1,796 research outputs found
Effective Dispersion in Computable Metric Spaces
We investigate the relationship between computable metric spaces and where is a given metric space. In the case of Euclidean space, and are equivalent up to isometry, which does not hold in general. We introduce the notion of effectively dispersed metric space. This notion is essential in the proof of the main result of this paper: is effectively totally bounded if and only if is effectively totally bounded, i.e. the property that a computable metric space is effectively totally bounded (and in particular effectively compact) depends only on the underlying metric space
Successive Refinement of Abstract Sources
In successive refinement of information, the decoder refines its
representation of the source progressively as it receives more encoded bits.
The rate-distortion region of successive refinement describes the minimum rates
required to attain the target distortions at each decoding stage. In this
paper, we derive a parametric characterization of the rate-distortion region
for successive refinement of abstract sources. Our characterization extends
Csiszar's result to successive refinement, and generalizes a result by Tuncel
and Rose, applicable for finite alphabet sources, to abstract sources. This
characterization spawns a family of outer bounds to the rate-distortion region.
It also enables an iterative algorithm for computing the rate-distortion
region, which generalizes Blahut's algorithm to successive refinement. Finally,
it leads a new nonasymptotic converse bound. In all the scenarios where the
dispersion is known, this bound is second-order optimal.
In our proof technique, we avoid Karush-Kuhn-Tucker conditions of optimality,
and we use basic tools of probability theory. We leverage the Donsker-Varadhan
lemma for the minimization of relative entropy on abstract probability spaces.Comment: Extended version of a paper presented at ISIT 201
Bimetric Theory of Fractional Quantum Hall States
We present a bimetric low-energy effective theory of fractional quantum Hall
(FQH) states that describes the topological properties and a gapped collective
excitation, known as Girvin-Macdonald-Platzman (GMP) mode. The theory consist
of a topological Chern-Simons action, coupled to a symmetric rank two tensor,
and an action \`a la bimetric gravity, describing the gapped dynamics of the
spin- GMP mode. The theory is formulated in curved ambient space and is
spatially covariant, which allows to restrict the form of the effective action
and the values of phenomenological coefficients. Using the bimetric theory we
calculate the projected static structure factor up to the order in the
momentum expansion. To provide further support for the theory, we derive the
long wave limit of the GMP algebra, the dispersion relation of the GMP mode,
and the Hall viscosity of FQH states. We also comment on the possible
applications to fractional Chern insulators, where closely related structures
arise. Finally, it is shown that the familiar FQH observables acquire a curious
geometric interpretation within the bimetric formalism.Comment: 14 pages, v2: Acknowledgments updated, v3: A few presentation
improvements, Published versio
Langevin and Hamiltonian based Sequential MCMC for Efficient Bayesian Filtering in High-dimensional Spaces
Nonlinear non-Gaussian state-space models arise in numerous applications in
statistics and signal processing. In this context, one of the most successful
and popular approximation techniques is the Sequential Monte Carlo (SMC)
algorithm, also known as particle filtering. Nevertheless, this method tends to
be inefficient when applied to high dimensional problems. In this paper, we
focus on another class of sequential inference methods, namely the Sequential
Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising
alternative to SMC methods. After providing a unifying framework for the class
of SMCMC approaches, we propose novel efficient strategies based on the
principle of Langevin diffusion and Hamiltonian dynamics in order to cope with
the increasing number of high-dimensional applications. Simulation results show
that the proposed algorithms achieve significantly better performance compared
to existing algorithms
Spectral dimensions and dimension spectra of quantum spacetimes
Different approaches to quantum gravity generally predict that the dimension
of spacetime at the fundamental level is not 4. The principal tool to measure
how the dimension changes between the IR and UV scales of the theory is the
spectral dimension. On the other hand, the noncommutative-geometric perspective
suggests that quantum spacetimes ought to be characterised by a discrete
complex set -- the dimension spectrum. Here we show that these two notions
complement each other and the dimension spectrum is very useful in unravelling
the UV behaviour of the spectral dimension. We perform an extended analysis
highlighting the trouble spots and illustrate the general results with two
concrete examples: the quantum sphere and the -Minkowski spacetime, for
a few different Laplacians. In particular, we find out that the spectral
dimensions of the former exhibit log-periodic oscillations, the amplitude of
which decays rapidly as the deformation parameter tends to the classical value.
In contrast, no such oscillations occur for either of the three considered
Laplacians on the -Minkowski spacetime.Comment: 35 pages, 7 figures, v2 some comments and references added, summary
extended, title change
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