123 research outputs found
Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities
Information-theoretic measures such as the entropy, cross-entropy and the
Kullback-Leibler divergence between two mixture models is a core primitive in
many signal processing tasks. Since the Kullback-Leibler divergence of mixtures
provably does not admit a closed-form formula, it is in practice either
estimated using costly Monte-Carlo stochastic integration, approximated, or
bounded using various techniques. We present a fast and generic method that
builds algorithmically closed-form lower and upper bounds on the entropy, the
cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate
the versatile method by reporting on our experiments for approximating the
Kullback-Leibler divergence between univariate exponential mixtures, Gaussian
mixtures, Rayleigh mixtures, and Gamma mixtures.Comment: 20 pages, 3 figure
Holomorphic Yukawa Couplings in Heterotic String Theory
This thesis is concerned with heterotic E8 x E8 string models that can
produce quasirealistic N = 1 supersymmetric extensions of the Standard Model in
the low-energy limit. We start rather generally by deriving the
four-dimensional spectrum and Lagrangian terms from the ten-dimensional theory,
through a process of compactification over six-dimensional Calabi-Yau
manifolds, upon which holomorphic poly-stable vector bundles are defined. We
then specialise to a class of heterotic string models for which the vector
bundle is split into a sum of line bundles and the Calabi-Yau manifold is
defined as a complete intersection in projective ambient spaces. We develop a
method for calculating holomorphic Yukawa couplings for such models, by
relating bundle-valued forms on the Calabi-Yau manifold to their ambient space
counterparts, so that the relevant integrals can be evaluated over projective
spaces. The method is applicable for any of the 7890 CICY manifolds known in
the literature, and we show that it can be related to earlier algebraic
techniques to compute holomorphic Yukawa couplings. We provide explicit
calculations of the holomorphic Yukawa couplings for models compactified on the
tetra-quadric and on a co-dimension two CICY. A vanishing theorem is
formulated, showing that in some cases, topology rather than symmetry is
responsible for the absence of certain trilinear couplings. In addition, some
Yukawa matrices are found to be dependent on the complex structure moduli and
their rank is reduced in certain regions of the moduli space. In the final
part, we focus on a method to evaluate the matter field Kahler potential
without knowing the Ricci-flat Calabi-Yau metric. This is possible for large
internal gauge fluxes, for which the normalisation integral localises around a
point on the compactification manifold.Comment: PhD thesi
A spacetime derivation of the Lorentzian OPE inversion formula
Caron-Huot has recently given an interesting formula that determines OPE data
in a conformal field theory in terms of a weighted integral of the four-point
function over a Lorentzian region of cross-ratio space. We give a new
derivation of this formula based on Wick rotation in spacetime rather than
cross-ratio space. The derivation is simple in two dimensions but more involved
in higher dimensions. We also derive a Lorentzian inversion formula in one
dimension that sheds light on previous observations about the chaos regime in
the SYK model.Comment: 26 pages plus appendice
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
Aspects of The First Law of Complexity
We investigate the first law of complexity proposed in arXiv:1903.04511,
i.e., the variation of complexity when the target state is perturbed, in more
detail. Based on Nielsen's geometric approach to quantum circuit complexity, we
find the variation only depends on the end of the optimal circuit. We apply the
first law to gain new insights into the quantum circuits and complexity models
underlying holographic complexity. In particular, we examine the variation of
the holographic complexity for both the complexity=action and complexity=volume
conjectures in perturbing the AdS vacuum with coherent state excitations of a
free scalar field. We also examine the variations of circuit complexity
produced by the same excitations for the free scalar field theory in a fixed
AdS background. In this case, our work extends the existing treatment of
Gaussian coherent states to properly include the time dependence of the
complexity variation. We comment on the similarities and differences of the
holographic and QFT results.Comment: 108 pages, 15 figures; v2: references adde
The geometry of colour
This thesis explores the geometric description of animal colour vision. It examines the relationship of colour spaces to behavior and to physiology. I provide a derivation of, and explore the limits of, geometric spaces derived from the notion of risk and uncertainty aversion as well as the geometric objects that enumerate the variety of achievable colours. Using these principles I go on to explore evolutionary questions concerning colourfulness, such as aposematism, mimicry and the idea of aesthetic preference
Twistor actions for gauge theory and gravity
This is a review of recent developments in the study of perturbative gauge
theory and gravity using action functionals on twistor space. It is intended to
provide a user-friendly introduction to twistor actions, geared towards
researchers or graduate students interested in learning something about the
utility, prospects, and shortcomings of this approach. For those already
familiar with the twistor approach, it should provide a condensed overview of
the literature as well as several novel results of potential interest. This
work is based primarily upon the author's D.Phil. thesis. We first consider
four-dimensional, maximally supersymmetric Yang-Mills theory as a gauge theory
in twistor space. We focus on the perturbation theory associated to this
action, which in an axial gauge leads to the MHV formalism. This allows us to
efficiently compute scattering amplitudes at tree-level (and beyond) in twistor
space. Other gauge theory observables such as local operators and null
polygonal Wilson loops can also be formulated twistorially, leading to proofs
for several correspondences between correlation functions and Wilson loops, as
well as a recursive formula for computing mixed Wilson loop / local operator
correlators. We then apply the twistor action approach to general relativity,
using the on-shell equivalence between conformal and Einstein gravity. This can
be extended to N=4 supersymmetry. The perturbation theory of the twistor action
leads to formulae for the MHV amplitude with and without cosmological constant,
yields a candidate for the Einstein twistor action, and induces a MHV formalism
on twistor space. Appendices include discussion of super-connections and
Coulomb branch regularization on twistor space.Comment: 178 pages, 30 figures. Review based on the author's D.Phil. thesis.
v2: references adde
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