Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints

Regularization of ill-posed linear inverse problems via $\ell_1$ penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an $\ell_1$ penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to $\ell_1$-constraints, using a gradient method, with projection on $\ell_1$-balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.Comment: 24 pages, 5 figures. v2: added reference, some amendments, 27 page

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 higher-level phylogeny of Archosauria (Tetrapoda:Diapsida)

Crown group Archosauria, which includes birds, dinosaurs, crocodylomorphs, and several extinct Mesozoic groups, is a primary division of the vertebrate tree of life. However, the higher-level phylogenetic relationships within Archosauria are poorly resolved and controversial, despite years of study. The phylogeny of crocodile-line archosaurs (Crurotarsi) is particularly contentious, and has been plagued by problematic taxon and character sampling. Recent discoveries and renewed focus on archosaur anatomy enable the compilation of a new dataset, which assimilates and standardizes character data pertinent to higher-level archosaur phylogeny, and is scored across the largest group of taxa yet analysed. This dataset includes 47 new characters (25% of total) and eight taxa that have yet to be included in an analysis, and total taxonomic sampling is more than twice that of any previous study. This analysis produces a well-resolved phylogeny, which recovers mostly traditional relationships within Avemetatarsalia, places Phytosauria as a basal crurotarsan clade, finds a close relationship between Aetosauria and Crocodylomorpha, and recovers a monophyletic Rauisuchia comprised of two major subclades. Support values are low, suggesting rampant homoplasy and missing data within Archosauria, but the phylogeny is highly congruent with stratigraphy. Comparison with alternative analyses identifies numerous scoring differences, but indicates that character sampling is the main source of incongruence. The phylogeny implies major missing lineages in the Early Triassic and may support a Carnian-Norian extinction event.Marshall Scholarship for study in the United KingdomJurassic FoundationUniversity of BristolPaleontological Societ

Geometric methods on low-rank matrix and tensor manifolds

In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors

Theropod Fauna from Southern Australia Indicates High Polar Diversity and Climate-Driven Dinosaur Provinciality

The Early Cretaceous fauna of Victoria, Australia, provides unique data on the composition of high latitude southern hemisphere dinosaurs. We describe and review theropod dinosaur postcranial remains from the Aptian–Albian Otway and Strzelecki groups, based on at least 37 isolated bones, and more than 90 teeth from the Flat Rocks locality. Several specimens of medium- and large-bodied individuals (estimated up to ∼8.5 metres long) represent allosauroids. Tyrannosauroids are represented by elements indicating medium body sizes (∼3 metres long), likely including the holotype femur of Timimus hermani, and a single cervical vertebra represents a juvenile spinosaurid. Single specimens representing medium- and small-bodied theropods may be referrable to Ceratosauria, Ornithomimosauria, a basal coelurosaur, and at least three taxa within Maniraptora. Thus, nine theropod taxa may have been present. Alternatively, four distinct dorsal vertebrae indicate a minimum of four taxa. However, because most taxa are known from single bones, it is likely that small-bodied theropod diversity remains underestimated. The high abundance of allosauroids and basal coelurosaurs (including tyrannosauroids and possibly ornithomimosaurs), and the relative rarity of ceratosaurs, is strikingly dissimilar to penecontemporaneous dinosaur faunas of Africa and South America, which represent an arid, lower-latitude biome. Similarities between dinosaur faunas of Victoria and the northern continents concern the proportional representatation of higher clades, and may result from the prevailing temperate–polar climate of Australia, especially at high latitudes in Victoria, which is similar to the predominant warm–temperate climate of Laurasia, but distinct from the arid climate zone that covered extensive areas of Gondwana. Most dinosaur groups probably attained a near-cosmopolitan distribution in the Jurassic, prior to fragmentation of the Pangaean supercontinent, and some aspects of the hallmark ‘Gondwanan’ fauna of South America and Africa may therefore reflect climate-driven provinciality, not vicariant evolution driven by continental fragmentation. However, vicariance may still be detected at lower phylogenetic levels

The Convex Geometry of Linear Inverse Problems

In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases such as sparse vectors and low-rank matrices, as well as several others including sums of a few permutations matrices, low-rank tensors, orthogonal matrices, and atomic measures. The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery. Thus this work extends the catalog of simple models that can be recovered from limited linear information via tractable convex programming

More than one way to be a giant: Convergence and disparity in the hip joints of saurischian dinosaurs

ABSTRACT Saurischian dinosaurs evolved seven orders of magnitude in body mass, as well as a wide diversity of hip joint morphology and locomotor postures. The very largest saurischians possess incongruent bony hip joints, suggesting that large volumes of soft tissues mediated hip articulation. To understand the evolutionary trends and functional relationships between body size and hip anatomy of saurischians, we tested the relationships among discrete and continuous morphological characters using phylogenetically corrected regression. Giant theropods and sauropods convergently evolved highly cartilaginous hip joints by reducing supraacetabular ossifications, a condition unlike that in early dinosauromorphs. However, transitions in femoral and acetabular soft tissues indicate that large sauropods and theropods built their hip joints in fundamentally different ways. In sauropods, the femoral head possesses irregularly rugose subchondral surfaces for thick hyaline cartilage. Hip articulation was achieved primarily using the highly cartilaginous femoral head and the supraacetabular labrum on the acetabular ceiling. In contrast, theropods covered their femoral head and neck with thinner hyaline cartilage and maintained extensive articulation between the fibrocartilaginous femoral neck and the antitrochanter. These findings suggest that the hip joints of giant sauropods were built to sustain large compressive loads whereas those of giant theropods experienced compression and shear forces

Identification and Classification of Conserved RNA Secondary Structures in the Human Genome

The discoveries of microRNAs and riboswitches, among others, have shown functional RNAs to be biologically more important and genomically more prevalent than previously anticipated. We have developed a general comparative genomics method based on phylogenetic stochastic context-free grammars for identifying functional RNAs encoded in the human genome and used it to survey an eight-way genome-wide alignment of the human, chimpanzee, mouse, rat, dog, chicken, zebra-fish, and puffer-fish genomes for deeply conserved functional RNAs. At a loose threshold for acceptance, this search resulted in a set of 48,479 candidate RNA structures. This screen finds a large number of known functional RNAs, including 195 miRNAs, 62 histone 3′UTR stem loops, and various types of known genetic recoding elements. Among the highest-scoring new predictions are 169 new miRNA candidates, as well as new candidate selenocysteine insertion sites, RNA editing hairpins, RNAs involved in transcript auto regulation, and many folds that form singletons or small functional RNA families of completely unknown function. While the rate of false positives in the overall set is difficult to estimate and is likely to be substantial, the results nevertheless provide evidence for many new human functional RNAs and present specific predictions to facilitate their further characterization