arXiv subjects

Jigen Peng

Publications and source records attributed to Jigen Peng.

At least 55 records · Page 3Linked to original sources

arXiv2017

Non-convex Fraction Function Penalty: Sparse Signals Recovered from Quasi-linear Systems

The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the $\ell_{0}$-norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set most of the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function $ρ_{a}$ in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem $(QP_{a}^λ)$ for all $a>0$. With the change of parameter $a>0$, our algorithm could get a promising result, which is one of the advantages for our algorithm compared with other algorithms. Numerical experiments show that our method performs much better compared with some state-of-art methods.

math.OC↗

arXiv2017

Recovering Sparse Nonnegative Signals via Non-convex Fraction Function Penalty

Many real world practical problems can be formulated as $\ell_{0}$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative signals to underdetermined linear systems. They have been widely applied in signal and image processing, machine learning, pattern recognition and computer vision. Unfortunately, this $\ell_{0}$-minimization problem with nonnegativity constraint is computational and NP-hard because of the discrete and discontinuous nature of the $\ell_{0}$-norm. In this paper, we replace the $\ell_{0}$-norm with a non-convex fraction function, and study the minimization problem of this non-convex fraction function in recovering the sparse nonnegative signals from an underdetermined linear system. Firstly, we discuss the equivalence between $(P_{0}^{\geq})$ and $(FP_{a}^{\geq})$, and the equivalence between $(FP_{a}^{\geq})$ and $(FP_{a,λ}^{\geq})$. It is proved that the optimal solution of the problem $(P_{0}^{\geq})$ could be approximately obtained by solving the regularization problem $(FP_{a,λ}^{\geq})$ if some specific conditions satisfied. Secondly, we propose a nonnegative iterative thresholding algorithm to solve the regularization problem $(FP_{a,λ}^{\geq})$ for all $a>0$. Finally, some numerical experiments on sparse nonnegative siganl recovery problems show that our method performs effective in finding sparse nonnegative signals compared with the linear programming.

math.OC↗

arXiv2017

Affine matrix rank minimization problem via non-convex fraction function penalty

Affine matrix rank minimization problem is a fundamental problem with a lot of important applications in many fields. It is well known that this problem is combinatorial and NP-hard in general. In this paper, a continuous promoting low rank non-convex fraction function is studied to replace the rank function in this NP-hard problem. Inspired by our former work in compressed sensing, an iterative singular value thresholding algorithm is proposed to solve the regularization transformed affine matrix rank minimization problem. For different $a>0$, we could get a much better result by adjusting the different value of $a$, which is one of the advantages for the iterative singular value thresholding algorithm compared with some state-of-art methods. Some convergence results are established and numerical experiments show that this thresholding algorithm is feasible for solving the regularization transformed affine matrix rank minimization problem. Moreover, we proved that the value of the regularization parameter $λ>0$ can not be chosen too large. Indeed, there exists $\barλ>0$ such that the optimal solution of the regularization transformed affine matrix rank minimization problem is equal to zero for any $λ>\barλ$. Numerical experiments on matrix completion problems show that our method performs powerful in finding a low-rank matrix and the numerical experiments about image inpainting problems show that our algorithm has better performances than some state-of-art methods.

math.OC↗

arXiv2017

Analysis of equivalence relation in joint sparse recovery

The joint sparse recovery problem is a generalization of the single measurement vector problem which is widely studied in Compressed Sensing and it aims to recovery a set of jointly sparse vectors. i.e. have nonzero entries concentrated at common location. Meanwhile l_p-minimization subject to matrices is widely used in a large number of algorithms designed for this problem. Therefore the main contribution in this paper is two theoretical results about this technique. The first one is to prove that in every multiple systems of linear equation, there exists a constant p* such that the original unique sparse solution also can be recovered from a minimization in l_p quasi-norm subject to matrices whenever 0< p<p*. The other one is to show an analysis expression of such p*. Finally, we display the results of one example to confirm the validity of our conclusions.

cs.IT↗

arXiv2016

Infinite-dimensional Bayesian approach for inverse scattering problems of a fractional Helmholtz equation

