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Motivated by the increasingly popular Score-based Generative Modeling (SGM), we study the Inexact Langevin Dynamics (ILD) and Inexact Langevin Algorithm (ILA) where a score function estimate is used in place of the exact score. We establish…
Structured Kernel Interpolation (SKI) (Wilson et al. 2015) helps scale Gaussian Processes (GPs) by approximating the kernel matrix via interpolation at inducing points, achieving linear computational complexity. However, it lacks rigorous…
The famous Davies-GKSL secular Markovian master equation is tremendously successful in approximating the evolution of open quantum systems in terms of just a few parameters. However, the fully-secular Davies-GKSL equation fails to…
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account nonsecular effects, but nevertheless has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan form (in particular, it…
We establish rigorous error bounds for prime counting using a truncated Gaussian (TG) kernel in the explicit formula framework. Our main theorem proves that the approximation error remains globally below 1/2 for all sufficiently large…
In many contexts Gaussian Mixtures (GM) are used to approximate probability distributions, possibly time-varying. In some applications the number of GM components exponentially increases over time, and reduction procedures are required to…
For time series with long-range temporal dependence, inference for covariance and precision matrices is non-trivial. We propose a Berry-Esseen type Gaussian approximation result that gives a finite-sample bound for the Kolmogorov distance…
We provide the rigorous justification of the NLS approximation, in Sobolev regularity, for a class of quasilinear Hamiltonian Klein Gordon equations with quadratic nonlinearities on large one-dimensional tori $\T_L:=\mathbb{R}/(2\pi L…
The thermodynamic uncertainty relation is a universal trade-off relation connecting the precision of a current with the average dissipation at large times. For continuous time Markov chains (also called Markov jump processes) this relation…
We consider an initial-boundary value problem for $\partial_tu-\partial_t^{-\alpha}\nabla^2u=f(t)$, that is, for a fractional diffusion ($-1<\alpha<0$) or wave ($0<\alpha<1$) equation. A numerical solution is found by applying a…
We establish novel and general high-dimensional concentration inequalities and Berry-Esseen bounds for vector-valued martingales induced by Markov chains. We apply these results to analyze the performance of the Temporal Difference (TD)…
Many learning algorithms can be represented as Markov processes, and understanding their generalization error is a central topic in learning theory. For specific continuous-time noisy algorithms, a prominent analysis technique relies on…
In this work, we consider the convergence analysis of time-splitting schemes for the nonlinear Klein--Gordon/wave equation under rough initial data. The optimal error bounds of the Lie splitting and the Strang splitting are established with…
Quantum master equations are commonly used to model the dynamics of open quantum systems, but their accuracy is rarely compared with the analytical solution of exactly solvable models. In this work, we perform such a comparison for the…
Model reduction is a powerful tool in dealing with numerical simulation of large scale dynamic systems for studying complex physical systems. Two major types of model reduction methods for linear time-invariant dynamic systems are Krylov…
Recent studies suggest utilizing generative models instead of traditional auto-regressive algorithms for time series forecasting (TSF) tasks. These non-auto-regressive approaches involving different generative methods, including GAN,…
Lumping a Markov process introduces a coarser level of description that is useful in many contexts and applications. The dynamics on the coarse grained states is often approximated by its Markovian component. In this letter we derive…
Distinct monitoring or intervention schemes can produce different conditioned stochastic quantum trajectories while sharing the same unconditional (ensemble-averaged) dynamics. This is the essence of unravelings of a given…
Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models. The aim of this paper is to provide a…
In 1976, Gorini, Kossakowski, Sudarshan and Lindblad independently discovered a general form of master equations for an open quantum Markovian dynamics. In honor of all the authors, the equation is nowadays called the GKLS master equation.…