Related papers: An analytic BPHZ theorem for regularity structures
We present a general regularization procedure for piecewise smooth vector fields whose discontinuity locus is a variety of normal crossings type. We show that such regularization can be smoothed through a finite sequence of blowings-up,…
Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this context, we consider ill-posed nonlinear inverse problems…
We investigate three types of averaging principles and the normal deviation for multi-scale stochastic differential equations (in short, SDEs) with polynomial nonlinearity. More specifically, we first demonstrate the strong convergence of…
Database theory and database practice are typically the domain of computer scientists who adopt what may be termed an algorithmic perspective on their data. This perspective is very different than the more statistical perspective adopted by…
In order to study in a regularisation free manner the renormalisability of d=2 supersymmetric non-linear $\si$ models, one has to use the algebraic BRS methods ; moreover, in the absence of an off-shell formulation, one has often to deal…
This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale…
We give an intuitive proof of a new non-renormalization theorem in supersymmetric field theories. It applies both perturbatively and non-perturbatively. The superpotential is not renormalized in perturbation theory but receives…
In a recent work a modified BPHZ scheme has been introduced and applied to one-loop Feynman graphs in non-commutative phi^4-theory. In the present paper, we first review the BPHZ method and then we apply the modified BPHZ scheme as well as…
An important part of problems in statistical physics and computer science can be expressed as the computation of marginal probabilities over a Markov Random Field. The belief propagation algorithm, which is an exact procedure to compute…
We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a…
In this paper, we propose an approximation method to study the regularity of solutions to the Isaacs equation. This class of problems plays a paramount role in the regularity theory for fully nonlinear elliptic equations. First, it is a…
Recurrent neural networks are a widely used class of neural architectures. They have, however, two shortcomings. First, they are often treated as black-box models and as such it is difficult to understand what exactly they learn as well as…
Recently, the stochastic asymptotical regularization (SAR) has been developed in (\emph{Inverse Problems}, 39: 015007, 2023) for the uncertainty quantification of the stable approximate solution of linear ill-posed inverse problems. In this…
This paper introduces and investigates a regularity condition in the asymptotic sense for optimization problems whose objective functions are polynomial. Under this regularity condition, the normalization argument in asymptotic analysis…
For stationary two-valued harmonic functions with H\"older regularity, we establish their Lipschitz regularity and prove that the nodal set consists of analytic hypersurfaces away from a singular set. The main tools are the Almgren…
We introduce a general framework for analyzing learning algorithms based on the notion of self-regularization, which captures implicit complexity control without requiring explicit regularization. This is motivated by previous observations…
Statistical models that possess symmetry arise in diverse settings such as random fields associated to geophysical phenomena, exchangeable processes in Bayesian statistics, and cyclostationary processes in engineering. We formalize the…
Renormalized perturbation theory \`a la BPHZ can be founded on causality as analyzed by H. Epstein and V. Glaser in the seventies. Here, we list and discuss a number of additional constraints of algebraic character some of which have to be…
Rate-independent systems arise in a number of applications. Usually, weak solutions to such problems with potentially very low regularity are considered, requiring mathematical techniques capable of handling nonsmooth functions. In this…
The concept of deterministic dynamical chaos has a long history and is well established by now. Nevertheless, its field theoretic essence and its stochastic generalization have been revealed only very recently. Within the newly found…