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Lihu Xu

Publications and source records attributed to Lihu Xu.

At least 37 records · Page 2Linked to original sources
arXiv2023

Distribution estimation and change-point estimation for time series via DNN-based GANs

The generative adversarial networks (GANs) have recently been applied to estimating the distribution of independent and identically distributed data, and have attracted a lot of research attention. In this paper, we use the blocking technique to demonstrate the effectiveness of GANs for estimating the distribution of stationary time series. Theoretically, we derive a non-asymptotic error bound for the Deep Neural Network (DNN)-based GANs estimator for the stationary distribution of the time series. Based on our theoretical analysis, we propose an algorithm for estimating the change point in time series distribution. The two main results are verified by two Monte Carlo experiments respectively, one is to estimate the joint stationary distribution of $5$-tuple samples of a 20 dimensional AR(3) model, the other is about estimating the change point at the combination of two different stationary time series. A real world empirical application to the human activity recognition dataset highlights the potential of the proposed methods.

cs.LG↗
arXiv2023

Multivariate stable approximation in Wasserstein distance by Stein's method

By a delicate analysis for the Stein's equation associated to the $α$-stable law approximation with $α\in (0,2)$, we prove a quantitative stable central limit theorem in Wasserstein type distance, which generalizes the results in the series of work \cite{Xu19, CNX19+,CNXYZ19+} from the univariate case to the multiple variate case. From an explicit computation for Pareto's distribution, we see that the rate of our approximation is sharp. The analysis of the Stein's equation is new and has independent interest.

math.PR↗
arXiv2023

Approximation of the ergodic measure of SDEs with singular drift by Euler-Maruyama scheme

We study the approximation of the ergodic measure of the following stochastic differential equation (SDE) on $\mathbb{R}^d$: \begin{eqnarray}\label{e:SDEE} d X_t &=& (b_1(X_t)+b_2(X_t)) d t+σ(X_t) d W_t, \end{eqnarray} where $W_t$ is a $d$-dimensional standard Brownian motion, and $b_1: \mathbb{R}^d \mapsto \mathbb{R}^d$, $b_2: \mathbb{R}^d \mapsto \mathbb{R}^d$ and $σ: \mathbb{R}^d \mapsto \mathbb{R}^{d\times d}$ are the functions to be specified in Assumption 2.1 below. In particular, $b_1$ satisfies $b_1\in \mathbb{L}^\infty(\mathbb{R}^d)\cap \mathbb{L}^1(\mathbb{R}^d)$ or $b_1 \in \mathcal{C}_b^α(\mathbb{R}^d)$ with $α\in (0,1)$, which makes the standard numerical schemes not work or fail to give a good convergence rate. In order to overcome these two difficulties, we first apply a Zvonkin's transform to SDE and obtain a new SDE which has coefficients with nice properties and admits a unique ergodic measure $\widehat μ$, then discretize the new equation by Euler-Maruyama scheme to approximate $\widehat μ$, and finally use the inverse Zvonkin's transform to get an approximation of the ergodic measure of SDE, denoted by $μ$. Our approximation method is inspired by Xie and Zhang [22]. The proof of our main result is based on the method of introducing a stationary Markov chain, a key ingredient in this method is establishing the regularity of a Poisson equation, which is done by combining the classical PDE local regularity and a nice extension trick introduced by Gurvich [10].

math.PR↗
arXiv2022

Distribution Estimation of Contaminated Data via DNN-based MoM-GANs

This paper studies the distribution estimation of contaminated data by the MoM-GAN method, which combines generative adversarial net (GAN) and median-of-mean (MoM) estimation. We use a deep neural network (DNN) with a ReLU activation function to model the generator and discriminator of the GAN. Theoretically, we derive a non-asymptotic error bound for the DNN-based MoM-GAN estimator measured by integral probability metrics with the $b$-smoothness Hölder class. The error bound decreases essentially as $n^{-b/p}\vee n^{-1/2}$, where $n$ and $p$ are the sample size and the dimension of input data. We give an algorithm for the MoM-GAN method and implement it through two real applications. The numerical results show that the MoM-GAN outperforms other competitive methods when dealing with contaminated data.

stat.ML↗
arXiv2022

Almost sure invariance principle of $β-$mixing time series in Hilbert space

Inspired by \citet{Berkes14} and \citet{Wu07}, we prove an almost sure invariance principle for stationary $β-$mixing stochastic processes defined on Hilbert space. Our result can be applied to Markov chain satisfying Meyn-Tweedie type Lyapunov condition and thus generalises the contraction condition in \citet[Example 2.2]{Berkes14}. We prove our main theorem by the big and small blocks technique and an embedding result in \citet{gotze2011estimates}. Our result is further applied to the ergodic Markov chain and functional autoregressive processes.

math.PR↗
arXiv2022

Non-Asymptotic Guarantees for Robust Statistical Learning under Infinite Variance Assumption

