An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the road from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates along with applications.Comment: 66 pages, 3 figure

Comment on "Minimal size of a barchan dune"

It is now an accepted fact that the size at which dunes form from a flat sand bed as well as their `minimal size' scales on the flux saturation length. This length is by definition the relaxation length of the slowest mode toward equilibrium transport. The model presented by Parteli, Duran and Herrmann [Phys. Rev. E 75, 011301 (2007)] predicts that the saturation length decreases to zero as the inverse of the wind shear stress far from the threshold. We first show that their model is not self-consistent: even under large wind, the relaxation rate is limited by grain inertia and thus can not decrease to zero. A key argument presented by these authors comes from the discussion of the typical dune wavelength on Mars (650 m) on the basis of which they refute the scaling of the dune size with the drag length evidenced by Claudin and Andreotti [Earth Pla. Sci. Lett. 252, 30 (2006)]. They instead propose that Martian dunes, composed of large grains (500 micrometers), were formed in the past under very strong winds. We show that this saltating grain size, estimated from thermal diffusion measurements, is not reliable. Moreover, the microscopic photographs taken by the rovers on Martian aeolian bedforms show a grain size of 87 plus or minus 25 micrometers together with hematite spherules at millimetre scale. As those so-called ``blueberries'' can not be entrained by reasonable winds, we conclude that the saltating grains on Mars are the small ones, which gives a second strong argument against the model of Parteli et al.Comment: A six page comment on ``Minimal size of a barchan dune'' by Parteli, Duran and Herrmann [Phys. Rev. E 75, 011301 (2007) arXiv:0705.1778

Spectral dimensionality reduction for HMMs

Hidden Markov Models (HMMs) can be accurately approximated using co-occurrence frequencies of pairs and triples of observations by using a fast spectral method in contrast to the usual slow methods like EM or Gibbs sampling. We provide a new spectral method which significantly reduces the number of model parameters that need to be estimated, and generates a sample complexity that does not depend on the size of the observation vocabulary. We present an elementary proof giving bounds on the relative accuracy of probability estimates from our model. (Correlaries show our bounds can be weakened to provide either L1 bounds or KL bounds which provide easier direct comparisons to previous work.) Our theorem uses conditions that are checkable from the data, instead of putting conditions on the unobservable Markov transition matrix

A Risk Comparison of Ordinary Least Squares vs Ridge Regression

We compare the risk of ridge regression to a simple variant of ordinary least squares, in which one simply projects the data onto a finite dimensional subspace (as specified by a Principal Component Analysis) and then performs an ordinary (un-regularized) least squares regression in this subspace. This note shows that the risk of this ordinary least squares method is within a constant factor (namely 4) of the risk of ridge regression.Comment: Appearing in JMLR 14, June 201

Harmonic analysis on the Möbius gyrogroup

In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$

Accounting for Cognitive Costs in On-Line Auction Design

Many auction mechanisms, including first and second price ascending and sealed bid auctions, have been proposed and analyzed in the economics literature. We compare the usefulness of different mechanisms for on-line auctions, focusing on the cognitive costs placed on users (e.g. the cost of determining the value of a good), the possibilities for agent mediation, and the trust properties of the auction. Different auction formats prove to be attractive for agent mediated on-line auctions than for traditional off-line auctions. For example, second price sealed bid auctions are attractive in traditional auctions because they avoid the communication cost of multiple bids in first price ascending auctions, and the “gaming” required to estimate the second highest bid in first price sealed bid auctions. However, when bidding agents are cheap, communication costs cease to be important, and a progressive auction mechanism is preferred over a closed bid auction mechanism, since users with semi-autonomous agents can avoid the cognitive cost of placing an accurate value on a good. As another example, when an on-line auction is being conducted by an untrusted auctioneer (e.g. the auctioneer is selling its own items), rational participants will build bidding agents that transform second price auctions into first price auctions.Engineering and Applied Science

Efficient Feature Selection in the Presence of Multiple Feature Classes

We present an information theoretic approach to feature selection when the data possesses feature classes. Feature classes are pervasive in real data. For example, in gene expression data, the genes which serve as features may be divided into classes based on their membership in gene families or pathways. When doing word sense disambiguation or named entity extraction, features fall into classes including adjacent words, their parts of speech, and the topic and venue of the document the word is in. When predictive features occur predominantly in a small number of feature classes, our information theoretic approach significantly improves feature selection. Experiments on real and synthetic data demonstrate substantial improvement in predictive accuracy over the standard L0 penalty-based stepwise and stream wise feature selection methods as well as over Lasso and Elastic Nets, all of which are oblivious to the existence of feature classes

Evaluation of nonmetallic thermal protection materials for the manned space shuttle. Volume 1, task 1: Assessment of technical risks associated with utilization of nonmetallic thermal protection system

Technical problems of design and flight qualification of the proposed classes of surface insulation materials and leading edge materials were reviewed. A screening test plan, a preliminary design data test plan and a design data test plan were outlined. This program defined the apparent critical differences between the surface insulators and the leading edge materials, structuring specialized screening test plans for each of these two classes of materials. Unique testing techniques were shown to be important in evaluating the structural interaction aspects of the surface insulators and a separate task was defined to validate the test plan. In addition, a compilation was made of available information on proposed material (including metallic TPS), previous shuttle programs, pertinent test procedures, and other national programs of merit. This material was collected and summarized in an informally structured workbook