arXiv subjects

Angang Cui

Publications and source records attributed to Angang Cui.

13 recordsLinked to original sources

arXiv2020

Adaptive iterative singular value thresholding algorithm to low-rank matrix recovery

The problem of recovering a low-rank matrix from the linear constraints, known as affine matrix rank minimization problem, has been attracting extensive attention in recent years. In general, affine matrix rank minimization problem is a NP-hard. In our latest work, a non-convex fraction function is studied to approximate the rank function in affine matrix rank minimization problem and translate the NP-hard affine matrix rank minimization problem into a transformed affine matrix rank minimization problem. A scheme of iterative singular value thresholding algorithm is generated to solve the regularized transformed affine matrix rank minimization problem. However, one of the drawbacks for our iterative singular value thresholding algorithm is that the parameter $a$, which influences the behaviour of non-convex fraction function in the regularized transformed affine matrix rank minimization problem, needs to be determined manually in every simulation. In fact, how to determine the optimal parameter $a$ is not an easy problem. Here instead, in this paper, we will generate an adaptive iterative singular value thresholding algorithm to solve the regularized transformed affine matrix rank minimization problem. When doing so, our new algorithm will be intelligent both for the choice of the regularized parameter $\lambda$ and the parameter $a$.

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arXiv2019

Nonconvex fraction function recovery sparse signal by convex optimization algorithm

In this paper, we will generate a convex iterative FP thresholding algorithm to solve the problem $(FP^{\lambda}_{a})$. Two schemes of convex iterative FP thresholding algorithms are generated. One is convex iterative FP thresholding algorithm-Scheme 1 and the other is convex iterative FP thresholding algorithm-Scheme 2. A global convergence theorem is proved for the convex iterative FP thresholding algorithm-Scheme 1. Under an adaptive rule, the convex iterative FP thresholding algorithm-Scheme 2 will be adaptive both for the choice of the regularized parameter $\lambda$ and parameter $a$. These are the advantages for our two schemes of convex iterative FP thresholding algorithm compared with our previous proposed two schemes of iterative FP thresholding algorithm. At last, we provide a series of numerical simulations to test the performance of the convex iterative FP thresholding algorithm-Scheme 2, and the simulation results show that our convex iterative FP thresholding algorithm-Scheme 2 performs very well in recovering a sparse signal.

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arXiv2018

A non-convex approach to low-rank and sparse matrix decomposition

In this paper, we develop a nonconvex approach to the problem of low-rank and sparse matrix decomposition. In our nonconvex method, we replace the rank function and the $l_{0}$-norm of a given matrix with a non-convex fraction function on the singular values and the elements of the matrix respectively. An alternative direction method of multipliers algorithm is utilized to solve our proposed nonconvex problem with the nonconvex fraction function penalty. Numerical experiments on some low-rank and sparse matrix decomposition problems show that our method performs very well in recovering low-rank matrices which are heavily corrupted by large sparse errors.

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arXiv2018

A New Nonconvex Strategy to Affine Matrix Rank Minimization Problem

The affine matrix rank minimization (AMRM) problem is to find a matrix of minimum rank that satisfies a given linear system constraint. It has many applications in some important areas such as control, recommender systems, matrix completion and network localization. However, the problem (AMRM) is NP-hard in general due to the combinational nature of the matrix rank function. There are many alternative functions have been proposed to substitute the matrix rank function, which lead to many corresponding alternative minimization problems solved efficiently by some popular convex or nonconvex optimization algorithms. In this paper, we propose a new nonconvex function, namely, $TL_{\alpha}^{\epsilon}$ function (with $0\leq\alpha<1$ and $\epsilon>0$), to approximate the rank function, and translate the NP-hard problem (AMRM) into the $TL_{p}^{\epsilon}$ function affine matrix rank minimization (TLAMRM) problem. Firstly, we study the equivalence of problem (AMRM) and (TLAMRM), and proved that the uniqueness of global minimizer of the problem (TLAMRM) also solves the NP-hard problem (AMRM) if the linear map $\mathcal{A}$ satisfies a restricted isometry property (RIP). Secondly, an iterative thresholding algorithm is proposed to solve the regularization problem (RTLAMRM) for all $0\leq\alpha<1$ and $\epsilon>0$. At last, some numerical results on low-rank matrix completion problems illustrated that our algorithm is able to recover a low-rank matrix, and the extensive numerical on image inpainting problems shown that our algorithm performs the best in finding a low-rank image compared with some state-of-art methods.