There has been a surge of interest in developing robust estimators for models with heavy-tailed and bounded variance data in statistics and machine learning, while few works impose unbounded variance. This paper proposes two type of robust estimators, the ridge log-truncated M-estimator and the elastic net log-truncated M-estimator. The first estimator is applied to convex regressions such as quantile regression and generalized linear models, while the other one is applied to high dimensional non-convex learning problems such as regressions via deep neural networks. Simulations and real data analysis demonstrate the {robustness} of log-truncated estimations over standard estimations.

stat.ML↗
arXiv2022

An approximation to the invariant measure of the limiting diffusion of G/Ph/n+GI queues in the Halfin-Whitt regime and related asymptotics

In this paper, we develop a stochastic algorithm based on the Euler--Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of $G/Ph/n+GI$ queues in the Halfin-Whitt regime. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein's method for multi-dimensional diffusions, in which the regularity of Stein's equation developed by Gurvich (2014, 2022) plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of $G/Ph/n+GI$ queues and its Euler-Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein's equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role.

math.PR↗
arXiv2022

A probability approximation framework: Markov process approach

We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated Itô's formula and Markov operator. As applications, we study the error bounds of the following three approximations: approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, Euler-Maruyama (EM) discretization for multi-dimensional Ornstein-Uhlenbeck stable process, and multivariate normal approximation. All these error bounds are in Wasserstein-1 distance.

math.PR↗
arXiv2022

Functional large deviations for Stroock's approximation to a class of Gaussian processes with application to small noise diffusions

Letting~$N=\left\{N(t), t\geq0\right\}$ be a standard Poisson process, Stroock~ \cite{Stroock-1981} constructed a family of continuous processes by $$Θ_ε(t)=\int_0^tθ_ε(r)dr, \ \ \ \ \ 0 \le t \le 1,$$ where $θ_ε(r)=\frac{1}ε(-1)^{N(ε^{-2}r)}$, and proved that it weakly converges to a standard Brownian motion under the continuous function topology. We establish the functional large deviations principle (LDP) for the approximations of a class of Gaussian processes constructed by integrals over $Θ_ε(t)$, and find the explicit form for rate function. As an application, we consider the following (non-Markovian) stochastic differential equation \begin{equation*} \begin{aligned} X^ε(t) &=x_{0}+\int^{t}_{0}b(X^ε(s))ds+λ(ε)\int^{t}_{0}σ(X^ε(s))dΘ_ε(s), \end{aligned} \end{equation*} where $b$ and $σ$ are both Lipschitz functions, and establish its Freidlin-Wentzell type LDP as $ε\rightarrow 0$. The rate function indicates a phase transition phenomenon as $λ(ε)$ moves from one region to the other.

math.PR↗
arXiv2022

Large deviations principle for 2D Navier-Stokes equations with space time localised noise

We consider a stochastic 2D Navier-Stokes equation in a bounded domain. The random force is assumed to be non-degenerate and periodic in time, its law has a support localised with respect to both time and space. Slightly strengthening the conditions in the pioneering work about exponential ergodicity by Shirikyan [Shi15], we prove that the stochastic system satisfies Donsker-Varadhan typle large deviations principle. Our proof is based on a criterion of [JNPS15] in which we need to verify uniform irreducibility and uniform Feller property for the related Feynman-Kac semigroup.

math.PR↗
arXiv2022

Large deviations principle via Malliavin calculus for the Navier-Stokes system driven by a degenerate white-in-time noise

The purpose of this paper is to establish the Donsker-Varadhan type large deviations principle (LDP) for the two-dimensional stochastic Navier-Stokes system. The main novelty is that the noise is assumed to be highly degenerate in the Fourier space. The proof is carried out by using a criterion for the LDP developed in arXiv:1410.6188 in a discrete-time setting and extended in arXiv:1505.03686 to the continuous-time. One of the main conditions of that criterion is the uniform Feller property for the Feynman-Kac semigroup, which we verify by using Malliavin calculus.

math.PR↗
arXiv2021

Singular integrals of subordinators with applications to structural properties of SPDEs

We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates. As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution $Z_t$, (2) a small ball probability of $Z_t$, (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs.

math.PR↗
arXiv2021

An approximation to steady-state of M/Ph/n+M queue

In this paper, we develop a stochastic algorithm based on Euler-Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of the $M/Ph/n+M$ queue. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion of the queueing model. Our result also provides an approximation to the steady-state of the diffusion-scaled queueing processes in the Halfin-Whitt regime given the well established interchange of limits property. To establish the error bound, we employ the recently developed Stein's method for multi-dimensional diffusions, in which the regularity of Stein's equation developed by Gurvich \cite{Gur1} plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of the $M/Ph/n+M$ queue and its Euler-Maruyama scheme. In particular, the variance of the CLT of the limiting queue is determined by using Stein's equation and Malliavin calculus.

math.PR↗
arXiv2021

Central limit theorem and Self-normalized Cramér-type moderate deviation for Euler-Maruyama Scheme