In this paper, we focus on a new wave equation described wave propagation in the attenuation medium. In the first part of this paper, based on the time-domain space fractional wave equation, we formulate the frequency-domain equation named as fractional Helmholtz equation. According to the physical interpretations, this new model could be divided into two separate models: loss-dominated model and dispersion-dominated model. For the loss-dominated model (it is an integer- and fractional-order mixed elliptic equation), a well-posedness theory has been established and the Lipschitz continuity of the scattering field with respect to the scatterer has also been established.Because the complexity of the dispersion-dominated model (it is an integer- and fractional-order mixed elliptic system), we only provide a well-posedness result for sufficiently small wavenumber. In the second part of this paper, we generalize the Bayesian inverse theory in infinite-dimension to allow a part of the noise depends on the target function (the function needs to be estimated). Then, we prove that the estimated function tends to be the true function if both the model reduction error and the white noise vanish. At last, our theory has been applied to the loss-dominated model with absorbing boundary condition.

math.AP↗

arXiv2016

A randomized intertial primal-dual fixed point algorithm for monotone inclusions

In this paper, we propose a randomized intertial block-coordinate primaldual fixed point algorithm to solve a wide array of monotone inclusion problems base on the modification of the heavy ball method of Nesterov. These methods rely on a sweep of blocks of variables which are activated at each iteration according to a random rule. To this end we formulate the inertial version of the Krasnosel'skii-Mann algorithm for approximating the set of fixed points of a quasinonexpansive operator, for which we also provide an exhaustive convergence analysis. As a by-product, we can obtain some intertial block-coordinate operator splitting methods for solving composite monotone inclusion and convex minimization problems.

math.OC↗

arXiv2016

A Carleman type estimate of variable order space-fractional diffusion equations and applications to some inverse problems

Variable order space-fractional diffusion equation derived as an important model to describe complex anomalous diffusion phenomenon. In this article, well-posedness theory has been constructed for equations with the "Dirichlet" or the "Neumann" type volume-constrained conditions by using the technique of the nonlocal vector calculus. Then some regularity properties have been obtained under the variable-order Sobolev space framework. By choosing a space-independent weight function and using the technique of the nonlocal vector calculus, a Carleman type estimate has been obtained. At last, the Carleman type estimate has been used both to a backward diffusion problem and an inverse source problem to obtain some uniqueness and stability results.

math.AP↗

arXiv2016

Studies on an inverse source problem for a space-time fractional diffusion equation by constructing a strong maximum principle

In this paper, we focus on a space-time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result for an inverse source problem on the determination of the temporal component of the inhomogeneous term.

math.AP↗

arXiv2016

Variable Total Variation Regularization for Backward Time-Space Fractional Diffusion Problem

In this paper, we consider a backward problem for a time-space fractional diffusion process. For this problem, we propose to construct the initial data by minimizing data residual error in fourier space domain and variable total variation (TV) regularizing term which can protect the edges as TV regularizing term and reduce staircasing effect. The well-posedness of this optimization problem is studied under a very general setting. Actually, we write the time-space fractional diffusion equation as an abstract fractional differential equation and get our results by using fractional semigroup theory, so our results can be applied to other backward problems for more general fractional differential equations. Then a modified Bregman iterative algorithm is proposed to approximate the minimizer. The new features of this algorithm is that the regularizing term changed in each step and we need not to solve the complexed Euler-Lagrange equations of variable TV regularizing term (just need to solve a simpler Euler-Lagrange equations). The convergence of this algorithm and the strategy of choosing parameters are also obtained. Numerical implementations are given to support our analysis to show the flexibility of our minimization model.

math.NA↗

arXiv2016

A stochastic coordinate descent inertial primal-dual algorithm for large-scale composite optimization

We consider an inertial primal-dual algorithm to compute the minimizations of the sum of two convex functions and the composition of another convex function with a continuous linear operator. With the idea of coordinate descent, we design a stochastic coordinate descent inertial primal-dual splitting algorithm. Moreover, in order to prove the convergence of the proposed inertial algorithm, we formulate first the inertial version of the randomized Krasnosel'skii-Mann iterations algorithm for approximating the set of fixed points of a nonexpansive operator and investigate its convergence properties. Then the convergence of stochastic coordinate descent inertial primal-dual splitting algorithm is derived by applying the inertial version of the randomized Krasnosel'skii-Mann iterations to the composition of the proximity operator.

math.OC↗

arXiv2016

Efficient primal-dual fixed point algorithm with dynamic stepsize for convex problems with applications to imaging restoration

