14,748 research outputs found
Statistical mechanical foundations of power-law distributions
The foundations of the Boltzmann-Gibbs (BG) distributions for describing
equilibrium statistical mechanics of systems are examined. Broadly, they fall
into: (i) probabilistic paaroaches based on the principle of equal a priori
probability (counting technique and method of steepest descents), law of large
numbers, or the state density considerations and (ii) a variational scheme --
maximum entropy principle (due to Gibbs and Jaynes) subject to certain
constraints. A minimum set of requirements on each of these methods are briefly
pointed out: in the first approach, the function space and the counting
algorithm while in the second, "additivity" property of the entropy with
respect to the composition of statistically independent systems. In the past
few decades, a large number of systems, which are not necessarily in
thermodynamic equilibrium (such as glasses, for example), have been found to
display power-law distributions, which are not describable by the
above-mentioned methods. In this paper, parallel to all the inquiries
underlying the BG program described above are given in a brief form. In
particular, in the probabilistic derivations, one employs a different function
space and one gives up "additivity" in the variational scheme with a different
form for the entropy. The requirement of stability makes the entropy choice to
be that proposed by Tsallis. From this a generalized thermodynamic description
of the system in a quasi-equilibrium state is derived. A brief account of a
unified consistent formalism associated with systems obeying power-law
distributions precursor to the exponential form associated with thermodynamic
equilibrium of systems is presented here.Comment: 19 pages, no figures. Invited talk at Anomalous Distributions,
Nonlinear Dynamics and Nonextensivity, Santa Fe, USA, November 6-9, 200
Convex Relaxations of SE(2) and SE(3) for Visual Pose Estimation
This paper proposes a new method for rigid body pose estimation based on
spectrahedral representations of the tautological orbitopes of and
. The approach can use dense point cloud data from stereo vision or an
RGB-D sensor (such as the Microsoft Kinect), as well as visual appearance data.
The method is a convex relaxation of the classical pose estimation problem, and
is based on explicit linear matrix inequality (LMI) representations for the
convex hulls of and . Given these representations, the relaxed
pose estimation problem can be framed as a robust least squares problem with
the optimization variable constrained to these convex sets. Although this
formulation is a relaxation of the original problem, numerical experiments
indicate that it is indeed exact - i.e. its solution is a member of or
- in many interesting settings. We additionally show that this method
is guaranteed to be exact for a large class of pose estimation problems.Comment: ICRA 2014 Preprin
On the impossibility of coin-flipping in generalized probabilistic theories via discretizations of semi-infinite programs
Coin-flipping is a fundamental cryptographic task where a spatially separated
Alice and Bob wish to generate a fair coin-flip over a communication channel.
It is known that ideal coin-flipping is impossible in both classical and
quantum theory. In this work, we give a short proof that it is also impossible
in generalized probabilistic theories under the Generalized No-Restriction
Hypothesis. Our proof relies crucially on a formulation of cheating strategies
as semi-infinite programs, i.e., cone programs with infinitely many
constraints. This introduces a new formalism which may be of independent
interest to the quantum community
Exploiting Social Network Structure for Person-to-Person Sentiment Analysis
Person-to-person evaluations are prevalent in all kinds of discourse and
important for establishing reputations, building social bonds, and shaping
public opinion. Such evaluations can be analyzed separately using signed social
networks and textual sentiment analysis, but this misses the rich interactions
between language and social context. To capture such interactions, we develop a
model that predicts individual A's opinion of individual B by synthesizing
information from the signed social network in which A and B are embedded with
sentiment analysis of the evaluative texts relating A to B. We prove that this
problem is NP-hard but can be relaxed to an efficiently solvable hinge-loss
Markov random field, and we show that this implementation outperforms text-only
and network-only versions in two very different datasets involving
community-level decision-making: the Wikipedia Requests for Adminship corpus
and the Convote U.S. Congressional speech corpus
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