2,261 research outputs found
Dynkin operators and renormalization group actions in pQFT
Renormalization techniques in perturbative quantum field theory were known,
from their inception, to have a strong combinatorial content emphasized, among
others, by Zimmermann's celebrated forest formula. The present article reports
on recent advances on the subject, featuring the role played by the Dynkin
operators (actually their extension to the Hopf algebraic setting) at two
crucial levels of renormalization, namely the Bogolioubov recursion and the
renormalization group (RG) equations. For that purpose, an iterated integrals
toy model is introduced to emphasize how the operators appear naturally in the
setting of renormalization group analysis. The toy model, in spite of its
simplicity, captures many key features of recent approaches to RG equations in
pQFT, including the construction of a universal Galois group for quantum field
theories
Brace algebras and the cohomology comparison theorem
The Gerstenhaber and Schack cohomology comparison theorem asserts that there
is a cochain equivalence between the Hochschild complex of a certain algebra
and the usual singular cochain complex of a space. We show that this comparison
theorem preserves the brace algebra structures. This result gives a structural
reason for the recent results establishing fine topological structures on the
Hochschild cohomology, and a simple way to derive them from the corresponding
properties of cochain complexes.Comment: Revised version of "The bar construction as a Hopf algebra", Dec.
200
Lie theory for Hopf operads
The present article takes advantage of the properties of algebras in the
category of S-modules (twisted algebras) to investigate further the fine
algebraic structure of Hopf operads. We prove that any Hopf operad P carries
naturally the structure of twisted Hopf P-algebra. Many properties of classical
Hopf algebraic structures are then shown to be encapsulated in the twisted Hopf
algebraic structure of the corresponding Hopf operad. In particular, various
classical theorems of Lie theory relating Lie polynomials to words (i.e.
elements of the tensor algebra) are lifted to arbitrary Hopf operads.Comment: 23 pages. Using xyPi
Incremental refinement of image salient-point detection
Low-level image analysis systems typically detect "points of interest", i.e., areas of natural images that contain corners or edges. Most of the robust and computationally efficient detectors proposed for this task use the autocorrelation matrix of the localized image derivatives. Although the performance of such detectors and their suitability for particular applications has been studied in relevant literature, their behavior under limited input source (image) precision or limited computational or energy resources is largely unknown. All existing frameworks assume that the input image is readily available for processing and that sufficient computational and energy resources exist for the completion of the result. Nevertheless, recent advances in incremental image sensors or compressed sensing, as well as the demand for low-complexity scene analysis in sensor networks now challenge these assumptions. In this paper, we investigate an approach to compute salient points of images incrementally, i.e., the salient point detector can operate with a coarsely quantized input image representation and successively refine the result (the derived salient points) as the image precision is successively refined by the sensor. This has the advantage that the image sensing and the salient point detection can be terminated at any input image precision (e.g., bound set by the sensory equipment or by computation, or by the salient point accuracy required by the application) and the obtained salient points under this precision are readily available. We focus on the popular detector proposed by Harris and Stephens and demonstrate how such an approach can operate when the image samples are refined in a bitwise manner, i.e., the image bitplanes are received one-by-one from the image sensor. We estimate the required energy for image sensing as well as the computation required for the salient point detection based on stochastic source modeling. The computation and energy required by the proposed incremental refinement approach is compared against the conventional salient-point detector realization that operates directly on each source precision and cannot refine the result. Our experiments demonstrate the feasibility of incremental approaches for salient point detection in various classes of natural images. In addition, a first comparison between the results obtained by the intermediate detectors is presented and a novel application for adaptive low-energy image sensing based on points of saliency is presented
Mirror, mirror on the wall, tell me, is the error small?
Do object part localization methods produce bilaterally symmetric results on
mirror images? Surprisingly not, even though state of the art methods augment
the training set with mirrored images. In this paper we take a closer look into
this issue. We first introduce the concept of mirrorability as the ability of a
model to produce symmetric results in mirrored images and introduce a
corresponding measure, namely the \textit{mirror error} that is defined as the
difference between the detection result on an image and the mirror of the
detection result on its mirror image. We evaluate the mirrorability of several
state of the art algorithms in two of the most intensively studied problems,
namely human pose estimation and face alignment. Our experiments lead to
several interesting findings: 1) Surprisingly, most of state of the art methods
struggle to preserve the mirror symmetry, despite the fact that they do have
very similar overall performance on the original and mirror images; 2) the low
mirrorability is not caused by training or testing sample bias - all algorithms
are trained on both the original images and their mirrored versions; 3) the
mirror error is strongly correlated to the localization/alignment error (with
correlation coefficients around 0.7). Since the mirror error is calculated
without knowledge of the ground truth, we show two interesting applications -
in the first it is used to guide the selection of difficult samples and in the
second to give feedback in a popular Cascaded Pose Regression method for face
alignment.Comment: 8 pages, 9 figure
Nonlocal, noncommutative diagrammatics and the linked cluster Theorems
Recent developments in quantum chemistry, perturbative quantum field theory,
statistical physics or stochastic differential equations require the
introduction of new families of Feynman-type diagrams. These new families arise
in various ways. In some generalizations of the classical diagrams, the notion
of Feynman propagator is extended to generalized propagators connecting more
than two vertices of the graphs. In some others (introduced in the present
article), the diagrams, associated to noncommuting product of operators inherit
from the noncommutativity of the products extra graphical properties. The
purpose of the present article is to introduce a general way of dealing with
such diagrams. We prove in particular a "universal" linked cluster theorem and
introduce, in the process, a Feynman-type "diagrammatics" that allows to handle
simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams
arising from the study of interacting systems (such as the ones where the
ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or
electric field, by impurities...) or Wightman fields (that is, expectation
values of products of interacting fields). Our diagrammatics seems to be the
first attempt to encode in a unified algebraic framework such a wide variety of
situations. In the process, we promote two ideas. First, Feynman-type
diagrammatics belong mathematically to the theory of linear forms on
combinatorial Hopf algebras. Second, linked cluster-type theorems rely
ultimately on M\"obius inversion on the partition lattice. The two theories
should therefore be introduced and presented accordingl
Right-handed Hopf algebras and the preLie forest formula
Three equivalent methods allow to compute the antipode of the Hopf algebras
of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson-Salam
formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the
first two hold generally for arbitrary connected graded Hopf algebras, the
third one requires extra structure properties of the underlying Hopf algebra
but has the nice property to reduce drastically the number of terms in the
expression of the antipode (it is optimal in that sense).The present article is
concerned with the forest formula: we show that it generalizes to arbitrary
right-handed polynomial Hopf algebras. These Hopf algebras are dual to the
enveloping algebras of preLie algebras -a structure common to many
combinatorial Hopf algebras which is carried in particular by the Hopf algebras
of Feynman diagrams
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