Learning Variational Models with Unrolling and Bilevel Optimization

In this paper we consider the problem of learning variational models in the context of supervised learning via risk minimization. Our goal is to provide a deeper understanding of the two approaches of learning of variational models via bilevel optimization and via algorithm unrolling. The former considers the variational model as a lower level optimization problem below the risk minimization problem, while the latter replaces the lower level optimization problem by an algorithm that solves said problem approximately. Both approaches are used in practice, but unrolling is much simpler from a computational point of view. To analyze and compare the two approaches, we consider a simple toy model, and compute all risks and the respective estimators explicitly. We show that unrolling can be better than the bilevel optimization approach, but also that the performance of unrolling can depend significantly on further parameters, sometimes in unexpected ways: While the stepsize of the unrolled algorithm matters a lot (and learning the stepsize gives a significant improvement), the number of unrolled iterations plays a minor role

Planet formation bursts at the borders of the dead zone in 2D numerical simulations of circumstellar disks

As accretion in protoplanetary disks is enabled by turbulent viscosity, the border between active and inactive (dead) zones constitutes a location where there is an abrupt change in the accretion flow. The gas accumulation that ensues triggers the Rossby wave instability, that in turn saturates into anticyclonic vortices. It was suggested that the trapping of solids within them leads to a burst of planet formation on very short timescales. We perform two-dimensional global simulations of the dynamics of gas and solids in a non-magnetized thin protoplanetary disk with the Pencil Code. We use multiple particle species of radius 1, 10, 30, and 100 cm, solving for the particles' gravitational interaction by a particle-mesh method. The dead zone is modeled as a region of low viscosity. Adiabatic and locally isothermal equations of state are used. We find that the Rossby wave instability is triggered under a variety of conditions, thus making vortex formation a robust process. Inside the vortices, fast accumulation of solids occurs and the particles collapse into objects of planetary mass in timescales as short as five orbits. Because the drag force is size-dependent, aerodynamical sorting ensues within the vortical motion, and the first bound structures formed are composed primarily of similarly-sized particles. In addition to erosion due to ram pressure, we identify gas tides from the massive vortices as a disrupting agent of formed protoplanetary embryos. We also estimate the collisional velocity history of the particles that compose the most massive embryo by the end of the simulation, finding that the vast majority of them never experienced a collision with another particle at speeds faster than 1 m/s.Comment: 19 pages, 15 figures + Appendices. Accepted by A&A. Nature of replacement: included a missing referenc

Neural Correlates of Spatial Attention and Target Detection in a Multi-Target Environment

Our ability to attend and respond in a multi-target environment is an essential and distinct human skill, as is dramatically demonstrated in stroke patients suffering from extinction. We performed an fMRI study to determine the neural anatomy associated with attending and responding to simultaneously presented targets. In healthy subjects, we tested the hypothesis that the right intraparietal sulcus (IPS) is associated both with the top-down direction of attention to multiple target locations and the bottom-up detection of multiple targets, whereas the temporo-parietal junction (TPJ) is predominantly associated with the bottom-up detection of multiple targets. We used a cued target detection task with a high proportion of catch trials to separately estimate top-down cue-related and bottom-up target-related neural activity. Both cues and targets could be presented unilaterally or bilaterally. We found no evidence of target-related neural activation specific to bilateral situations in the TPJ, but observed both cue-related and target-related neural activation specific to bilateral situations in the right IPS and target-related neural activity specific to bilateral situations in the right inferior frontal gyrus (IFG). We conclude that the IPS and the IFG of the right hemisphere underlie our ability to attend and respond in a multi-target environment.This work was supported by the fortüne-Programm of the Medical Faculty of the University of Tübingen (2003-0-0) and the Deutsche Forschungsgemeinschaft (KA 1258/10-1 and HA 5839/3-1)

Feigin-Frenkel center in types B, C and D

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras V_{crit}(g) associated with the simple Lie algebras g of types B, C and D. The construction relies on the Schur-Weyl duality involving the Brauer algebra, and the generators are expressed as weighted traces over tensor spaces and, equivalently, as traces over the spaces of singular vectors for the action of the Lie algebra sl_2 in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra g. We also introduce analogues of the Bethe subalgebras of the Yangians Y(g) and show that their graded images coincide with the respective commutative subalgebras of U(g[t]).Comment: 29 pages, constructions of Pfaffian-type Sugawara operators and commutative subalgebras in universal enveloping algebras are adde

The Arabidopsis thaliana knockout mutant for phytochelatin synthase1 (cad1-3) Is defective in callose deposition, bacterial pathogen defense and auxin content, but shows an increased stem lignification

