arXiv subjects

Haijuan Hu

Publications and source records attributed to Haijuan Hu.

9 recordsLinked to original sources

arXiv2023

Comparison on the criticality parameters for two supercritical branching processes in random environments

Let $\{Z_{1,n} , n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$ be two supercritical branching processes in different random environments, with criticality parameters $μ_1$ and $μ_2$ respectively. It is known that $\frac{1}{n} \ln Z_{1,n} \rightarrow μ_1$ and $\frac{1}{m} \ln Z_{2,m} \rightarrow μ_2$ in probability as $m, n \rightarrow \infty.$ In this paper, we are interested in the comparison on the two criticality parameters. To this end, we prove a non-uniform Berry-Esseen's bound and Cramér's moderate deviations for $\frac{1}{n} \ln Z_{1,n} - \frac{1}{m} \ln Z_{2,m}$ as $m, n \rightarrow \infty.$ An application is also given for constructing confidence interval for $μ_1-μ_2$.

math.PR↗

arXiv2023

Cramér-type moderate deviations for Euler-Maruyama scheme for SDE

In this paper, we establish normalized and self-normalized Cramér-type moderate deviations for Euler-Maruyama scheme for SDE. As a consequence of our results, Berry-Esseen's bounds and moderate deviation principles are also obtained. Our normalized Cramér-type moderate deviations refines the recent work of [Lu, J., Tan, Y., Xu, L., 2022. Central limit theorem and self-normalized Cramér-type moderate deviation for Euler-Maruyama scheme. Bernoulli 28(2): 937--964].

math.PR↗

arXiv2023

Rates of convergence in the central limit theorem for the elephant random walk with random step sizes

In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically distributed random variables instead of a fixed constant. For this model, we establish the law of the iterated logarithm, the central limit theorem, and we obtain rates of convergence in the central limit theorem with respect to the Kologmorov, Zolotarev and Wasserstein distances. We emphasize that, even in case of the usual elephant random walk, our results concerning the rates of convergence in the central limit theorem are new.

math.PR↗

arXiv2022

Low rank prior and l0 norm to remove impulse noise in images

Patch-based low rank is an important prior assumption for image processing. Moreover, according to our calculation, the optimization of l0 norm corresponds to the maximum likelihood estimation under random-valued impulse noise. In this article, we thus combine exact rank and l0 norm for removing the noise. It is solved formally using the alternating direction method of multipliers (ADMM), with our previous patch-based weighted filter (PWMF) producing initial images. Since this model is not convex, we consider it as a Plug-and-Play ADMM, and do not discuss theoretical convergence properties. Experiments show that this method has very good performance, especially for weak or medium contrast images.

cs.CV↗

arXiv2021

Cramér moderate deviations for the elephant random walk

We establish some limit theorems for one-dimensional elephant random walk, including Berry-Esseen bounds, Cramér moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorems for the elephant random walk. Moreover, by these limit theorems, we conclude that the domain of attraction of normal distribution mainly depends on a memory parameter $p$ which lies between $0$ and $3/4.$

math.PR↗

arXiv2020

Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment

Let $\{Z_n, n\geq 0\}$ be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for $\ln (Z_{n+n_0}/Z_{n_0})$ % under the annealed law, uniformly in $n_0 \in \mathbb{N}$, which extend the corresponding results by Grama et al. (Stochastic Process.\ Appl. 2017) established for $n_0=0$. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. (2017) are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of $\ln(Z_{n+n_0}/Z_{n_0})$ and $n$.

math.PR↗

arXiv2019

Spatial-Frequency Domain Nonlocal Total Variation for Image Denoising

Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a large number of denoising methods mixing these two approaches, starting with the nonlocal total variation (NLTV) model. The present article proposes an analysis of the NLTV model for image denoising as well as a number of improvements, the most important of which being to apply the denoising both in the space domain and in the Fourier domain, in order to exploit the complementarity of the representation of image data in both domains. A local version obtained by a regionwise implementation followed by an aggregation process, called Local Spatial-Frequency NLTV (L- SFNLTV) model, is finally proposed as a new reference algorithm for image denoising among the family of approaches mixing TV and NL operators. The experiments show the great performance of L-SFNLTV, both in terms of image quality and of computational speed, comparing with other recently proposed NLTV-related methods.

eess.IV↗

arXiv2018

A note on patch-based low-rank minimization for fast image denoising

Patch-based low-rank minimization for image processing attracts much attention in recent years. The minimization of the matrix rank coupled with the Frobenius norm data fidelity can be solved by the hard thresholding filter with principle component analysis (PCA) or singular value decomposition (SVD). Based on this idea, we propose a patch-based low-rank minimization method for image denoising. The main denoising process is stated in three equivalent way: PCA, SVD and low-rank minimization. Compared to recent patch-based sparse representation methods, experiments demonstrate that the proposed method is rather rapid, and it is effective for a variety of natural grayscale images and color images, especially for texture parts in images. Further improvements of this method are also given. In addition, due to the simplicity of this method, we could provide an explanation of the choice of the threshold parameter, estimation of PSNR values, and give other insights into this method.

cs.CV↗

arXiv2014

Removing Mixture of Gaussian and Impulse Noise by Patch-Based Weighted Means

We first establish a law of large numbers and a convergence theorem in distribution to show the rate of convergence of the non-local means filter for removing Gaussian noise. We then introduce the notion of degree of similarity to measure the role of similarity for the non-local means filter. Based on the convergence theorems, we propose a patch-based weighted means filter for removing impulse noise and its mixture with Gaussian noise by combining the essential idea of the trilateral filter and that of the non-local means filter. Our experiments show that our filter is competitive compared to recently proposed methods.

cs.CV↗