The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing

This paper deals with the computational complexity of conditions which guarantee that the NP-hard problem of finding the sparsest solution to an underdetermined linear system can be solved by efficient algorithms. In the literature, several such conditions have been introduced. The most well-known ones are the mutual coherence, the restricted isometry property (RIP), and the nullspace property (NSP). While evaluating the mutual coherence of a given matrix is easy, it has been suspected for some time that evaluating RIP and NSP is computationally intractable in general. We confirm these conjectures by showing that for a given matrix A and positive integer k, computing the best constants for which the RIP or NSP hold is, in general, NP-hard. These results are based on the fact that determining the spark of a matrix is NP-hard, which is also established in this paper. Furthermore, we also give several complexity statements about problems related to the above concepts.Comment: 13 pages; accepted for publication in IEEE Trans. Inf. Theor

DOLPHIn - Dictionary Learning for Phase Retrieval

We propose a new algorithm to learn a dictionary for reconstructing and sparsely encoding signals from measurements without phase. Specifically, we consider the task of estimating a two-dimensional image from squared-magnitude measurements of a complex-valued linear transformation of the original image. Several recent phase retrieval algorithms exploit underlying sparsity of the unknown signal in order to improve recovery performance. In this work, we consider such a sparse signal prior in the context of phase retrieval, when the sparsifying dictionary is not known in advance. Our algorithm jointly reconstructs the unknown signal - possibly corrupted by noise - and learns a dictionary such that each patch of the estimated image can be sparsely represented. Numerical experiments demonstrate that our approach can obtain significantly better reconstructions for phase retrieval problems with noise than methods that cannot exploit such "hidden" sparsity. Moreover, on the theoretical side, we provide a convergence result for our method

An Infeasible-Point Subgradient Method Using Adaptive Approximate Projections

We propose a new subgradient method for the minimization of nonsmooth convex functions over a convex set. To speed up computations we use adaptive approximate projections only requiring to move within a certain distance of the exact projections (which decreases in the course of the algorithm). In particular, the iterates in our method can be infeasible throughout the whole procedure. Nevertheless, we provide conditions which ensure convergence to an optimal feasible point under suitable assumptions. One convergence result deals with step size sequences that are fixed a priori. Two other results handle dynamic Polyak-type step sizes depending on a lower or upper estimate of the optimal objective function value, respectively. Additionally, we briefly sketch two applications: Optimization with convex chance constraints, and finding the minimum l1-norm solution to an underdetermined linear system, an important problem in Compressed Sensing.Comment: 36 pages, 3 figure

Joint Antenna Selection and Phase-Only Beamforming Using Mixed-Integer Nonlinear Programming

In this paper, we consider the problem of joint antenna selection and analog beamformer design in downlink single-group multicast networks. Our objective is to reduce the hardware costs by minimizing the number of required phase shifters at the transmitter while fulfilling given distortion limits at the receivers. We formulate the problem as an L0 minimization problem and devise a novel branch-and-cut based algorithm to solve the resulting mixed-integer nonlinear program to optimality. We also propose a suboptimal heuristic algorithm to solve the above problem approximately with a low computational complexity. Computational results illustrate that the solutions produced by the proposed heuristic algorithm are optimal in most cases. The results also indicate that the performance of the optimal methods can be significantly improved by initializing with the result of the suboptimal method.Comment: to be presented at WSA 201

Extended Successive Convex Approximation for Phase Retrieval with Dictionary Learning

