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We propose a new framework to design and analyze accelerated methods that solve general monotone equation (ME) problems $F(x)=0$. Traditional approaches include generalized steepest descent methods and inexact Newton-type methods. If $F$ is…
We establish a large-deviations principle for the largest eigenvalue of a generalized sample covariance matrix, meaning a matrix proportional to $Z^T \Gamma Z$, where $Z$ has i.i.d. real or complex entries and $\Gamma$ is not necessarily…
We determine the kind of parametric oscillators that are generated in the usual factorization procedure of second-order linear differential equations when one introduces a constant shift of the Riccati solution of the classical harmonic…
Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices is formed by the adjacency matrix of an Erd\H{o}s-R\'{e}nyi graph $\mathcal{G}_{n,p}$ equipped with i.i.d.…
We propose a novel estimation framework for quadratic functionals of precision matrices in high-dimensional settings, particularly in regimes where the feature dimension $p$ exceeds the sample size $n$. Traditional moment-based estimators…
We compare the long-time error bounds and spatial resolution of finite difference methods with different spatial discretizations for the Dirac equation with small electromagnetic potentials characterized by $\varepsilon \in (0, 1]$ a…
We study two novel approaches to efficiently encoding universal constraints imposed by conformal symmetry, and describe applications to quantum chaos in higher dimensional CFTs. The first approach consists of a reformulation of the shadow…
Differential Riccati equations (DREs) are semilinear matrix- or operator-valued differential equations with quadratic non-linearities. They arise in many different areas, and are particularly important in optimal control of linear quadratic…
Modelling multivariate spatio-temporal data with complex dependency structures is a challenging task but can be simplified by assuming that the original variables are generated from independent latent components. If these components are…
Large deviation functions contain information on the stability and response of systems driven into nonequilibrium steady states, and in such a way are similar to free energies for systems at equilibrium. As with equilibrium free energies,…
It is a common phenomenon that for high-dimensional and nonparametric statistical models, rate-optimal estimators balance squared bias and variance. Although this balancing is widely observed, little is known whether methods exist that…
We generalize the Brezzi-Rappaz-Raviart approximation theorem, which allows to obtain existence and a priori error estimates for approximations of solutions to some nonlinear partial differential equations. Our contribution lies in the fact…
We present a comparative study of two methods for the reduction of the dimensionality of a system of ordinary differential equations that exhibits time-scale separation. Both methods lead to a reduced system of stochastic differential…
We prove a reducibility result for a linear Klein-Gordon equation with a quasi-periodic driving on a compact interval with Dirichlet boundary conditions. No assumptions are made on the size of the driving, however we require it to be fast…
We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to 0 and $+\infty$.…
The use of high order fully implicit Runge-Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space…
High dimensional random dynamical systems are ubiquitous, including -- but not limited to -- cyber-physical systems, daily return on different stocks of S&P 1500 and velocity profile of interacting particle systems around McKeanVlasov…
Recent studies show that transformer-based architectures emulate gradient descent during a forward pass, contributing to in-context learning capabilities - an ability where the model adapts to new tasks based on a sequence of prompt…
This paper studies large-scale optimization problems on Riemannian manifolds whose objective function is a finite sum of negative log-probability losses. Such problems arise in various machine learning and signal processing applications. By…
When using a finite difference method to solve an initial--boundary--value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze…