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This work considers two popular minimization problems: (i) the minimization of a general convex function f (X) with the domain being positive semi-definite matrices; (ii) the minimization of a general convex function f (X) regularized by the matrix nuclear norm X * with the domain being general matrices. Despite their optimal statistical performance in the literature, these two optimization problems have a high computational complexity even when solved using tailored fast convex solvers. To develop faster and more scalable algorithms, we follow the proposal of Burer and Monteiro to factor the low-rank variable X = UU (for semi-definite matrices) or X = UV (for general matrices) and also replace the nuclear norm X * with ( U 2 F + V 2 F )/2. In spite of the non-convexity of the resulting factored formulations, we prove that each critical point either corresponds to the global optimum of the original convex problems or is a strict saddle where the Hessian matrix has a strictly negative eigenvalue. Such a nice geometric structure of the factored formulations allows many local search algorithms to find a global optimizer even with random initializations.

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In this paper we study the problem of recovering a low-rank matrix from a number of random linear measurements that are corrupted by outliers taking arbitrary values. We consider a nonsmooth nonconvex formulation of the problem, in which we explicitly enforce the low-rank property of the solution by using a factored representation of the matrix variable and employ an 1loss function to robustify the solution against outliers. We show that even when a constant fraction (which can be up to almost half) of the measurements are arbitrarily corrupted, as long as certain measurement operators arising from the measurement model satisfy the so-called 1 / 2 -restricted isometry property, the ground-truth matrix can be exactly recovered from any global minimum of the resulting optimization problem. Furthermore, we show that the objective function of the optimization problem is sharp and weakly convex. Consequently, a subgradient Method (SubGM) with geometrically diminishing step sizes will converge linearly to the ground-truth matrix when suitably initialized. We demonstrate the efficacy of the SubGM for the nonconvex robust low-rank matrix recovery problem with various numerical experiments.

Many signal processing problems-such as analysis, compression, denoising, and reconstruction-can be facilitated by expressing the signal as a linear combination of atoms from a well-chosen dictionary. In this paper, we study possible dictionaries for representing the discrete vector one obtains when collecting a finite set of uniform samples from a multiband analog signal. By analyzing the spectrum of combined time-and multiband-limiting operations in the discrete-time domain, we conclude that the information level of the sampled multiband vectors is essentially equal to the time-frequency area. For representing these vectors, we consider a dictionary formed by concatenating a collection of modulated Discrete Prolate Spheroidal Sequences (DPSS's). We study the angle between the subspaces spanned by this dictionary and an optimal dictionary, and we conclude that the multiband modulated DPSS dictionary-which is simple to construct and more flexible than the optimal dictionary in practical applications-is nearly optimal for representing multiband sample vectors. We also show that the multiband modulated DPSS dictionary not only provides a very high degree of approximation accuracy in an MSE sense for multiband sample vectors (using a number of atoms comparable to the information level), but also that it can provide high-quality approximations of all sampled sinusoids within the bands of interest.

Our objective is to efficiently design a robust projection matrix Φ Φ Φ for the Compressive Sensing (CS) systems when applied to the signals that are not exactly sparse. The optimal projection matrix is obtained by mainly minimizing the average coherence of the equivalent dictionary. In order to drop the requirement of the sparse representation error (SRE) for a set of training data as in [15] [16], we introduce a novel penalty function independent of a particular SRE matrix. Without requiring of training data, we can efficiently design the robust projection matrix and apply it for most of CS systems, like a CS system for image processing with a conventional wavelet dictionary in which the SRE matrix is generally not available. Simulation results demonstrate the efficiency and effectiveness of the proposed approach compared with the state-of-the-art methods. In addition, we experimentally demonstrate with natural images that under similar compression rate, a CS system with a learned dictionary in high dimensions outperforms the one in low dimensions in terms of reconstruction accuracy. This together with the fact that our proposed method can efficiently work in high dimension suggests that a CS system can be potentially implemented beyond the small patches in sparsity-based image processing.

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Arctigenin is a bioactive lignan isolated from the seeds of Arctium lappa L. which has been widely used as a diuretic and a diaphoretic in Traditional Chinese Medicine. In the present study, the authors investigated the effects of arctigenin on tumor migration and invasion in aggressive human breast cancer cells. The MTT assay results showed that arctigenin did not show a significant cytotoxic effect on the cell viability of MDA-MB-231 cells. However, wound healing migration and Boyden chamber invasion assays demonstrated that arctigenin significantly inhibited in vitro migration and invasion of the MDA-MB-231 cells. Furthermore, gelatin zymography results showed that arctigenin reduced the activity of MMP-2 and MMP-9. Western blot analysis results demonstrated that the expression of MMP-2, MMP-9 and heparanase proteins was significantly downregulated following the treatment of arctigenin. Finally, the antiangiogenic activity of arctigenin was also examined by the chick embryo chorioallantoic membrane (CAM) assay. Arctigenin treatment significantly inhibited angiogenesis in the CAM. In conclusion, the results revealed that arctigenin significantly inhibited the migration and invasion of MDA-MB-231 cells by downregulating MMP-2, MMP-9 and heparanase expression. However, further studies are still necessary to investigate the exact mechanisms involved and to explore signal transduction pathways to better understand the biological mechanisms.