Hardware Based Projection onto The Parity Polytope and Probability Simplex

This paper is concerned with the adaptation to hardware of methods for Euclidean norm projections onto the parity polytope and probability simplex. We first refine recent efforts to develop efficient methods of projection onto the parity polytope. Our resulting algorithm can be configured to have either average computational complexity $\mathcal{O}\left(d\right)$ or worst case complexity $\mathcal{O}\left(d\log{d}\right)$ on a serial processor where $d$ is the dimension of projection space. We show how to adapt our projection routine to hardware. Our projection method uses a sub-routine that involves another Euclidean projection; onto the probability simplex. We therefore explain how to adapt to hardware a well know simplex projection algorithm. The hardware implementations of both projection algorithms achieve area scalings of $\mathcal{O}(d\left(\log{d}\right)^2)$ at a delay of $\mathcal{O}(\left(\log{d}\right)^2)$. Finally, we present numerical results in which we evaluate the fixed-point accuracy and resource scaling of these algorithms when targeting a modern FPGA

LP-decodable multipermutation codes

In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are permutations of a multiset that may consist of duplicate entries. We first introduce a new class of matrices called multipermutation matrices. We characterize the convex hull of multipermutation matrices. Based on this characterization, we propose a new class of codes that we term LP-decodable multipermutation codes. Then, we derive two LP decoding algorithms. We first formulate an LP decoding problem for memoryless channels. We then derive an LP algorithm that minimizes the Chebyshev distance. Finally, we show a numerical example of our algorithm.Comment: This work was supported by NSF and NSERC. To appear at the 2014 Allerton Conferenc

Efficient learning of neighbor representations for boundary trees and forests

We introduce a semiparametric approach to neighbor-based classification. We build off the recently proposed Boundary Trees algorithm by Mathy et al.(2015) which enables fast neighbor-based classification, regression and retrieval in large datasets. While boundary trees use an Euclidean measure of similarity, the Differentiable Boundary Tree algorithm by Zoran et al.(2017) was introduced to learn low-dimensional representations of complex input data, on which semantic similarity can be calculated to train boundary trees. As is pointed out by its authors, the differentiable boundary tree approach contains a few limitations that prevents it from scaling to large datasets. In this paper, we introduce Differentiable Boundary Sets, an algorithm that overcomes the computational issues of the differentiable boundary tree scheme and also improves its classification accuracy and data representability. Our algorithm is efficiently implementable with existing tools and offers a significant reduction in training time. We test and compare the algorithms on the well known MNIST handwritten digits dataset and the newer Fashion-MNIST dataset by Xiao et al.(2017).Comment: 9 pages, 2 figure

Collisionless distribution function for the relativistic force-free Harris sheet

A self-consistent collisionless distribution function for the relativistic analogue of the force-free Harris sheet is presented. This distribution function is the relativistic generalization of the distribution function for the non-relativistic collisionless force-free Harris sheet recently found by Harrison and Neukirch [Phys. Rev. Lett. 102, 135003 (2009)], as it has the same dependence on the particle energy and canonical momenta. We present a detailed calculation which shows that the proposed distribution function generates the required current density profile (and thus magnetic field profile) in a frame of reference in which the electric potential vanishes identically. The connection between the parameters of the distribution function and the macroscopic parameters such as the current sheet thickness is discussed. (C) 2012 American Institute of Physics. [doi: 10.1063/1.3677268]PostprintPeer reviewe

Dust cloud evolution in sub-stellar atmospheres via plasma deposition and plasma sputtering

Context. In contemporary sub-stellar model atmospheres, dust growth occurs through neutral gas-phase surface chemistry. Recently, there has been a growing body of theoretical and observational evidence suggesting that ionisation processes can also occur. As a result, atmospheres are populated by regions composed of plasma, gas and dust, and the consequent influence of plasma processes on dust evolution is enhanced.Aim. This paper aims to introduce a new model of dust growth and destruction in sub-stellar atmospheres via plasma deposition and plasma sputtering.Methods. Using example sub-stellar atmospheres from DRIFT-PHOENIX, we have compared plasma deposition and sputtering timescales to those from neutral gas-phase surface chemistry to ascertain their regimes of influence. We calculated the plasma sputtering yield and discuss the circumstances where plasma sputtering dominates over deposition.Results. Within the highest dust density cloud regions, plasma deposition and sputtering dominates over neutral gas-phase surface chemistry if the degree of ionisation is ≳10−4. Loosely bound grains with surface binding energies of the order of 0.1–1 eV are susceptible to destruction through plasma sputtering for feasible degrees of ionisation and electron temperatures; whereas, strong crystalline grains with binding energies of the order 10 eV are resistant to sputtering.Conclusions. The mathematical framework outlined sets the foundation for the inclusion of plasma deposition and plasma sputtering in global dust cloud formation models of sub-stellar atmospheres

Collisional Grooming Models of the Kuiper Belt Dust Cloud

We modeled the 3-D structure of the Kuiper Belt dust cloud at four different dust production rates, incorporating both planet-dust interactions and grain-grain collisions using the collisional grooming algorithm. Simulated images of a model with a face-on optical depth of ~10^-4 primarily show an azimuthally-symmetric ring at 40-47 AU in submillimeter and infrared wavelengths; this ring is associated with the cold classical Kuiper Belt. For models with lower optical depths (10^-6 and 10^-7), synthetic infrared images show that the ring widens and a gap opens in the ring at the location of of Neptune; this feature is caused by trapping of dust grains in Neptune's mean motion resonances. At low optical depths, a secondary ring also appears associated with the hole cleared in the center of the disk by Saturn. Our simulations, which incorporate 25 different grain sizes, illustrate that grain-grain collisions are important in sculpting today's Kuiper Belt dust, and probably other aspects of the Solar System dust complex; collisions erase all signs of azimuthal asymmetry from the submillimeter image of the disk at every dust level we considered. The model images switch from being dominated by resonantly-trapped small grains ("transport dominated") to being dominated by the birth ring ("collision dominated") when the optical depth reaches a critical value of tau ~ v/c, where v is the local Keplerian speed.Comment: 31 pages, including 9 figure