We consider the problem of finding the minimization of the sum of a convex function and the composition of another convex function with a continuous linear operator from the view of fixed point algorithms based on proximity operators. We design a primal-dual fixed point algorithm with dynamic stepsize based on the proximity operator and obtain a scheme with a closed form solution for each iteration. Based on Modified Mann iteration and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithm.

math.OC↗

arXiv2016

An inertial primal-dual fixed point algorithm for composite optimization problems

We consider an inertial primal-dual fixed point algorithm (IPDFP) to compute the minimizations of the following Problem (1.1). This is a full splitting approach, in the sense that the nonsmooth functions are processed individually via their proximity operators. The convergence of the IPDFP is obtained by reformulating the Problem (1.1) to the sum of three convex functions. This work brings together and notably extends several classical splitting schemes, like the primaldual method proposed by Chambolle and Pock, and the recent proximity algorithms of Charles A. et al designed for the L1/TV image denoising model. The iterative algorithm is used for solving nondifferentiable convex optimization problems arising in image processing. The experimental results indicate that the proposed IPDFP iterative algorithm performs well with respect to state-of-the-art methods.

math.OC↗

arXiv2016

A stochastic coordinate descent primal-dual algorithm with dynamic stepsize for large-scale composite optimization

In this paper we consider the problem of finding the minimizations of the sum of two convex functions and the composition of another convex function with a continuous linear operator. With the idea of coordinate descent, we design a stochastic coordinate descent primal-dual splitting algorithm with dynamic stepsize. Based on randomized Modified Krasnosel'skii-Mann iterations and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithms. Moreover, we give two applications of our method.

math.OC↗

arXiv2016

A stochastic coordinate descent splitting primal-dual fixed point algorithm and applications to large-scale composite optimization

We consider the problem of finding the minimizations of the sum of two convex functions and the composition of another convex function with a continuous linear operator from the view of fixed point algorithms based on proximity operators, which is is inspired by recent results of Chen, Huang and Zhang. With the idea of coordinate descent, we design a stochastic coordinate descent splitting primal- dual fixed point algorithm. Based on randomized krasnosel'skii mann iterations and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithms.

math.OC↗

arXiv2016

Explosive solutions of parabolic stochastic partial differential equations with L$\acute{e}$vy noise

In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a L$\acute{\mbox{e}}$vy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that the positive solutions will blow up in finite time in mean $L^{p}$-norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrated the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by L$\acute{\mbox{e}}$vy noise has a global solution.

math.PR↗

arXiv2015

When is P such that l_0-minimization Equals to l_p-minimization

In this paper, we present an analysis expression of p(A,b) such that the unique solution to l_0-minimization also can be the unique solution to l_p-minimization for any 0<p<p(A,b). Furthermore, the main contribution of this paper isn't only the analysis expressed of such p^(A,b) but also its proof. Finally, we display the results of two examples to confirm the validity of our conclusions

cs.IT↗

arXiv2015

Bayesian approach to inverse problems for functions with variable index Besov prior

We adopt Bayesian approach to consider the inverse problem of estimate a function from noisy observations. One important component of this approach is the prior measure. Total variation prior has been proved with no discretization invariant property, so Besov prior has been proposed recently. Different prior measures usually connect to different regularization terms. Variable index TV, variable index Besov regularization terms have been proposed in image analysis, however, there are no such prior measure in Bayesian theory. So in this paper, we propose a variable index Besov prior measure which is a Non-Guassian measure. Based on the variable index Besov prior measure, we build the Bayesian inverse theory. Then applying our theory to integer and fractional order backward diffusion problems. Although there are many researches about fractional order backward diffusion problems, we firstly apply Bayesian inverse theory to this problem which provide an opportunity to quantify the uncertainties for this problem.

math.ST↗

arXiv2015

Well-posedness for compressible MHD system with highly oscillating initial data

In this paper, we transform compressible MHD system written in Euler coordinate to Lagrange coordinate in critical Besov space. Then we construct unique local solutions for compressible MHD system. Our results improve the range of Lebesgue exponent in Besov space from $[2, N)$ to $[2, 2N)$ with $N$ stands for dimension. In addition, we give a lower bound for the maximal existence time which is important for our construction of global solutions. Based on the local solution, we obtain a unique global solution with high oscillating initial velocity and density by using effective viscous flux and Hoff's energy methods to explore the structure of compressible MHD system.

math.AP↗