The enzyme phytochelatin synthase (PCS) has long been studied with regard to its role in metal(loid) detoxification in several organisms, i.e., plants, yeasts, and nematodes. It is in fact widely recognized that PCS detoxifies a number of heavy metals by catalyzing the formation of thiol-rich oligomers, namely phytochelatins, from glutathione and related peptides. However, recent investigations have highlighted other possible roles played by the PCS enzyme in the plant cell, e.g., the control of pathogen-triggered callose deposition. In order to examine novel aspects of Arabidopsis thaliana PCS1 (AtPCS1) functions and to elucidate its possible roles in the secondary metabolism, metabolomic data of A. thaliana wild-type and cad1-3 mutant were compared, the latter lacking AtPCS1. HPLC-ESI-MS analysis showed differences in the relative levels of metabolites from the glucosinolate and phenylpropanoid pathways between cad1-3 and wild-type plants. Specifically, in control (Cd-untreated) plants, higher levels of 4-methoxy-indol 3-ylmethylglucosinolate were found in cad1-3 plants vs. wild-type. Moreover, the cad1-3 mutant showed to be impaired in the deposit of callose after Cd exposure, suggesting that AtPCS1 protects the plant against the toxicity of heavy metals not only by synthesizing PCs, but also by contributing to callose deposition. In line with the contribution of callose in counteracting Cd toxicity, we found that another callose-defective mutant, pen2-1, was more sensitive to high concentrations of Cd than wild-type plants. Moreover, cad1-3 plants were more susceptible than wild-type to the hemibiotrophic bacterial pathogen Pseudomonas syringae. The metabolome also revealed differences in the relative levels of hydroxycinnamic acids and flavonols,with consequences on cell wall properties and auxin content, respectively. First, increased lignification in the cad1-3 stems was found, probably aimed at counteracting the entry of Cd into the inner tissues. Second, in cad1-3 shoots, increased relative levels of kaempferol 3,7 dirhamnoside and quercetin hexoside rhamnoside were detected. These flavonols are endogenous inhibitors of auxin transport in planta; auxin levels in both roots and shoots of the cad1-3 mutant were in fact lower than those of the wild-type. Overall, our data highlight novel aspects of AtPCS1 functions in A. thaliana

Persistence, extinction and spatio-temporal synchronization of SIRS cellular automata models

Spatially explicit models have been widely used in today's mathematical ecology and epidemiology to study persistence and extinction of populations as well as their spatial patterns. Here we extend the earlier work--static dispersal between neighbouring individuals to mobility of individuals as well as multi-patches environment. As is commonly found, the basic reproductive ratio is maximized for the evolutionary stable strategy (ESS) on diseases' persistence in mean-field theory. This has important implications, as it implies that for a wide range of parameters that infection rate will tend maximum. This is opposite with present results obtained in spatial explicit models that infection rate is limited by upper bound. We observe the emergence of trade-offs of extinction and persistence on the parameters of the infection period and infection rate and show the extinction time having a linear relationship with respect to system size. We further find that the higher mobility can pronouncedly promote the persistence of spread of epidemics, i.e., the phase transition occurs from extinction domain to persistence domain, and the spirals' wavelength increases as the mobility increasing and ultimately, it will saturate at a certain value. Furthermore, for multi-patches case, we find that the lower coupling strength leads to anti-phase oscillation of infected fraction, while higher coupling strength corresponds to in-phase oscillation.Comment: 12page

Introductory clifford analysis

In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications

International Veterinary Epilepsy Task Force Consensus Proposal: Diagnostic approach to epilepsy in dogs

This article outlines the consensus proposal on diagnosis of epilepsy in dogs by the International Veterinary Epilepsy Task Force. The aim of this consensus proposal is to improve consistency in the diagnosis of epilepsy in the clinical and research settings. The diagnostic approach to the patient presenting with a history of suspected epileptic seizures incorporates two fundamental steps: to establish if the events the animal is demonstrating truly represent epileptic seizures and if so, to identify their underlying cause. Differentiation of epileptic seizures from other non-epileptic episodic paroxysmal events can be challenging. Criteria that can be used to make this differentiation are presented in detail and discussed. Criteria for the diagnosis of idiopathic epilepsy (IE) are described in a three-tier system. Tier I confidence level for the diagnosis of IE is based on a history of two or more unprovoked epileptic seizures occurring at least 24 h apart, age at epileptic seizure onset of between six months and six years, unremarkable inter-ictal physical and neurological examination, and no significant abnormalities on minimum data base blood tests and urinalysis. Tier II confidence level for the diagnosis of IE is based on the factors listed in tier I and unremarkable fasting and post-prandial bile acids, magnetic resonance imaging (MRI) of the brain (based on an epilepsy-specific brain MRI protocol) and cerebrospinal fluid (CSF) analysis. Tier III confidence level for the diagnosis of IE is based on the factors listed in tier I and II and identification of electroencephalographic abnormalities characteristic for seizure disorders. The authors recommend performing MRI of the brain and routine CSF analysis, after exclusion of reactive seizures, in dogs with age at epileptic seizure onset 6 years, inter-ictal neurological abnormalities consistent with intracranial neurolocalisation, status epilepticus or cluster seizure at epileptic seizure onset, or a previous presumptive diagnosis of IE and drug-resistance with a single antiepileptic drug titrated to the highest tolerable dose