Related papers: Volterra Equations Driven by Rough Signals
We study the surface quasi-geostrophic equation driven by a generic additive noise process $W$. By means of convex integration techniques, we establish existence of weak solutions whenever the stochastic convolution $z$ associated with $W$…
Signature is an infinite graded sequence of statistics known to characterize geometric rough paths, which includes the paths with bounded variation. This object has been studied successfully for machine learning with mostly applications in…
We study two complementary methodologies for calibrating implied volatility surfaces: analytical approximations and data-driven models based on rough path theory. On the analytical side, we revisit a second-order asymptotic expansion for…
In this paper, we deal with the convolution series that are a far reaching generalization of the conventional power series and the power series with the fractional exponents including the Mittag-Leffler type functions. Special attention is…
The theory of rough paths arose from a desire to establish continuity properties of ordinary differential equations involving terms of low regularity. While essentially an analytic theory, its main motivation and applications are in…
Dot product kernels, such as polynomial and exponential (softmax) kernels, are among the most widely used kernels in machine learning, as they enable modeling the interactions between input features, which is crucial in applications like…
An invex function generalizes a convex function in the sense that every stationary point is a global minimizer. Recently, invex functions and their subclasses have attracted attention in signal processing and machine learning. However,…
Space-time regularity of linear stochastic partial differential equations is studied. The solution is defined in the mild sense in the state space $L^p$. The corresponding regularity is obtained by showing that the stochastic convolution…
This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the…
We study classical solutions (existence, uniqueness, and explicit solution operator) for homogeneous, linear, and semilinear abstract Volterra integral equations of wave type with almost sectorial operators. We use a functional calculus for…
The monotone path polytope of a polytope $P$ encapsulates the combinatorial behavior of the shadow vertex rule (a pivot rule used in linear programming) on $P$. Computing monotone path polytopes is the entry door to the larger subject of…
We study signature tensors of paths from a geometric viewpoint. The signatures of a given class of paths parametrize an algebraic variety inside the space of tensors, and these signature varieties provide both new tools to investigate paths…
We examine the duality theory for a class of non-convex functions obtained by composing a convex function with a continuous one. Using Fenchel duality, we derive a dual problem that satisfies weak duality under general assumptions. To…
The notion of fractional Fourier transform (FrFT) has been used and investigated for many years by various research communities, which finds widespread applications in many diverse fields of research study. The potential applications…
Nonlinearity of power amplifier is one of the major limitations to the achievable capacity in wireless transmission systems. Nonlinear impairments are determined by the nonlinear distortions of the power amplifier and modulator…
Convolution, a cornerstone of signal processing and optical neural networks, has traditionally been implemented by mapping mathematical operations onto complex hardware. Here, we overcome this challenge by revealing that wave dynamics in…
We study small-time central limit theorems for stochastic Volterra integral equations with H\"older continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional…
We are concerned with nonparametric hypothesis testing of time series functionals. It is known that the popular autoregressive sieve bootstrap is, in general, not valid for statistics whose (asymptotic) distribution depends on moments of…
Deep convolutional neural networks are hindered by training instability and feature redundancy towards further performance improvement. A promising solution is to impose orthogonality on convolutional filters. We develop an efficient…
We investigate the use of models from the theory of regularity structures as features in machine learning tasks. A model is a polynomial function of a space-time signal designed to well-approximate solutions to partial differential…