CN108920777A - A kind of Homotopy singular problem processing method based on random perturbation - Google Patents

Abstract

The Homotopy singular problem processing method based on random perturbation that the invention discloses a kind of, including step 1：Assuming that observation vector y is generated by a unknown signaling x by linear transformation y=Ax+n, wherein A is known observing matrix, and n is noise vector；Step 2：Solution vector x, regularization parameter λ, random perturbation coefficient η and auxiliary set ε are initialized,The number of iterations counting variable l=0 is set；Step 3：The KKT equation group under the influence of random perturbation is constructed, and solves equation group, calculates maximum step-length value Δ λ；Step 4：The Δ λ acquired according to step 3 updates λ, x and ε,Step 5：Enable l=l+1, return step 3, until λ < 10^‑3It terminates, exports complete regularization path.The present invention is based on randomized methods, propose a kind of novel Homotopy singular problem processing method, this method realization is more simple, and computation complexity is lower, while can obtain correct solution path again.

Description

A kind of Homotopy singular problem processing method based on random perturbation

Technical field

The invention belongs to field of signal processing, are related to the processing of singular problem in homotopy (Homotopy) algorithm, specifically Say it is a kind of Homotopy singular problem processing method based on random perturbation.

Background technique

In recent years, with the development of rarefaction representation and compressive sensing theory, l₁Norm minimum problem has been increasingly becoming state Inside and outside research hotspot, and it is widely used in signal processing and optimization field.Homotopy is to solve l₁Norm minimum Problem provides an important active set method, and singular problem is one of the main problem that Homotopy faces, unusual to ask Topic refers to that the coefficient matrix of iterative equation group is irreversible when observing matrix includes repetition, approximation or linearly related column, thus Method is caused to fail.At this stage, less in relation to handling the effective ways of Homotopy singular problem both at home and abroad, in document：Dai J,Xu W,Zhang J,et al.Homotopy algorithm for l1-norm minimisation problems[J] .IET Signal Processing,2015,9(1):In 1-9, a kind of disturbance loading method is proposed, square is observed by modification Each member of battle array usually solves singular problem, however this method realization is complex, and computation complexity is higher.

Summary of the invention

For the deficiency of existing method, ask the invention proposes a kind of novel Homotopy based on random perturbation is unusual Processing method is inscribed, this method improves on existing disturbance loading method, the Karush-only met to each index A small random perturbation is introduced in Kuhn-Tucker (KKT) condition so that every time in iteration only one index enter or from Open active set, it is ensured that coefficient matrix is reversible to handle singular problem.

Include the following steps for realizing technical solution of the invention：

Step 1：Assuming that observation vector y is generated by a unknown signaling x by linear transformation y=Ax+n, wherein A is known Observing matrix, n are noise vector；

Step 2：Solution vector x, regularization parameter λ, random perturbation coefficient η and auxiliary set ε are initialized,Setting changes Generation counting number variable l=0；

Step 3：The KKT equation group under the influence of random perturbation is constructed, and solves equation group, calculates maximum step-length value Δ λ；

Step 4：The Δ λ acquired according to step 3 updates λ, x and ε,

Step 5：Enable l=l+1, return step 3, until λ < 10^-3It terminates, exports complete regularization path.

Beneficial effects of the present invention：

The present invention is based on randomized methods, propose a kind of novel Homotopy singular problem processing method, we Method realization is more simple, and computation complexity is lower, while can obtain correct solution path again.

Detailed description of the invention

Fig. 1 is implementation flow chart of the present invention.

Specific embodiment

1 the invention will be further described with reference to the accompanying drawing.

(1) assume observation vectorBy a unknown signalingIt is generated by linear transformation y=Ax+n, wherein Dimension of the P for observation vector y, dimension of the N for vector x, P < N,For known observing matrix, n is noise vector；

(2) the unknown signaling x solution vector is initializedFor full 0 element, initialize regularization parameter λ=| | A^Ty||_∞, wherein ()^TIndicate transposition operation, | | | |_∞The Infinite Norm of representing matrix or vector initializes random perturbation system NumberEach element be Gauss number (mean value 1, a variance 10^-8), initialization auxiliary setWhereinIt indicates empty set, primary iteration counting how many times variable l=0 is set；

(3) the KKT equation group under the influence of random perturbation is constructed：

Wherein, sgn () indicates sign function,Indicate Hadamard product, () ε is indicated in vector or matrix by collecting The subvector or submatrix of ε index corresponding element or column composition are closed,Similarly.Equation group is solved, is obtained：

Wherein, x '=dx/d λ indicates vector x to the first derivative of λ.Calculate maximum step-length value Δ λ：

Δ λ=max { Δ λ₁,Δλ₂,Δλ₃}

Wherein：

A_iThe i-th column of representing matrix A.

