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