We study the complexity of expressing the greatest common divisor of n positive numbers as a linear combination of the numbers. We prove the NP-completeness of finding an optimal set of multipliers with respect to either the L0 metric or the L∞ norm. We present and analyze a new method for expressing the gcd of n numbers as their linear combination and give an upper bound on the size of the lar...

Introduction: In dynamic MRI, spatio-temporal resolution is a very important issue. Recently, compressed sensing approach has become a highly attracted imaging technique since it enables accelerated acquisition without aliasing artifacts. Our group has proposed an l1-norm based compressed sensing dynamic MRI called k-t FOCUSS which outperforms the existing methods. However, it is known that the...

The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models, including balanced approach [18, 7] and analysis based approach [11, 31, 50], is due to their capability of sparsely approximating piecewise smooth function...

We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed l1 (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates l0 and l1 norms through a nonnegative parameter a ∈ (0,+∞), similar to lp with p ∈ (0, 1]. TL1 is known in the statistics literature to enjoy three desired properties: unbiasedness, sparsity and Lipschitz co...

We consider a reaction-diffusion equation in a cellular flow. We prove that in the strong flow regime there are two possible scenario for the initial data that is compactly supported and the size of the support is large enough. If the flow cells are large compared to the reaction length scale, propagating fronts will always form. For the small cell size, any finitely supported initial data will...

Abstract Compressed sensing technology is currently a relatively mature channel estimation technology, and the sparsity of underwater acoustic provides basis for application compressed to channel. Channel one most commonly used signal reconstruction methods. The reconstructed based on improve reliability transmission. key SL0 algorithm. Based algorithm, new objective function proposed better ap...

The problem of outlier detection consists in finding data that is not representative of the population from which it was ostensibly derived. Recently, to solve this problem, Liu et al. [1] proposed a two steps hypersphere-based approach, taking into account a confidence score pre-calculated for each input data. Defining these scores in a first step, independently from the second one, makes this...

Recent advances in stochastic optimization and regularized dual averaging approaches revealed a substantial interest for a simple and scalable stochastic method which is tailored to some more specific needs. Among the latest one can find sparse signal recovery and l0-based sparsity inducing approaches. These methods in particular can force many components of the solution shrink to zero thus cla...

This paper considers the underdetermined blind separation of multiple input multiple output (MIMO) radar signals that are insufficiently sparse in both time and frequency domains under noisy conditions, while traditional algorithms are usually applied in the ideal sparse environment. An effective separation method based on single source point (SSP) identification and time-frequency smoothed l0 ...