Smoothed Analysis of Tensor Decompositions

Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and tensors analogs of much of the matrix algebra toolkit are unlikely to exist because of hardness results. Efficient decomposition in the overcomplete case (where rank exceeds dimension) is particularly challenging. We introduce a smoothed analysis model for studying these questions and develop an efficient algorithm for tensor decomposition in the highly overcomplete case (rank polynomial in the dimension). In this setting, we show that our algorithm is robust to inverse polynomial error -- a crucial property for applications in learning since we are only allowed a polynomial number of samples. While algorithms are known for exact tensor decomposition in some overcomplete settings, our main contribution is in analyzing their stability in the framework of smoothed analysis. Our main technical contribution is to show that tensor products of perturbed vectors are linearly independent in a robust sense (i.e. the associated matrix has singular values that are at least an inverse polynomial). This key result paves the way for applying tensor methods to learning problems in the smoothed setting. In particular, we use it to obtain results for learning multi-view models and mixtures of axis-aligned Gaussians where there are many more "components" than dimensions. The assumption here is that the model is not adversarially chosen, formalized by a perturbation of model parameters. We believe this an appealing way to analyze realistic instances of learning problems, since this framework allows us to overcome many of the usual limitations of using tensor methods.Comment: 32 pages (including appendix

How does flow in a pipe become turbulent?

The transition to turbulence in pipe flow does not follow the scenario familiar from Rayleigh-Benard or Taylor-Couette flow since the laminar profile is stable against infinitesimal perturbations for all Reynolds numbers. Moreover, even when the flow speed is high enough and the perturbation sufficiently strong such that turbulent flow is established, it can return to the laminar state without any indication of the imminent decay. In this parameter range, the lifetimes of perturbations show a sensitive dependence on initial conditions and an exponential distribution. The turbulence seems to be supported by three-dimensional travelling waves which appear transiently in the flow field. The boundary between laminar and turbulent dynamics is formed by the stable manifold of an invariant chaotic state. We will also discuss the relation between observations in short, periodically continued domains, and the dynamics in fully extended puffs.Comment: for the proceedings of statphys 2

Invasive Andropogon gayanus (gamba grass) is an ecosystem transformer of nitrogen relations in Australian savanna

The African grass Andropogon gayanus Kunth. is invading Australian savannas, altering their ecological and biogeochemical function. To assess impacts on nitrogen (N) cycling, we quantified litter decomposition and N dynamics of grass litter in native grass and A. gayanus invaded savanna using destructive in situ grass litter harvests and litterbag incubations (soil surface and aerial position). Only 30% of the A. gayanus in situ litter decomposed, compared to 61% of the native grass litter, due to the former being largely comprised of highly resistant A. gayanus stem. In contrast to the stem, A. gayanus leaf decomposition was approximately 3- and 2-times higher than the dominant native grass, Alloteropsis semilata at the surface and aerial position, respectively. Lower initial lignin concentrations, and higher consumption by termites, accounted for the greater surface decomposition rate of A. gayanus. N flux estimates suggest the N release of A. gayanus litter is insufficient to compensate for increased N uptake and N loss via fire in invaded plots. Annually burnt invaded savanna may lose up to 8.2% of the upper soil N pool over a decade. Without additional inputs via biological N fixation, A. gayanus invasion is likely to diminish the N capital of Australia's frequently burnt savannas

Order-of-magnitude speedup for steady states and traveling waves via Stokes preconditioning in Channelflow and Openpipeflow

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton's method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.Comment: in Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics, ed. Alexander Gelfgat (Springer, 2018

CLUES TO THE MEDIEVAL DESTABILIZATION OF THE NEBRASKA SAND HILLS, USA, FROM ANCIENT POCKET GOPHER BURROWS

The Nebraska Sand Hills are a stabilized dune field in the central United States that reflect past conditions of drought. The most recent drought, known as the Medieval Climatic Anomaly, occurred from A.D. 900 to A.D. 1300 and had an enormous effect on the thriving prairie ecosystem, which included large populations of the plains pocket gopher (Geomys bursarius). Burrows of these organisms across a paleosol-eolian sand boundary in the Sand Hills indicate abrupt climate change during the transition from stabilized to active dune field and from humid to arid conditions. Medieval gophers tunneled at greater depths below the surface than do modern gophers, indicating the behavioral changes these animals underwent to survive during the transition. The gophers were likely surviving on roots remaining in the underlying soil as it was buried by sand; they tunneled .1 m up to the surface to deposit mounds of excavated soil and sand. Most of the burrows occur in areas of low-angle bedding, suggesting loss of vegetation occurred first on the crests of the newly formed dunes while vegetation persisted in the interdunes. Optically stimulated luminescence dates from a dune containing ancient gopher burrows are nearly identical throughout the height of the dune, indicating rapid accumulation of sand. As accumulation of sand was rapid, vegetative loss must also have occurred quickly, though not in a uniform pattern across the region. Pocket gophers were apparently able to survive in areas of remaining vegetation for a short time, but in a relatively short period of time, they were unable to reach their food sources and were forced ultimately to abandon the uplands in the region

Linking Grassland and Range National Goals with the Grazing Lands Conservation Initiative for National Benefit

Grassland and Range National Goals (GRNGs) were developed to identify national research and education needs for these resources, both public and private. The Grazing Lands Conservation Initiative (GLCI) developed within the NRCS to address the need for technical assistance for the management of privately held grazing lands. These activities are complimentary and, when combined, will address the national need for research, education, and technical assistance leading to better management of grassland and range resources

Side effects of analgesia may significantly reduce quality of life in symptomatic multiple myeloma: a cross-sectional prevalence study

Background Pain is a common symptom in patients with multiple myeloma (MM). Many patients are dependent on analgesics and in particular opioids, but there is limited information on the impact of these drugs and their side effects on health-related quality of life (HRQoL). Method In a cross-sectional study, semi-structured interviews were performed in 21 patients attending the hospital with symptomatic MM on pain medications. HRQoL was measured using items 29 and 30 of the European Organisation for Research and Treatment of Cancer (EORTC) QLQ-C30. Results Patients were able to recall a median of two (range 0–4) analgesics. They spontaneously identified a median of two (range 1–5) side effects attributable to their analgesic medications. Patients’ assessment of HRQoL based on the EORTC QLQ-C30 questions 29/30 was mean 48.3 (95 % CI; 38.7–57.9) out of 100. Patients’ assessment of their HRQoL in the hypothetical situation, in which they would not experience any side effects from analgesics, was significantly higher: 62.6 (53.5–71.7) (t test, p=0.001). Conclusion This study provides, for the first time, evidence that side effects of analgesics are common in symptomatic MM and may result in a statistically and clinically significant reduction of self-reported HRQoL

Grassland productivity limited by multiple nutrients

Terrestrial ecosystem productivity is widely accepted to be nutrient limited1. Although nitrogen (N) is deemed a key determinant of aboveground net primary production (ANPP)2,3, the prevalence of co-limitation by N and phosphorus (P) is increasingly recognized4,​5,​6,​7,​8. However, the extent to which terrestrial productivity is co-limited by nutrients other than N and P has remained unclear. Here, we report results from a standardized factorial nutrient addition experiment, in which we added N, P and potassium (K) combined with a selection of micronutrients (K+μ), alone or in concert, to 42 grassland sites spanning five continents, and monitored ANPP. Nutrient availability limited productivity at 31 of the 42 grassland sites. And pairwise combinations of N, P, and K+μ co-limited ANPP at 29 of the sites. Nitrogen limitation peaked in cool, high latitude sites. Our findings highlight the importance of less studied nutrients, such as K and micronutrients, for grassland productivity, and point to significant variations in the type and degree of nutrient limitation. We suggest that multiple-nutrient constraints must be considered when assessing the ecosystem-scale consequences of nutrient enrichment

Diversity and Productivity in a Long-Term Grassland Experiment

Plant diversity and niche complementarity had progressively stronger effects on ecosystem functioning during a 7-year experiment, with 16-species plots attaining 2.7 times greater biomass than monocultures. Diversity effects were neither transients nor explained solely by a few productive or unviable species. Rather, many higher-diversity plots outperformed the best monoculture. These results help resolve debate over biodiversity and ecosystem functioning, show effects at higher than expected diversity levels, and demonstrate, for these ecosystems, that even the best-chosen monocultures cannot achieve greater productivity or carbon stores than higher-diversity sites. Includes Supplementary Material