Sparse Recovery from Combined Fusion Frame Measurements

Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as Compressed Sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead of vectors to represent signals. This work combines these exciting fields to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standard CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications. With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it does not need to be sparse within each of the subspaces it occupies. This sparsity model is captured using a mixed l1/l2 norm for fusion frames. A signal sparse in a fusion frame can be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixed l1/l2 norm. The provided sampling conditions generalize coherence and RIP conditions used in standard CS theory. It is demonstrated that they are sufficient to guarantee sparse recovery of any signal sparse in our model. Moreover, a probabilistic analysis is provided using a stochastic model on the sparse signal that shows that under very mild conditions the probability of recovery failure decays exponentially with increasing dimension of the subspaces

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling

Identification of tissue-specific microRNAs from mouse

MicroRNAs (miRNAs) are a new class of noncoding RNAs, which are encoded as short inverted repeats in the genomes of invertebrates and vertebrates [1, 2]. It is believed that miRNAs are modulators of target mRNA translation and stability, although most target mRNAs remain to be identified. Here we describe the identification of 34 novel miRNAs by tissue- specific cloning of approximately 21-nucleotide RNAs from mouse. Almost all identified miRNAs are conserved in the human genome and are also frequently found in nonmammalian vertebrate genomes, such as pufferfish. In heart, liver, or brain, it is found that a single, tissue-specifically expressed miRNA dominates the population of expressed miRNAs and suggests a role for these miRNAs in tissue specification or cell lineage decisions. Finally, a miRNA was identified that appears to be the fruitfly and mammalian ortholog of C. elegans lin-4 stRNA

On the linear independence of spikes and sines

The purpose of this work is to survey what is known about the linear independence of spikes and sines. The paper provides new results for the case where the locations of the spikes and the frequencies of the sines are chosen at random. This problem is equivalent to studying the spectral norm of a random submatrix drawn from the discrete Fourier transform matrix. The proof involves depends on an extrapolation argument of Bourgain and Tzafriri.Comment: 16 pages, 4 figures. Revision with new proof of major theorem

Wisdom of groups promotes cooperation in evolutionary social dilemmas

Whether or not to change strategy depends not only on the personal success of each individual, but also on the success of others. Using this as motivation, we study the evolution of cooperation in games that describe social dilemmas, where the propensity to adopt a different strategy depends both on individual fitness as well as on the strategies of neighbors. Regardless of whether the evolutionary process is governed by pairwise or group interactions, we show that plugging into the "wisdom of groups" strongly promotes cooperative behavior. The more the wider knowledge is taken into account the more the evolution of defectors is impaired. We explain this by revealing a dynamically decelerated invasion process, by means of which interfaces separating different domains remain smooth and defectors therefore become unable to efficiently invade cooperators. This in turn invigorates spatial reciprocity and establishes decentralized decision making as very beneficial for resolving social dilemmas.Comment: 8 two-column pages, 7 figures; accepted for publication in Scientific Report

The road to deterministic matrices with the restricted isometry property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page

A new rhynchocephalian from the late jurassic of Germany with a dentition that is unique amongst tetrapods.

Rhynchocephalians, the sister group of squamates (lizards and snakes), are only represented by the single genus Sphenodon today. This taxon is often considered to represent a very conservative lineage. However, rhynchocephalians were common during the late Triassic to latest Jurassic periods, but rapidly declined afterwards, which is generally attributed to their supposedly adaptive inferiority to squamates and/or Mesozoic mammals, which radiated at that time. New finds of Mesozoic rhynchocephalians can thus provide important new information on the evolutionary history of the group. A new fossil relative of Sphenodon from the latest Jurassic of southern Germany, Oenosaurus muehlheimensis gen. et sp. nov., presents a dentition that is unique amongst tetrapods. The dentition of this taxon consists of massive, continuously growing tooth plates, probably indicating a crushing dentition, thus representing a previously unknown trophic adaptation in rhynchocephalians. The evolution of the extraordinary dentition of Oenosaurus from the already highly specialized Zahnanlage generally present in derived rhynchocephalians demonstrates an unexpected evolutionary plasticity of these animals. Together with other lines of evidence, this seriously casts doubts on the assumption that rhynchocephalians are a conservative and adaptively inferior lineage. Furthermore, the new taxon underlines the high morphological and ecological diversity of rhynchocephalians in the latest Jurassic of Europe, just before the decline of this lineage on this continent. Thus, selection pressure by radiating squamates or Mesozoic mammals alone might not be sufficient to explain the demise of the clade in the Late Mesozoic, and climate change in the course of the fragmentation of the supercontinent of Pangaea might have played a major role