the ASL0 algorithm is higher than that of the SL0 algorithm, and the reconstruction error is lower than that of the SL0 algorithm. The inventive method has adopted a kind of revised approximate hyperbolic tangent function, and the degree of its approximation _l0 norm is better than the smooth function in NSL0 algorithm and ASL0 algorithm, as can be seen from Fig. 4 and Fig. 5, the reconstruction of the present invention's algorithm Best performance.

Simulation Experiment 3: The relationship between the reconstruction performance of each algorithm and the degree of sparsity

Figure 6 and Figure 7 show the relationship between the reconstruction signal-to-noise ratio, reconstruction error and sparsity of each algorithm, respectively. Assume that the variation range of the sparsity K is 20-100, and the signal-to-noise ratio is 30dB. It can be seen from Fig. 6 and Fig. 7 that the reconstruction performance of the algorithm of the present invention is always better than that of SL0 algorithm, NSL0 algorithm and ASL0 algorithm under different signal sparsity.

The basic principles and main features of the present invention and the advantages of the present invention have been shown and described above. Those skilled in the industry should understand that the present invention is not limited by the above-mentioned embodiments. What are described in the above-mentioned embodiments and the description only illustrate the principle of the present invention. Without departing from the spirit and scope of the present invention, the present invention will also have Variations and improvements are possible, which fall within the scope of the claimed invention. The protection scope of the present invention is defined by the appended claims and their equivalents.

Claims (6)

1. a smooth ₁₀ norm method based on the modified approximate hyperbolic tangent function, is characterized in that, comprises the following steps: (1) a modified approximate hyperbolic tangent function designed to approach the _l0 norm; (2) Establish a sparse problem model based on the modified approximate hyperbolic tangent function; (3) Using the modified Newton method to solve the sparse problem represented by the modified function when the parameter σ decreases successively, σ is the shape parameter of the function, and the sparse signal is reconstructed. 2. a kind of smooth ₁₀ norm method based on modified approximate hyperbolic tangent function according to claim 1, is characterized in that: the expression of the modified approximate hyperbolic tangent function in described step (1) is: <mrow><msub><mi>f</mi><mi>&amp;sigma;</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mi>exp</mi><mrow><mo>(</mo><mo>-</mo><mo>|</mo><mi>x</mi><msup><mo>|</mo><mrow><mn>1</mn><mo>//</mo><mn>4</mn></mrow></msup><mo>-</mo><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mrow><mi>exp</mi><mrow><mo>(</mo><mo>-</mo><mo>|</mo><mi>x</mi><msup><mo>|</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>/</mo><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mo>)</mo></mrow><mo>+</mo><mi>exp</mi><mrow><mo>(</mo><mo>|</mo><mi>x</mi><msup><mo>|</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo>/</mo><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mi>mfrac></mrow> Among them, x is a variable; σ is the shape parameter of the function. 3. a kind of smooth ₁₀ norm method based on revising approximate hyperbolic tangent function according to claim 2, is characterized in that: described step (2) is specially: Using the smooth function f _σ (x) to approximate the l ₀ norm, let: <mrow><msub><mi>F</mi><mi>&amp;sigma;</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>N</mi><mo>-</mo><munderover><mo>&amp;Sigma;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>f</mi><mi>&amp;sigma;</mi></msub><mrow><mo>(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow> in, is a sparse vector; x _i (i=1,2,...,N) is the i-th element in the sparse vector x. can get: <mrow><mo>|</mo><mo>|</mo><mi>x</mi><mo>|</mo><msub><mo>|</mo><mn>0</mn></msub><mo>=</mo><munder><mi>lim</mi><mrow><mi>&amp;sigma;</mi><mo>&amp;RightArrow;</mo><mn>0</mn></mrow></munder><msub><mi>F</mi><mi>&amp;sigma;</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow> Among them, ||·|| ₀ represents the l ₀ norm, then the sparse problem model based on the modified approximate hyperbolic tangent function is: <mfenced open = "" close = ""><mtable><mtr><mtd><mrow><munder><mi>min</mi><mi>x</mi></munder><msub><mi>F</mi><mi>&amp;sigma;</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mtd><mtd><mrow><mi>s</mi><mo>.</mo><mi>t</mi><mo>.</mo></mrow></mtd><mtd><mrow><mi>y</mi><mo>=</mo><mi>D</mi><mi>x</mi></mrow></mtd></mtr></mtable></mfenced> in, is the measurement matrix; is the perception matrix. 4. a kind of smooth ₁₀ norm method based on revising approximate hyperbolic tangent function according to claim 3, is characterized in that: described step (3) is specially: (3.1) Take σ as a set of descending sequences [σ ₁ >σ ₂ >,...>σ _J ], where σ _J is a positive number close to zero; (3.2) Use the modified Newton method to sequentially analyze the time when σ=σ _i (1≤i≤J) sty=Dx to solve, so that the vector x can gradually approach the global optimal solution. 5. a kind of smooth ₁₀ norm method based on revising approximate hyperbolic tangent function according to claim 4, is characterized in that: described step (3.1) is specially: initialization: (a) Set the initial value (b) Select a suitable sequence [σ ₁ ,σ ₂ ,...,σ _J ], and σ _j+1 =ρσ _j , where ρ(0<ρ<1) is the attenuation factor, 6. according to claim 4 or 5 described a kind of smooth ₁₀ norm method based on modified approximate hyperbolic tangent function, it is characterized in that: described step (3.2) is specially: Algorithm iteration: forj=1,2,...,J ①Let σ=σ _j , ②Search for the global minimum value of the function F _σ (x) successively in the modified Newton direction, and project the minimum value onto the feasible set; for l=1,2,...,L (a) The modified Newton direction is (b) (c) will projected onto the feasible set ③ order output sparse vector solution