Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm

The nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing. However, solving the nuclear norm based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem. In this paper, we propose to perform a family of nonconvex surrogates of $L_0$-norm on the singular values of a matrix to approximate the rank function. This leads to a nonconvex nonsmooth minimization problem. Then we propose to solve the problem by Iteratively Reweighted Nuclear Norm (IRNN) algorithm. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem, which has a closed form solution due to the special properties of the nonconvex surrogate functions. We also extend IRNN to solve the nonconvex problem with two or more blocks of variables. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthesized data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms

Generalized Nonconvex Nonsmooth Low-Rank Minimization

As surrogate functions of $L_0$-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex low-rank minimization problem is much more challenging than the nonconvex sparse minimization problem. We observe that all the existing nonconvex penalty functions are concave and monotonically increasing on $[0,\infty)$. Thus their gradients are decreasing functions. Based on this property, we propose an Iteratively Reweighted Nuclear Norm (IRNN) algorithm to solve the nonconvex nonsmooth low-rank minimization problem. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem. By setting the weight vector as the gradient of the concave penalty function, the WSVT problem has a closed form solution. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthetic data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.Comment: IEEE International Conference on Computer Vision and Pattern Recognition, 201

Adaptive detection and ISI mitigation for mobile molecular communication

Current studies on modulation and detection schemes in molecular communication mainly focus on the scenarios with static transmitters and receivers. However, mobile molecular communication is needed in many envisioned applications, such as target tracking and drug delivery. Until now, investigations about mobile molecular communication have been limited. In this paper, a static transmitter and a mobile bacterium-based receiver performing random walk are considered. In this mobile scenario, the channel impulse response changes due to the dynamic change of the distance between the transmitter and the receiver. Detection schemes based on fixed distance fail in signal detection in such a scenario. Furthermore, the intersymbol interference (ISI) effect becomes more complex due to the dynamic character of the signal which makes the estimation and mitigation of the ISI even more difficult. In this paper, an adaptive ISI mitigation method and two adaptive detection schemes are proposed for this mobile scenario. In the proposed scheme, adaptive ISI mitigation, estimation of dynamic distance and the corresponding impulse response reconstruction are performed in each symbol interval. Based on the dynamic channel impulse response in each interval, two adaptive detection schemes, concentration-based adaptive threshold detection (CATD) and peak-time-based adaptive detection (PAD), are proposed for signal detection. Simulations demonstrate that, the ISI effect is significantly reduced and the adaptive detection schemes are reliable and robust for mobile molecular communication

Treating senile dementia with traditional Chinese medicine

Senile dementia is a syndrome in the elderly involving deficits in memory and cognition. There has been a long history of research and medical practice in dementia in China, during which the ancient Chinese people have formed a whole theory and accumulated abundant experience in the treatment of dementia. During recent decades, with new theories and technologies being digested and integrated, progress has been made in the medical and pharmacy research on senile dementia in China. In this review, we will focus on the traditional opinion, clinical practice, and recent progress in pharmacological research in China towards the treatment of dementia. We also discuss the potential trends of global convergence