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arXiv2018

Iterative thresholding algorithm based on non-convex method for modified lp-norm regularization minimization

Recently, the $\l_{p}$-norm regularization minimization problem $(P_{p}^{\lambda})$ has attracted great attention in compressed sensing. However, the $\l_{p}$-norm $\|x\|_{p}^{p}$ in problem $(P_{p}^{\lambda})$ is nonconvex and non-Lipschitz for all $p\in(0,1)$, and there are not many optimization theories and methods are proposed to solve this problem. In fact, it is NP-hard for all $p\in(0,1)$ and $\lambda>0$. In this paper, we study two modified $\l_{p}$ regularization minimization problems to approximate the NP-hard problem $(P_{p}^{\lambda})$. Inspired by the good performance of Half algorithm and $2/3$ algorithm in some sparse signal recovery problems, two iterative thresholding algorithms are proposed to solve the problems $(P_{p,1/2,\epsilon}^{\lambda})$ and $(P_{p,2/3,\epsilon}^{\lambda})$ respectively. Numerical results show that our algorithms perform effectively in finding the sparse signal in some sparse signal recovery problems for some proper $p\in(0,1)$.

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arXiv2018

Sparse Portfolio Selection via Non-convex Fraction Function

In this paper, a continuous and non-convex promoting sparsity fraction function is studied in two sparse portfolio selection models with and without short-selling constraints. Firstly, we study the properties of the optimal solution to the problem $(FP_{a,\lambda,\eta})$ including the first-order and the second optimality condition and the lower and upper bound of the absolute value for its nonzero entries. Secondly, we develop the thresholding representation theory of the problem $(FP_{a,\lambda,\eta})$. Based on it, we prove the existence of the resolvent operator of gradient of $P_{a}(x)$, calculate its analytic expression, and propose an iterative fraction penalty thresholding (IFPT) algorithm to solve the problem $(FP_{a,\lambda,\eta})$. Moreover, we also prove that the value of the regularization parameter $\lambda>0$ can not be chosen too large. Indeed, there exists $\bar{\lambda}>0$ such that the optimal solution to the problem $(FP_{a,\lambda,\eta})$ is equal to zero for any $\lambda>\bar{\lambda}$. At last, inspired by the thresholding representation theory of the problem $(FP_{a,\lambda,\eta})$, we propose an iterative nonnegative fraction penalty thresholding (INFPT) algorithm to solve the problem $(FP_{a,\lambda,\eta}^{\geq})$. Empirical results show that our methods, for some proper $a>0$, perform effective in finding the sparse portfolio weights with and without short-selling constraints.

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arXiv2018

Modified lp-norm regularization minimization for sparse signal recovery

In numerous substitution models for the $\l_{0}$-norm minimization problem $(P_{0})$, the $\l_{p}$-norm minimization $(P_{p})$ with $0 0$, this modified function would like to interpolate the $\l_{p}$-norm $\|x\|_{p}^{p}$. By this transformation, we translated the $\l_{p}$-norm regularization minimization $(P_{p}^{\lambda})$ into a modified $\l_{p}$-norm regularization minimization $(P_{p}^{\lambda,\epsilon})$. Then, we develop the thresholding representation theory of the problem $(P_{p}^{\lambda,\epsilon})$, and based on it, the IT algorithm is proposed to solve the problem $(P_{p}^{\lambda,\epsilon})$ for all $0<p<1$. Indeed, we could get some much better results by choosing proper $p$, which is one of the advantages for our algorithm compared with other methods. Numerical results also show that, for some proper $p$, our algorithm performs the best in some sparse signal recovery problems compared with some state-of-art methods.