We consider a stochastic differential equation and its Euler-Maruyama (EM) scheme, under some appropriate conditions, they both admit a unique invariant measure, denoted by $π$ and $π_η$ respectively ($η$ is the step size of the EM scheme). We construct an empirical measure $Π_η$ of the EM scheme as a statistic of $π_η$, and use Stein's method developed in \citet{FSX19} to prove a central limit theorem of $Π_η$. The proof of the self-normalized Cramér-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chain, splitting $η^{-1/2}(Π_η(.)-π(.))$ into a martingale difference series sum $\mcl H_η$ and a negligible remainder $\mcl R_η$. We handle $\mcl H_η$ by the time-change technique for martingale, while prove that $\mcl R_η$ is exponentially negligible by concentration inequalities, which have their independent interest. Moreover, we show that SNCMD holds for $x = o(η^{-1/6})$, which has the same order as that of the classical result in \citet{shao1999cramer,JSW03}.

math.PR↗
arXiv2021

Scaling limit of a directed polymer among a Poisson field of independent walks

We consider a directed polymer model in dimension $1+1$, where the disorder is given by the occupation field of a Poisson system of independent random walks on $\mathbb Z$. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation (SHE) with a Gaussian noise, whose space-time covariance is given by the heat kernel. In contrast to the case with space-time white noise where the solution of the SHE admits a Wiener-Itô chaos expansion, we establish an $L^1$-convergent chaos expansions of iterated integrals generated by Picard iterations. Using this expansion and its discrete counterpart for the polymer partition functions, the convergence of the terms in the expansion is proved via functional analytic arguments and heat kernel estimates. The Poisson random walk system is amenable to careful moment analysis, which is an important input to our arguments.

math.PR↗
arXiv2020

A generalized Catoni's ${\rm M}$-estimator under finite {$α$-th moment assumption} with $α\in (1,2)$

We generalize the { ${\rm M}$-estimator} put forward by Catoni in his seminal paper [C12] to the case in which samples can have finite $α$-th moment with $α\in (1,2)$ rather than finite variance, our approach is by slightly modifying the influence function $φ$ therein. The choice of the new influence function is inspired by the Taylor-like expansion developed in [C-N-X]. We obtain a deviation bound of the estimator, as $α\rightarrow 2$, this bound is the same as that in [C12]. Experiment shows that our generalized ${\rm M}$-estimator performs better than the empirical mean estimator, the smaller the $α$ is, the better the performance will be. As an application, we study an $\ell_{1}$ regression considered by Zhang et al. [Z-Z] who assumed that samples have finite variance, and relax their assumption to be finite {$α$-th} moment with $α\in (1,2)$.

math.ST↗
arXiv2020

Large Deviations of the Entropy Production Rate for a Class of Gaussian Processes

We prove a large deviation principle (LDP) and a fluctuation theorem (FT) for the entropy production rate (EPR) of the following $d$ dimensional stochastic differential equation \begin{equation*} d X_{t}=AX_{t} d t+\sqrt{Q} d B_{t} \end{equation*} where $A$ is a real normal stable matrix, $Q$ is positive definite, and the matrices $A$ and $Q$ commute. The rate function for the EPR takes the following explicit form: \begin{equation*} I(x)=\left\{ \begin{array}{ll} x\frac{\sqrt{1+\ell_0(x)}-1}{2}+\frac 12\sum\limits_{k=1}^{d} \left(\sqrt{α_k^2- β_k^2\ell_0(x)}+α_k\right) , & x\ge 0 , \\ -x\frac{\sqrt{1+\ell_0(x)}+1}{2} +\frac 12\sum\limits_{k=1}^{d} \left(\sqrt{α_k^2- β_k^2\ell_0(x)}+α_k\right) , &x<0, \end{array} \right. \end{equation*} where $α_{k}\pm {\rm i} β_{k}$ are the eigenvalues of $A$, and $\ell_0(x)$ is the unique solution of the equation: \begin{align*} |x|={\sqrt{1+\ell}} \times \sum_{k=1}^{d} \frac{β_k^2}{\sqrt{α_k^2 -\ellβ_k^2} },\qquad -1 \le \ell< \min_{k=1,...,d}\{\frac{α_k^2}{β_k^2}\}. \end{align*} Simple closed form formulas for rate functions are rare and our work identifies an important class of large deviation problems where such formulas are available. The logarithmic moment generating function (the fluctuation function) $Λ$ associated with the LDP has a closed form (see it in the paper). The functions $Λ(λ)$ and $ I(x)$ satisfy the Cohen-Gallavotti symmetry properties. In particular, the functions $I$ and $Λ$ do not depend on the diffusion matrix $Q$, and are determined completely by the real and imaginary parts of the eigenvalues of $A$. Formally, the deterministic system with $Q=0$ has zero EPR and thus the model exhibits a phase transition in that the EPR changes discontinuously at $Q = 0$.

math.PR↗