Phase retrieval aims at reconstructing unknown signals from magnitude measurements of linear mixtures. In this paper, we consider the phase retrieval with dictionary learning problem, which includes an additional prior information that the measured signal admits a sparse representation over an unknown dictionary. The task is to jointly estimate the dictionary and the sparse representation from magnitude-only measurements. To this end, we study two complementary formulations and develop efficient parallel algorithms by extending the successive convex approximation framework using a smooth majorization. The first algorithm is termed compact-SCAphase and is preferable in the case of less diverse mixture models. It employs a compact formulation that avoids the use of auxiliary variables. The proposed algorithm is highly scalable and has reduced parameter tuning cost. The second algorithm, referred to as SCAphase, uses auxiliary variables and is favorable in the case of highly diverse mixture models. It also renders simple incorporation of additional side constraints. The performance of both methods is evaluated when applied to blind sparse channel estimation from subband magnitude measurements in a multi-antenna random access network. Simulation results demonstrate the efficiency of the proposed techniques compared to state-of-the-art methods.Comment: This work has been submitted to the IEEE Transactions on Signal Processing for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Extending the rapeseed gene pool with resynthesized Brassica napus II: Heterosis

Hybrid breeding relies on the combination of parents from two differing heterotic groups. However, the genetic diversity in adapted oilseed rape breeding material is rather limited. Therefore, the use of resynthesized Brassica napus as a distant gene pool was investigated. Hybrids were derived from crosses between 44 resynthesized lines with a diverse genetic background and two male sterile winter oilseed rape tester lines. The hybrids were evaluated together with their parents and check cultivars in 2 years and five locations in Germany. Yield, plant height, seed oil, and protein content were monitored, and genetic distances were estimated with molecular markers (127 polymorphic RFLP fragments). Resynthesized lines varied in yield between 40.9 dt/ha and 21.5 dt/ha, or between 85.1 and 44.6% of check cultivar yields. Relative to check cultivars, hybrids varied from 91.6 to 116.6% in yield and from 94.5 to 103.3% in seed oil content. Mid-parent heterosis varied from −3.5 to 47.2% for yield. The genetic distance of parental lines was not significantly correlated with heterosis or hybrid yield. Although resynthesized lines do not meet the elite rapeseed standards, they are a valuable source for hybrid breeding due to their large distance from present breeding material and their high heterosis when combined with European winter oilseed rape

Simvastatin add-on to escitalopram in patients with comorbid obesity and major depression (SIMCODE): study protocol of a multicentre, randomised, double-blind, placebo-controlled trial

Introduction: Major depressive disorder (MDD) and obesity are both common disorders associated with significant burden of disease worldwide. Importantly, MDD and obesity often co-occur, with each disorder increasing the risk for developing the other by about 50%-60%. Statins are among the most prescribed medications with well-established safety and efficacy. Statins are recommended in primary prevention of cardiovascular disease, which has been linked to both MDD and obesity. Moreover, statins are promising candidates to treat MDD because a meta-analysis of pilot randomised controlled trials has found antidepressive effects of statins as adjunct therapy to antidepressants. However, no study so far has tested the antidepressive potential of statins in patients with MDD and comorbid obesity. Importantly, this is a difficult-to-treat population that often exhibits a chronic course of MDD and is more likely to be treatment resistant. Thus, in this confirmatory randomised controlled trial, we will determine whether add-on simvastatin to standard antidepressant medication with escitalopram is more efficacious than add-on placebo over 12 weeks in 160 patients with MDD and comorbid obesity. Methods and analysis: This is a protocol for a randomised, placebo-controlled, double-blind multicentre trial with parallel-group design (phase II). One hundred and sixty patients with MDD and comorbid obesity will be randomised 1:1 to simvastatin or placebo as add-on to standard antidepressant medication with escitalopram. The primary outcome is change in the Montgomery-angstrom sberg Depression Rating Scale (MADRS) score from baseline to week 12. Secondary outcomes include MADRS response (defined as 50% MADRS score reduction from baseline), MADRS remission (defined as MADRS score <10), mean change in patients' self-reported Beck Depression Inventory (BDI-II) and mean change in high-density lipoprotein, low-density lipoprotein and total cholesterol from baseline to week 12. Ethics and dissemination: This protocol has been approved by the ethics committee of the federal state of Berlin (Ethik-Kommission des Landes Berlin, reference: 19/0226-EK 11) and by the relevant federal authority (Bundesinstitut fur Arzneimittel und Medizinprodukte (BfArM), reference: 4043387). Study findings will be published in peer-reviewed journals and will be presented at (inter)national conferences