On the stable recovery of the sparsest overcomplete representations in presence of noise

Let x be a signal to be sparsely decomposed over a redundant dictionary A, i.e., a sparse coefficient vector s has to be found such that x=As. It is known that this problem is inherently unstable against noise, and to overcome this instability, the authors of [Stable Recovery; Donoho et.al., 2006] have proposed to use an "approximate" decomposition, that is, a decomposition satisfying ||x - A s|| < \delta, rather than satisfying the exact equality x = As. Then, they have shown that if there is a decomposition with ||s||_0 < (1+M^{-1})/2, where M denotes the coherence of the dictionary, this decomposition would be stable against noise. On the other hand, it is known that a sparse decomposition with ||s||_0 < spark(A)/2 is unique. In other words, although a decomposition with ||s||_0 < spark(A)/2 is unique, its stability against noise has been proved only for highly more restrictive decompositions satisfying ||s||_0 < (1+M^{-1})/2, because usually (1+M^{-1})/2 << spark(A)/2. This limitation maybe had not been very important before, because ||s||_0 < (1+M^{-1})/2 is also the bound which guaranties that the sparse decomposition can be found via minimizing the L1 norm, a classic approach for sparse decomposition. However, with the availability of new algorithms for sparse decomposition, namely SL0 and Robust-SL0, it would be important to know whether or not unique sparse decompositions with (1+M^{-1})/2 < ||s||_0 < spark(A)/2 are stable. In this paper, we show that such decompositions are indeed stable. In other words, we extend the stability bound from ||s||_0 < (1+M^{-1})/2 to the whole uniqueness range ||s||_0 < spark(A)/2. In summary, we show that "all unique sparse decompositions are stably recoverable". Moreover, we see that sparser decompositions are "more stable".Comment: Accepted in IEEE Trans on SP on 4 May 2010. (c) 2010 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other work

Investigating the Relationship Between Mental Health and Insomnia in Pregnant Women Referred to Health Centers in Estahban

Pregnancy is the most sensitive period in women's life which makes many physical and mental changes. Sleep problems are one of the issues that are reported by pregnant women; it appears to be associated with psychological consequences in pregnant women. This study aims to investigate the relationship between mental health and insomnia in pregnant women referred to health centers in estahban. This descriptive-analytic study has been done on 182 pregnant women referred to health centers of Estahban in 2015 by available sampling method. Research tools used in this study were general health questionnaire 28 (GHQ 28) and insomnia severity index (ISI).Data were analyzed using Chi-Square and Pearson Correlation tests in SPSS 22 software. Research findings showed that 46.2% of women were suspected of mental disorders, and 58.8% of them suffered from insomnia. According to Chi-square test, there was a significant relationship between total score of mental health and a total score of insomnia(r=0.58, p&lt;0.05). Also, a significant relationship was observed in all variables of mental health and insomnia (p&lt;0.05). Results indicate a high level of mental disorders as well as insomnia among pregnant women; also, the mutual effect of these diseases on each other. As a result, sleep hygiene education as well as appropriate consideration and counseling to pregnant women to treat disorders for achieving a safe pregnancy are recommended

A fast approach for overcomplete sparse decomposition based on smoothed L0 norm

In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined Sparse Component Analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Contrary to previous methods, which usually solve this problem by minimizing the L1 norm using Linear Programming (LP) techniques, our algorithm tries to directly minimize the L0 norm. It is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state-of-the-art interior-point LP solvers, while providing the same (or better) accuracy.Comment: Accepted in IEEE Transactions on Signal Processing. For MATLAB codes, see (http://ee.sharif.ir/~SLzero). File replaced, because Fig. 5 was missing erroneousl

True high-order VCO-based ADC

A novel approach to use a voltage-controlled oscillator (VCO) as the first integrator of a high-order continuous-time delta-sigma modulator (CT-DSM) is presented. In the proposed architecture, the VCO is combined with a digital up-down counter to implement the first integrator of the CT-DSM. Thus, the first integrator is digital-friendly and hence can maximally benefit from technological scaling