Related papers: Rough paths and regularization
A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and…
The goal of regression and classification methods in supervised learning is to minimize the empirical risk, that is, the expectation of some loss function quantifying the prediction error under the empirical distribution. When facing scarce…
This paper deals with stochastic integrals of form $\int_0^T f(X_u)d Y_u$ in a case where the function $f$ has discontinuities, and hence the process $f(X)$ is usually of unbounded $p$-variation for every $p\geq 1$. Consequently,…
We embed the rough integration in a larger geometrical/algebraic framework of integrating one-forms against group-valued paths, and reduce the rough integral to an inhomogeneous analogue of the classical Young integral. We define dominated…
Weighted counting problems are a natural generalization of counting problems where a weight is associated with every computational path of polynomial-time non-deterministic Turing machines and the goal is to compute the sum of the weights…
We define a deterministic integral with respect to irregular paths as a limit of standard line integrals and completely describe a class of all paths for which this integral exists for functions with H\"older exponent in the range of (0,1].…
Solutions to linear controlled differential equations can be expressed in terms of iterated path integrals of the driving path. This collection of iterated integrals encodes essentially all information about the driving path. While upper…
We propose regularization methods for linear models based on the $L_q$-likelihood, which is a generalization of the log-likelihood using a power function. Some heavy-tailed distributions are known as $q$-normal distributions. We find that…
The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential…
We use a path integral approach for solving the stochastic equations underlying the financial markets, and we show the equivalence between the path integral and the usual SDE and PDE methods. We analyze both the one-dimensional and the…
In this paper we explain how the notion of ''weak Dirichlet process'' is the suitable generalization of the one of semimartingale with jumps. For such a process we provide a unique decomposition which is new also for semimartingales: in…
We study the problem of pathwise stochastic optimal control, where the optimization is performed for each fixed realisation of the driving noise, by phrasing the problem in terms of the optimal control of rough differential equations. We…
Rough path analysis is developed in the full Besov scale. This extends, and essentially concludes, an investigation started by [Pr\"omel--Trabs, Rough differential equations driven by signals in {B}esov spaces. J. Diff. Equ. 2016], further…
This paper deals with computation trees over an arbitrary structure consisting of a set along with collections of functions and predicates that are defined on it. It is devoted to the comparative analysis of three parameters of problems…
We formulate indefinite integration with respect to an irregular function as an algebraic problem and provide a criterion for the existence and uniqueness of a solution. This allows us to define a good notion of integral with respect to…
We propose novel smooth approximations to the classical rounding function, suitable for differentiable optimization and machine learning applications. Our constructions are based on two approaches: (1) localized sigmoid window functions…
We construct a procedure for Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization of a rough path in view of the relation between rough path theory and regularity structure. We also provide a plain expression of the BPHZ-renormalized…
This paper analyzes regularization terms proposed recently for improving the adversarial robustness of deep neural networks (DNNs), from a theoretical point of view. Specifically, we study possible connections between several effective…
Stochastic Rounding is a probabilistic rounding mode that is surprisingly effective in large-scale computations and low-precision arithmetic. Its random nature promotes error cancellation rather than error accumulation, resulting in slower…
We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the…