Related papers: The Tamed Unadjusted Langevin Algorithm
We demonstrate that for strongly log-convex densities whose potentials are discontinuous on manifolds, the ULA algorithm converges with stepsize bias of order $1/2$ in Wasserstein-p distance. Our resulting bound is then of the same order as…
We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution $\nu = e^{-f}$ on $\mathbb{R}^n$ under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when $\nu$…
We study sampling as optimization in the space of measures. We focus on gradient flow-based optimization with the Langevin dynamics as a case study. We investigate the source of the bias of the unadjusted Langevin algorithm (ULA) in…
This paper introduces a randomized tamed Euler scheme tailored for L\'evy-driven stochastic differential equations (SDEs) with superlinear random coefficients and Carath\'eodory-type drift. Under assumptions that allow for time-irregular…
This paper is concerned with sampling from probability distributions $\pi$ on $\mathbb{R}^d$ admitting a density of the form $\pi(x) \propto e^{-U(x)}$, where $U(x)=F(x)+G(Kx)$ with $K$ being a linear operator and $G$ being…
Local increases in the mean of a random field are detected (conservatively) by thresholding a field of test statistics at a level $u$ chosen to control the tail probability or $p$-value of its maximum. This $p$-value is approximated by the…
We study the underdamped Langevin dynamics with invariant measure $\mu(\,\mathrm{d}x\,\mathrm{d}v)\propto \mathrm{e}^{-U(x)-\lvert v\rvert^2/2}\,\mathrm{d}x\,\mathrm{d}v$. Assume that the position marginal $\mu_x(\,\mathrm{d}x)\propto…
The classical (overdamped) Langevin dynamics provide a natural algorithm for sampling from its invariant measure, which uniquely minimizes an energy functional over the space of probability measures, and which concentrates around the…
It is of significant interest in many applications to sample from a high-dimensional target distribution $\pi$ with the density $\pi(\text{d} x) \propto e^{-U(x)} (\text{d} x) $, based on the temporal discretization of the Langevin…
As a well-known fact, the classical Euler scheme works merely for SDEs with coefficients of linear growth. In this paper, we study a general framework of modified Euler schemes, which is applicable to SDEs with super-linear drifts and…
In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly…
Understanding the dimension dependency of computational complexity in high-dimensional sampling problem is a fundamental problem, both from a practical and theoretical perspective. Compared with samplers with unbiased stationary…
In this paper, we examine the problem of sampling from log-concave distributions with (possibly) superlinear gradient growth under kinetic (underdamped) Langevin algorithms. Using a carefully tailored taming scheme, we propose two novel…
We examine the Langevin diffusion confined to a closed, convex domain $D\subset\mathbb{R}^d$, represented as a reflected stochastic differential equation. We introduce a sequence of penalized stochastic differential equations and prove that…
In this paper, we are concerned with a modified Euler scheme for the SDE under consideration, where the drift is of super-linear growth and dissipative merely outside a closed ball. By adopting the synchronous coupling, along with the…
We investigate the estimates of the density for the traditional Euler-Maruyama discretization of stochastic differential equations (SDEs) with multiplicative noise. Our estimates focus on two key aspects: (1) the $L^p$-upper bounds for…
This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler…
Given $n$ samples from a population of individuals belonging to different species, what is the number $U$ of hitherto unseen species that would be observed if $\lambda n$ new samples were collected? This is an important problem in many…
In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as a first order optimization algorithm of an objective functional defined on the Wasserstein space of order $2$. Using…
The Ensemble Kalman methodology in an inverse problems setting can be viewed as an iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE). Assuming a suitable approximation…