Polynomial-time Tensor Decompositions with Sum-of-Squares

We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete 3-tensors and learning overcomplete dictionaries with constant relative sparsity. We also give the first robust analysis for decomposing overcomplete 4-tensors in the smoothed analysis model. A key ingredient of our analysis is to establish small spectral gaps in moment matrices derived from solutions to sum-of-squares relaxations. To enable this analysis we augment sum-of-squares relaxations with spectral analogs of maximum entropy constraints.Comment: to appear in FOCS 201

Brook trout response to canopy and large woody debris manipulations in Appalachian streams

The forested riparian area along many central Appalachian streams contains large volumes of harvestable timber. Best management practices (BMP) and streamside management zones (SMZ) have been developed to minimize the impacts of riparian timber harvest. Large woody debris (LWD) is an important component of forested streams and its role in chemical, biological, and physical processes in streams is complex. The extraction of timber within the streamside management zone reduces the amount of material available for aquatic structure.;Three-250 m study reaches were established on eight Appalachian headwater streams. Four of the streams were assigned the treatment of having a 50% basal area removal of SMZ timber and four were assigned a 90% basal area removal of SMZ timber. The down and up sections of each stream were then randomly assigned to be either basal area removal (removal) treatment or basal area removal plus instream LWD addition (removal + LWD) treatment, with reference sections located upstream of the treatment sections.;Pool habitat features changed substantially in all three sections, with variation between post-harvest study years. However, pool area did not increase after the addition of LWD. Post-harvest stream temperature exhibited a constant pattern of increased warming as water moved downstream through the harvest zones. Mean maximum daily temperature downstream of timber harvest in 90% removal streams was an average of 3.1°C warmer than above harvest sections, and mean daily temperature was 1.1°C warmer. The 50% removal streams did not exhibit the large increases in stream warming seen in the 90% removal streams.;Seasonal population estimates of brook trout were conducted in 2005 (pre-treatment) 2007 and 2008 (post-treatment). Brook trout populations fluctuated over time, but did not show a consistent increase following treatment. Mean total length of YOY brook trout did vary across some streams and sections but was not significant among treatments. The condition (Wr) of age 1+ brook trout (\u3e120 mm) did not differ between treatment and reference sections in 50% or 90% streams. Overall percent retention of resident fish differed significantly between sections. Percent immigration was high in all sections (60--71%) suggesting high rates of movement.;Consumption estimates by origin of prey varied significantly within sections over the course of the study. Brook trout consumed a greater proportion of terrestrial invertebrates in reference sections than in timber removal sections during the study. Increased timber harvest intensity resulted in decreased consumption of terrestrial invertebrates by brook trout. Terrestrial invertebrates represent a greater proportion of the abundance, biomass and energy for brook trout in reference sites and may be greatly reduced in timbered areas. Brook trout in removal and reference sections exploited particular prey taxa at significantly different rates.;The results of our study show that it is necessary to assess trends in habitat changes, and brook trout populations over several years as there are several unknowns associated with the possible response to varying basal area removal. In addition, our study suggests that there could be changes in brook trout diet in the removal sections and a potential shift in the feeding habits of brook trout, and a reduction of terrestrial invertebrate availability to brook trout may result in decreased growth of Appalachian brook trout in these sections

Modeling Agglomeration of Dust Particles in Plasma

The charge on an aggregate immersed in a plasma environment distributes itself over the aggregate's surface; this can be approximated theoretically by assuming a multipole distribution. The dipole-dipole (or higher order) charge interactions between fractal aggregates lead to rotations of the grains as they interact. Other properties of the dust grains also influence the agglomeration process, such as the monomer shape (spherical or ellipsoidal) or the presence of magnetic material. Finally, the plasma and grain properties also determine the morphology of the resultant aggregates. Porous and fluffy aggregates are more strongly coupled to the gas, leading to reduced collisional velocities, and greater collisional cross sections. These factors in turn can determine the growth rate of the aggregates and evolution of the dust cloud. This paper gives an overview of the numerical and experimental methods used to study dust agglomeration at CASPER and highlights some recent results