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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 $(FP_{a,\lambda}^{\geq})$. It is proved that the optimal solution of the problem $(P_{0}^{\geq})$ could be approximately obtained by solving the regularization problem $(FP_{a,\lambda}^{\geq})$ if some specific conditions satisfied. Secondly, we propose a nonnegative iterative thresholding algorithm to solve the regularization problem $(FP_{a,\lambda}^{\geq})$ 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.

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arXiv2017

Generalized singular value thresholding operator to affine matrix rank minimization problem

It is well known that the affine matrix rank minimization problem is NP-hard and all known algorithms for exactly solving it are doubly exponential in theory and in practice due to the combinational nature of the rank function. In this paper, a generalized singular value thresholding operator is generated to solve the affine matrix rank minimization problem. Numerical experiments show that our algorithm performs effectively in finding a low-rank matrix compared with some state-of-art methods.

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arXiv2017

Minimization of fraction function penalty in compressed sensing

In the paper, we study the minimization problem of a non-convex sparsity promoting penalty function $$P_{a}(x)=\sum_{i=1}^{n}p_{a}(x_{i})=\sum_{i=1}^{n}\frac{a|x_{i}|}{1+a|x_{i}|}$$ in compressed sensing, which is called fraction function. Firstly, we discuss the equivalence of $\ell_{0}$ minimization and fraction function minimization. It is proved that there corresponds a constant $a^{**}>0$ such that, whenever $a>a^{**}$, every solution to $(FP_{a})$ also solves $(P_{0})$, that the uniqueness of global minimizer of $(FP_{a})$ and its equivalence to $(P_{0})$ if the sensing matrix $A$ satisfies a restricted isometry property (RIP) and, last but the most important, that the optimal solution to the regularization problem $(FP_{a}^\lambda)$ also solves $(FP_{a})$ if the certain condition is satisfied, which is similar to the regularization problem in convex optimal theory. Secondly, we study the properties of the optimal solution to the regularization problem $(FP^{\lambda}_{a})$ including the first-order and the second optimality condition and the lower and upper bound of the absolute value for its nonzero entries. Finally, we derive the closed form representation of the optimal solution to the regularization problem ($FP_{a}^{\lambda}$) for all positive values of parameter $a$, and propose an iterative $FP$ thresholding algorithm to solve the regularization problem $(FP_{a}^{\lambda})$. We also provide a series of experiments to assess performance of the $FP$ algorithm, and the experiment results show that, compared with soft thresholding algorithm and half thresholding algorithms, the $FP$ algorithm performs the best in sparse signal recovery with and without measurement noise.

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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 $\rho_{a}$ in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem $(QP_{a}^{\lambda})$ 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.

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arXiv2016

Exact recovery low-rank matrix via transformed affine matrix rank minimization

The goal of affine matrix rank minimization problem is to reconstruct a low-rank or approximately low-rank matrix under linear constraints. In general, this problem is combinatorial and NP-hard. In this paper, a nonconvex fraction function is studied to approximate the rank of a matrix and translate this NP-hard problem into a transformed affine matrix rank minimization problem. The equivalence between these two problems is established, and we proved that the uniqueness of the global minimizer of transformed affine matrix rank minimization problem also solves affine matrix rank minimization problem if some conditions are satisfied. Moreover, we also proved that the optimal solution to the transformed affine matrix rank minimization problem can be approximately obtained by solving its regularization problem for some proper smaller $\lambda>0$. Lastly, the DC algorithm is utilized to solve the regularization transformed affine matrix rank minimization problem and the numerical experiments on image inpainting problems show that our method performs effectively in recovering low-rank images compared with some state-of-art algorithms.

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arXiv2016

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 $\lambda>0$ can not be chosen too large. Indeed, there exists $\bar{\lambda}>0$ such that the optimal solution of the regularization transformed affine matrix rank minimization problem is equal to zero for any $\lambda>\bar{\lambda}$. 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.

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