arXiv subjects

Lijian Yang

Publications and source records attributed to Lijian Yang.

11 recordsLinked to original sources

arXiv2026

Joint Distributions of Minimum and Maximum Angles on High-Dimensional Spheres

Consider $n$ independent random vectors sampled from uniform distribution on $(p-1)$-dimensional unit sphere. This paper investigates the limiting joint distribution for the minimum and the maximum values of their pairwise angles. It proves that the minimum and the maximum angles are asymptotically independent when both $n$ and $p$ tend to infinity, which solves an open problem raised in Cai, Fan and Jiang (2013) [\emph{Journal of Machine Learning Research} 14, 1837-1864]. Cai, Fan and Jiang (2013) obtained the limiting marginal distributions for both the minimum and the maximum angles under assumption $\lim_{n\to\infty}{\ln n}/p=\beta$ according to whether $\beta=0$, $\beta\in (0,\infty)$, or $\beta=\infty$. This paper presents unified limits for both joint distributions and marginal distributions regardless of the relative divergence rate of $n$ and $p$. The paper also derives the limiting distributions for some statistics based on the minimum and the maximum angles,

math.ST↗

arXiv2026

Fourier analytic variants of the Furstenberg and Kakeya problems

We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given $0<s,t<1$, we call a set $K \subseteq \mathbb{R}^2$ an $(s,t)$-Kakeya set if there exists a set of directions $E \subseteq S^1$ with Hausdorff dimension at least $t$ such that, for each $e \in E$, the set $K$ contains a subset of a unit line segment in direction $e$ whose Fourier dimension, viewed as a subset of $\mathbb{R}$, is at least $s$. For $\Delta(s,t)$ defined to be the infimum of the Fourier dimension among all $(s,t)$-Kakeya sets in $\mathbb{R}^2$, we prove that \[ \frac{2st}{s+2t} \leq \Delta(s,t) \leq \min\{s,2t\}. \] These bounds, though distinct, are asymptotically equivalent as either $s$ or $t$ tends to zero. We also obtain upper and lower bounds in the Furstenberg set version of the problem and in the case where the Hausdorff dimension of the collection of lines is replaced by the Fourier dimension.

math.CA↗

arXiv2025

Hausdorff dimension of restricted Kakeya sets

A Kakeya set in $\mathbb{R}^n$ is a compact set that contains a unit line segment $I_e$ in each direction $e \in S^{n-1}$. The Kakeya conjecture states that any Kakeya set in $\mathbb{R}^n$ has Hausdorff dimension $n$. We consider a restricted case where the midpoint of each line segment $I_e$ must belong to a fixed set $A$ with packing dimension at most $s \in [0, n]$. In this case, we show that the Hausdorff dimension of the Kakeya set is at least $n - s$. Furthermore, using the "bush argument", we improve the lower bound to $\max \{ n - s, n - g_n(s)\}$, where $g_n(s)$ is defined inductively. For example, when $n = 4$, we prove that the Hausdorff dimension is at least $\max\{\frac{19}{5} - \frac{3}{5}s,4-s\}$. We also establish Kakeya maximal function analogues of these results.

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arXiv2025

Generative AI for Film Creation: A Survey of Recent Advances

Generative AI (GenAI) is transforming filmmaking, equipping artists with tools like text-to-image and image-to-video diffusion, neural radiance fields, avatar generation, and 3D synthesis. This paper examines the adoption of these technologies in filmmaking, analyzing workflows from recent AI-driven films to understand how GenAI contributes to character creation, aesthetic styling, and narration. We explore key strategies for maintaining character consistency, achieving stylistic coherence, and ensuring motion continuity. Additionally, we highlight emerging trends such as the growing use of 3D generation and the integration of real footage with AI-generated elements. Beyond technical advancements, we examine how GenAI is enabling new artistic expressions, from generating hard-to-shoot footage to dreamlike diffusion-based morphing effects, abstract visuals, and unworldly objects. We also gather artists' feedback on challenges and desired improvements, including consistency, controllability, fine-grained editing, and motion refinement. Our study provides insights into the evolving intersection of AI and filmmaking, offering a roadmap for researchers and artists navigating this rapidly expanding field.

cs.CV↗

arXiv2024

Rethinking Normalization Strategies and Convolutional Kernels for Multimodal Image Fusion

Multimodal image fusion (MMIF) integrates information from different modalities to obtain a comprehensive image, aiding downstream tasks. However, existing research focuses on complementary information fusion and training strategies, overlooking the critical role of underlying architectural components like normalization and convolution kernels. We reevaluate the UNet architecture for end-to-end MMIF, identifying that widely used batch normalization limits performance by smoothing crucial sparse features. To address this, we propose a hybrid of instance and group normalization to maintain sample independence and reinforce intrinsic feature correlations. Crucially, this strategy facilitates richer feature maps, enabling large kernel convolution to fully leverage its receptive field, enhancing detail preservation. Furthermore, the proposed multi-path adaptive fusion module dynamically calibrates features from varying scales and receptive fields, ensuring effective information transfer. Our method achieves SOTA objective performance on MSRS, M$^3$FD, TNO, and Harvard datasets, producing visually clearer salient objects and lesion areas. Notably, it improves MSRS segmentation mIoU by 8.1\% over the infrared image. This performance stems from a synergistic design of normalization and convolution kernels, which preserves critical sparse features. The code is available at https://github.com/HeDan-11/LKC-FUNet.