(4) λ, x are updated：

λ=λ+Δ λ

X=x+ Δ λ x '

The corresponding index of the Δ λ acquired is denoted as i^*, ε is updated,

If Δ λ=Δ λ₁, then ε=ε { i^*},

If Δ λ=Δ λ₂Or Δ λ₃, then ε=ε ∪ { i^*},Wherein, indicate set difference operation.

(5) l=l+1, return step (3), until λ < 10- are enabled³It terminates, finally exports complete regularization path.

Effect of the invention is described further below with reference to emulation experiment.

In order to assess the performance of this method, we imitate following observing matrix A and observation vector y using this method True experiment：

Y=[- 0.7,0.1, -1.1, -1.9,0.5]^T

Wherein, the 2nd column with the 6th column of matrix A repeat.Regularization parameter λ is recorded in whole experiment process from initial value To 10^-3All values and corresponding cost function value (Wherein, | | | |₂With | | | |₁Respectively The l of representing matrix or vector₂Norm and l₁Norm) and when each iteration manipulative indexing moving direction, simulation result such as table 1 Shown, table 1 is to correspond to rope in all regularization parameter values obtained in experiment and corresponding cost function value and each iteration The moving direction drawn.

Table 1

As it can be seen from table 1 only one element enters or leaves active set ε, no surprise in each iteration of experimentation Different problem occurs, therefore the present invention can effectively solve singular problem.

The series of detailed descriptions listed above only for feasible embodiment of the invention specifically Protection scope bright, that they are not intended to limit the invention, it is all without departing from equivalent implementations made by technical spirit of the present invention Or change should all be included in the protection scope of the present invention.

Claims (8)

1. a kind of Homotopy singular problem processing method based on random perturbation, which is characterized in that include the following steps：

Step 1：Assuming that observation vector y is generated by a unknown signaling x by linear transformation y=Ax+n, wherein A is known observation Matrix, n are noise vector；

Step 2：The unknown signaling x solution vector, regularization parameter λ, random perturbation coefficient η and auxiliary set ε are initialized,Primary iteration counting how many times variable l=0 is set；

Step 3：The KKT equation group under the influence of random perturbation is constructed, and solves equation group, calculates maximum step-length value Δ λ；

Step 4：The Δ λ acquired according to step 3 updates λ, x and ε,

Step 5：Enable l=l+1, return step 3, until λ < 10^-3It terminates, exports complete regularization path.

2. a kind of Homotopy singular problem processing method based on random perturbation according to claim 1, feature exist In, in the step 1,P is the dimension of observation vector y,N is the dimension of vector x, and has P < N,

3. a kind of Homotopy singular problem processing method based on random perturbation according to claim 1, feature exist In initializing the unknown signaling x solution vector in the step 2For full 0 element, initialize regularization parameter λ= ||A^Ty||_∞, wherein ()^TIndicate transposition operation, | | | |_∞The Infinite Norm of representing matrix or vector initializes random perturbation CoefficientEach element be a Gauss number, initialization auxiliary setWhereinIndicate empty set.

4. a kind of Homotopy singular problem processing method based on random perturbation according to claim 3, feature exist In the mean value of the Gauss number is 1, variance 10^-8。

5. a kind of Homotopy singular problem processing method based on random perturbation according to claim 1, feature exist In in the step 3, KKT equation group is：

Wherein, sgn () indicates sign function,Indicate Hadamard product, ()_εIt indicates in vector or matrix by set ε rope Draw the subvector or submatrix of corresponding element or column composition,It indicates in vector or matrix by gatheringIndex corresponding element or Arrange the subvector or submatrix of composition.

6. a kind of Homotopy singular problem processing method based on random perturbation according to claim 5, feature exist In being obtained by the KKT solving equations：

Wherein, x '=dx/d λ indicates vector x to the first derivative of λ.

7. a kind of Homotopy singular problem processing method based on random perturbation according to claim 6, feature exist In the method for calculating maximum step-length value Δ λ is：

Δ λ=max { Δ λ₁,Δλ₂,Δλ₃}

Wherein：

A_iThe i-th column of representing matrix A.

8. a kind of Homotopy singular problem processing method based on random perturbation according to claim 7, feature exist In the specific implementation of the step 4：

Update λ, x：

λ=λ+Δ λ

X=x+ Δ λ x '

The corresponding index of the Δ λ acquired is denoted as i^*, ε is updated,

If Δ λ=Δ λ₁, then ε=ε { i^*},

If Δ λ=Δ λ₂Or Δ λ₃, then ε=ε ∪ { i^*}Wherein, indicate set difference operation.