Measuring Visual Complexity of Cluster-Based Visualizations

Handling visual complexity is a challenging problem in visualization owing to the subjectiveness of its definition and the difficulty in devising generalizable quantitative metrics. In this paper we address this challenge by measuring the visual complexity of two common forms of cluster-based visualizations: scatter plots and parallel coordinatess. We conceptualize visual complexity as a form of visual uncertainty, which is a measure of the degree of difficulty for humans to interpret a visual representation correctly. We propose an algorithm for estimating visual complexity for the aforementioned visualizations using Allen's interval algebra. We first establish a set of primitive 2-cluster cases in scatter plots and another set for parallel coordinatess based on symmetric isomorphism. We confirm that both are the minimal sets and verify the correctness of their members computationally. We score the uncertainty of each primitive case based on its topological properties, including the existence of overlapping regions, splitting regions and meeting points or edges. We compare a few optional scoring schemes against a set of subjective scores by humans, and identify the one that is the most consistent with the subjective scores. Finally, we extend the 2-cluster measure to k-cluster measure as a general purpose estimator of visual complexity for these two forms of cluster-based visualization

Extended Self-similarity in Kinetic Surface Roughening

We show from numerical simulations that a limited mobility solid-on-solid model of kinetically rough surface growth exhibits extended self-similarity analogous to that found in fluid turbulence. The range over which scale-independent power-law behavior is observed is significantly enhanced if two correlation functions of different order, such as those representing two different moments of the difference in height between two points, are plotted against each other. This behavior, found in both one and two dimensions, suggests that the `relative' exponents may be more fundamental than the `absolute' ones.Comment: 4 pages, 4 postscript figures included (some changes made according to referees' comments. accepted for publication in PRE Rapid Communication

Dimension-adaptive bounds on compressive FLD Classification

Efficient dimensionality reduction by random projections (RP) gains popularity, hence the learning guarantees achievable in RP spaces are of great interest. In finite dimensional setting, it has been shown for the compressive Fisher Linear Discriminant (FLD) classifier that forgood generalisation the required target dimension grows only as the log of the number of classes and is not adversely affected by the number of projected data points. However these bounds depend on the dimensionality d of the original data space. In this paper we give further guarantees that remove d from the bounds under certain conditions of regularity on the data density structure. In particular, if the data density does not fill the ambient space then the error of compressive FLD is independent of the ambient dimension and depends only on a notion of ‘intrinsic dimension'

Entropic Origin of the Growth of Relaxation Times in Simple Glassy Liquids

Transitions between ``glassy'' local minima of a model free-energy functional for a dense hard-sphere system are studied numerically using a ``microcanonical'' Monte Carlo method that enables us to obtain the transition probability as a function of the free energy and the Monte Carlo ``time''. The growth of the height of the effective free energy barrier with density is found to be consistent with a Vogel-Fulcher law. The dependence of the transition probability on time indicates that this growth is primarily due to entropic effects arising from the difficulty of finding low-free-energy saddle points connecting glassy minima.Comment: Four pages, plus three postscript figure

Entanglement as a source of black hole entropy

We review aspects of black hole thermodynamics, and show how entanglement of a quantum field between the inside and outside of a horizon can account for the area-proportionality of black hole entropy, provided the field is in its ground state. We show that the result continues to hold for Coherent States and Squeezed States, while for Excited States, the entropy scales as a power of area less than unity. We also identify location of the degrees of freedom which give rise to the above entropy.Comment: 12 pages, latex, 5 figures. Invited talk by SD at `Recent Developments in Gravity' (NEB XII), Nafplion, Greece, 30 June 2006. To appear in Journal of Physics: Conference Series; V2: References added, Minor changes to match published versio

Cooperative orbital ordering and Peierls instability in the checkerboard lattice with doubly degenerate orbitals

It has been suggested that the metal-insulator transitions in a number of spinel materials with partially-filled t_2g d-orbitals can be explained as orbitally-driven Peierls instabilities. Motivated by these suggestions, we examine theoretically the possibility of formation of such orbitally-driven states within a simplified theoretical model, a two-dimensional checkerboard lattice with two directional metal orbitals per atomic site. We include orbital ordering and inter-atom electron-phonon interactions self-consistently within a semi-classical approximation, and onsite intra- and inter-orbital electron-electron interactions at the Hartree-Fock level. We find a stable, orbitally-induced Peierls bond-dimerized state for carrier concentration of one electron per atom. The Peierls bond distortion pattern continues to be period 2 bond-dimerization even when the charge density in the orbitals forming the one-dimensional band is significantly smaller than 1. In contrast, for carrier density of half an electron per atom the Peierls instability is absent within one-electron theory as well as mean-field theory of electron-electron interactions, even for nearly complete orbital ordering. We discuss the implications of our results in relation to complex charge, bond, and orbital-ordering found in spinels.Comment: 8 pages, 5 figures; revised versio