On Model-Based RIP-1 Matrices

The Restricted Isometry Property (RIP) is a fundamental property of a matrix enabling sparse recovery. Informally, an m x n matrix satisfies RIP of order k in the l_p norm if ||Ax||_p \approx ||x||_p for any vector x that is k-sparse, i.e., that has at most k non-zeros. The minimal number of rows m necessary for the property to hold has been extensively investigated, and tight bounds are known. Motivated by signal processing models, a recent work of Baraniuk et al has generalized this notion to the case where the support of x must belong to a given model, i.e., a given family of supports. This more general notion is much less understood, especially for norms other than l_2. In this paper we present tight bounds for the model-based RIP property in the l_1 norm. Our bounds hold for the two most frequently investigated models: tree-sparsity and block-sparsity. We also show implications of our results to sparse recovery problems.Comment: Version 3 corrects a few errors present in the earlier version. In particular, it states and proves correct upper and lower bounds for the number of rows in RIP-1 matrices for the block-sparse model. The bounds are of the form k log_b n, not k log_k n as stated in the earlier versio

Sparsity and cosparsity for audio declipping: a flexible non-convex approach

This work investigates the empirical performance of the sparse synthesis versus sparse analysis regularization for the ill-posed inverse problem of audio declipping. We develop a versatile non-convex heuristics which can be readily used with both data models. Based on this algorithm, we report that, in most cases, the two models perform almost similarly in terms of signal enhancement. However, the analysis version is shown to be amenable for real time audio processing, when certain analysis operators are considered. Both versions outperform state-of-the-art methods in the field, especially for the severely saturated signals

First Report of Little Cherry Virus 1 Infecting Apricot (Prunus armeniaca) in Africa

peer reviewedLittle cherry disease (LChD) is an important viral disease of many stone fruit species (Prunus spp.), sweet cherry (Prunus avium L.) being the most common host. It is associated with two different virus species belonging to the family Closteroviridae, namely, Little cherry virus 1 (LChV-1, Velarivirus) and Little cherry virus 2 (LChV-2, Ampelovirus). The impact of LChD on sweet cherry production consists in the decrease of yield and fruit quality, which is mainly associated with LChV-2, whereas most of LChV-1 reported infections remain associated with an unclear etiology. Other stone fruit species, such as peach and plum, hosting LChV-1 have been reported (Matic et al. 2007; Šafářová et al. 2017). LChV-1 is mainly transmitted through propagation of infected plant material, and no vector transmission is known (Jelkmann and Eastwell 2011). In 2018, during the early vegetative season, a limited survey was carried out for virus detection in apricot and sweet cherry orchards in the main southern Moroccan stone fruit-producing regions of Agadir, Agdez, and Dayat Aoua. Two sweet cherry trees (P. avium ‘Coeur de Pigeon’ and ‘Bigarreau’) and three apricot trees (Prunus armeniaca L.), all asymptomatic, were sampled (five branches with leaves) from three different orchards. RNA was extracted (both leaves and cambial scrapings) using the Spectrum Plant Total RNA kit (Sigma-Aldrich, Belgium), prior to cDNA synthesis using the iScript Reverse Transcription Kit (Bio-Rad, Belgium). LChV-1 detection was done by reverse transcription PCR (RT-PCR) using the specific primers LCUW7090 (5′-GGTTGTCCTCGGTTGATTAC-3′)/LCUWc7389 (5′-GGCTTGGTTCCATACATCTC-3′) (Bajet et al. 2008), amplifying a 300-bp fragment spanning the ORF1b encoding the RdRp gene, and 1LC_12776F (5′-TCAAGAAAAGTTCTGGTGTGC-3′)/1LC_13223R (5′-CGAGCTAGACGTATCAGTATC-3′) (Glasa et al. 2015), targeting a 456-bp fragment of the CP gene. LChV-2 specific primers were used according to Eastwell and Bernardy (2001). RT-PCR results revealed the presence of LChV-1 in two apricot samples from Agdez. No LChV-1 was detected in the sweet cherry samples. The presence of LChV-1 was confirmed by means of the LChV-1 specific reverse transcription loop-mediated isothermal amplification approach as described by Tahzima et al. (2019). No LChV-2 was detected in any of the samples. The RdRp and CP specific amplification products were bidirectionally sequenced (Genewiz, Leipzig, Germany) and assembled. RdRp and CP partial nucleotide sequences of the Moroccan LChV-1 isolates MOT2 and MOA1 were deposited in GenBank (accession nos. MK905349, MK905350; and MK905351, MK905352, respectively). Based on BLAST analysis of RdRp and CP, the Moroccan LChV-1 sequences shared 99% nucleotide identity (99.55% amino acids) with the No2ISTO isolate (HG792418) from Greece and 97.96% (98.64% amino acids) with the Spanish Ponferrada isolate (KX192367), respectively. Although the presence of LChV-1 has previously been reported in many countries in different continents, to our knowledge, this represents the first detection of LChV-1 in Africa

Estimation in high dimensions: a geometric perspective

This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of asymptotic convex geometry, and demonstrates connections between geometric results and estimation problems. The theory is illustrated with applications to sparse recovery, matrix completion, quantization, linear and logistic regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change

Structured Sparsity: Discrete and Convex approaches

Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure

Tensor completion in hierarchical tensor representations

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the reconstruction of tensors of low multi-linear rank in recently introduced hierarchical tensor formats from a small number of measurements. Hierarchical tensors are a flexible generalization of the well-known Tucker representation, which have the advantage that the number of degrees of freedom of a low rank tensor does not scale exponentially with the order of the tensor. While corresponding tensor decompositions can be computed efficiently via successive applications of (matrix) singular value decompositions, some important properties of the singular value decomposition do not extend from the matrix to the tensor case. This results in major computational and theoretical difficulties in designing and analyzing algorithms for low rank tensor recovery. For instance, a canonical analogue of the tensor nuclear norm is NP-hard to compute in general, which is in stark contrast to the matrix case. In this book chapter we consider versions of iterative hard thresholding schemes adapted to hierarchical tensor formats. A variant builds on methods from Riemannian optimization and uses a retraction mapping from the tangent space of the manifold of low rank tensors back to this manifold. We provide first partial convergence results based on a tensor version of the restricted isometry property (TRIP) of the measurement map. Moreover, an estimate of the number of measurements is provided that ensures the TRIP of a given tensor rank with high probability for Gaussian measurement maps.Comment: revised version, to be published in Compressed Sensing and Its Applications (edited by H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral

Constraint propagation equations of the 3+1 decomposition of f(R) gravity

Theories of gravity other than general relativity (GR) can explain the observed cosmic acceleration without a cosmological constant. One such class of theories of gravity is f(R). Metric f(R) theories have been proven to be equivalent to Brans-Dicke (BD) scalar-tensor gravity without a kinetic term. Using this equivalence and a 3+1 decomposition of the theory it has been shown that metric f(R) gravity admits a well-posed initial value problem. However, it has not been proven that the 3+1 evolution equations of metric f(R) gravity preserve the (hamiltonian and momentum) constraints. In this paper we show that this is indeed the case. In addition, we show that the mathematical form of the constraint propagation equations in BD-equilavent f(R) gravity and in f(R) gravity in both the Jordan and Einstein frames, is exactly the same as in the standard ADM 3+1 decomposition of GR. Finally, we point out that current numerical relativity codes can incorporate the 3+1 evolution equations of metric f(R) gravity by modifying the stress-energy tensor and adding an additional scalar field evolution equation. We hope that this work will serve as a starting point for relativists to develop fully dynamical codes for valid f(R) models.Comment: 25 pages, matches published version in CQG, references update