arXiv subjects

Lingchen Kong

Publications and source records attributed to Lingchen Kong.

22 records · Page 2Linked to original sources

arXiv2014

Exact Recovery for Sparse Signal via Weighted $l_1$ Minimization

Numerical experiments in literature on compressed sensing have indicated that the reweighted $l_1$ minimization performs exceptionally well in recovering sparse signal. In this paper, we develop exact recovery conditions and algorithm for sparse signal via weighted $l_1$ minimization from the insight of the classical NSP (null space property) and RIC (restricted isometry constant) bound. We first introduce the concept of WNSP (weighted null space property) and reveal that it is a necessary and sufficient condition for exact recovery. We then prove that the RIC bound by weighted $l_1$ minimization is $δ_{ak}<\sqrt{\frac{a-1}{a-1+γ^2}}$, where $a>1$, $0<γ\leq1$ is determined by an optimization problem over the null space. When $γ< 1$ this bound is greater than $\sqrt{\frac{a-1}{a}}$ from $l_1$ minimization. In addition, we also establish the bound on $δ_k$ and show that it can be larger than the sharp one 1/3 via $l_1$ minimization and also greater than 0.4343 via weighted $l_1$ minimization under some mild cases. Finally, we achieve a modified iterative reweighted $l_1$ minimization (MIRL1) algorithm based on our selection principle of weight, and the numerical experiments demonstrate that our algorithm behaves much better than $l_1$ minimization and iterative reweighted $l_1$ minimization (IRL1) algorithm.

cs.IT↗

arXiv2013

New RIC Bounds via l_q-minimization with 0<q<=1 in Compressed Sensing

The restricted isometry constants (RICs) play an important role in exact recovery theory of sparse signals via l_q(0 =4/3 to guarantee the exact recovery of k sparse signals through the l_1 minimization. This paper aims to establish new RICs bounds via l_q(0 1, (ii)several sufficient conditions can be derived, such as for any 0 =2, for any 1/2 =1, (iii) the bound on δ_k is given as well for any 0 =2) is even or δ_k<0.3203 when k(>=3) is odd.

cs.IT↗

arXiv2011

Sufficient Conditions for Low-rank Matrix Recovery, Translated from Sparse Signal Recovery

The low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, system identification and control. This class of optimization problems is $\N¶$-hard and a popular approach replaces the rank function with the nuclear norm of the matrix variable. In this paper, we extend the concept of $s$-goodness for a sensing matrix in sparse signal recovery (proposed by Juditsky and Nemirovski [Math Program, 2011]) to linear transformations in LMR. Then, we give characterizations of $s$-goodness in the context of LMR. Using the two characteristic $s$-goodness constants, $γ_s$ and $\hatγ_s$, of a linear transformation, not only do we derive necessary and sufficient conditions for a linear transformation to be $s$-good, but also provide sufficient conditions for exact and stable $s$-rank matrix recovery via the nuclear norm minimization under mild assumptions. Moreover, we give computable upper bounds for one of the $s$-goodness characteristics which leads to verifiable sufficient conditions for exact low-rank matrix recovery.

cs.IT↗

arXiv2010

Homogeneous Cone Complementarity Problems and $P$ Properties

We consider existence and uniqueness properties of a solution to homogeneous cone complementarity problem (HCCP). Employing the $T$-algebraic characterization of homogeneous cones, we generalize the $P, P_0, R_0$ properties for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of HCCP. We prove that if a continuous function has either the order-$P_0$ and $R_0$, or the $P_0$ and $R_0$ properties then all the associated HCCPs have solutions. In particular, if a continuous function has the trace-$P$ property then the associated HCCP has a unique solution (if any); if it has the uniform-trace-$P$ property then the associated HCCP has the global uniqueness (of the solution) property (GUS). We present a necessary condition for a nonlinear transformation to have the GUS property. Moreover, we establish a global error bound for the HCCP with the uniform-trace-$P$ property. Finally, we study the HCCP with the relaxation transformation on a $T$-algebra and automorphism invariant properties for homogeneous cone linear complementarity problem.

math.OC↗