cs.CV↗

arXiv2024

Score-based Generative Priors Guided Model-driven Network for MRI Reconstruction

Score matching with Langevin dynamics (SMLD) method has been successfully applied to accelerated MRI. However, the hyperparameters in the sampling process require subtle tuning, otherwise the results can be severely corrupted by hallucination artifacts, especially with out-of-distribution test data. To address the limitations, we proposed a novel workflow where naive SMLD samples serve as additional priors to guide model-driven network training. First, we adopted a pretrained score network to generate samples as preliminary guidance images (PGI), obviating the need for network retraining, parameter tuning and in-distribution test data. Although PGIs are corrupted by hallucination artifacts, we believe they can provide extra information through effective denoising steps to facilitate reconstruction. Therefore, we designed a denoising module (DM) in the second step to coarsely eliminate artifacts and noises from PGIs. The features are extracted from a score-based information extractor (SIE) and a cross-domain information extractor (CIE), which directly map to the noise patterns. Third, we designed a model-driven network guided by denoised PGIs (DGIs) to further recover fine details. DGIs are densely connected with intermediate reconstructions in each cascade to enrich the information and are periodically updated to provide more accurate guidance. Our experiments on different datasets reveal that despite the low average quality of PGIs, the proposed workflow can effectively extract valuable information to guide the network training, even with severely reduced training data and sampling steps. Our method outperforms other cutting-edge techniques by effectively mitigating hallucination artifacts, yielding robust and high-quality reconstruction results.

cs.CV↗

arXiv2021

Multivariate Spline Estimation and Inference for Image-On-Scalar Regression

Motivated by recent data analyses in biomedical imaging studies, we consider a class of image-on-scalar regression models for imaging responses and scalar predictors. We propose using flexible multivariate splines over triangulations to handle the irregular domain of the objects of interest on the images, as well as other characteristics of images. The proposed estimators of the coefficient functions are proved to be root-n consistent and asymptotically normal under some regularity conditions. We also provide a consistent and computationally efficient estimator of the covariance function. Asymptotic pointwise confidence intervals and data-driven simultaneous confidence corridors for the coefficient functions are constructed. Our method can simultaneously estimate and make inferences on the coefficient functions while incorporating spatial heterogeneity and spatial correlation. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method, which is then applied to the spatially normalized positron emission tomography data of the Alzheimer's Disease Neuroimaging Initiative.

stat.ME↗

arXiv2019

Simultaneous Confidence Band for Stationary Covariance Function of Dense Functional Data

Inference via simultaneous confidence band is studied for stationary covariance function of dense functional data. A two-stage estimation procedure is proposed based on spline approximation, the first stage involving estimation of all the individual trajectories and the second stage involving estimation of the covariance function through smoothing the empirical covariance function. The proposed covariance estimator is smooth and as efficient as the oracle estimator when all individual trajectories are known. An asymptotic simultaneous confidence band (SCB) is developed for the true covariance function, and the coverage probabilities are shown to be asymptotically correct. Simulation experiments are conducted on the numerical performance of the proposed estimator and SCB. The proposed method is also illustrated by two real data examples.

stat.ME↗

arXiv2014

Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band

We propose kernel estimator for the distribution function of unobserved errors in autoregressive time series, based on residuals computed by estimating the autoregressive coefficients with the Yule-Walker method. Under mild assumptions, we establish oracle efficiency of the proposed estimator, that is, it is asymptotically as efficient as the kernel estimator of the distribution function based on the unobserved error sequence itself. Applying the result of Wang, Cheng and Yang [J. Nonparametr. Stat. 25 (2013) 395-407], the proposed estimator is also asymptotically indistinguishable from the empirical distribution function based on the unobserved errors. A smooth simultaneous confidence band (SCB) is then constructed based on the proposed smooth distribution estimator and Kolmogorov distribution. Simulation examples support the asymptotic theory.

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arXiv2008

Spline-backfitted kernel smoothing of nonlinear additive autoregression model

Application of nonparametric and semiparametric regression techniques to high-dimensional time series data has been hampered due to the lack of effective tools to address the ``curse of dimensionality.'' Under rather weak conditions, we propose spline-backfitted kernel estimators of the component functions for the nonlinear additive time series data that are both computationally expedient so they are usable for analyzing very high-dimensional time series, and theoretically reliable so inference can be made on the component functions with confidence. Simulation experiments have provided strong evidence that corroborates the asymptotic theory.

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arXiv2007

Spline Single-Index Prediction Model

For the past two decades, single-index model, a special case of projection pursuit regression, has proven to be an efficient way of coping with the high dimensional problem in nonparametric regression. In this paper, based on weakly dependent sample, we investigate the single-index prediction (SIP) model which is robust against deviation from the single-index model. The single-index is identified by the best approximation to the multivariate prediction function of the response variable, regardless of whether the prediction function is a genuine single-index function. A polynomial spline estimator is proposed for the single-index prediction coefficients, and is shown to be root-n consistent and asymptotically normal. An iterative optimization routine is used which is sufficiently fast for the user to analyze large data of high dimension within seconds. Simulation experiments have provided strong evidence that corroborates with the asymptotic theory. Application of the proposed procedure to the rive flow data of Iceland has yielded superior out-of-sample rolling forecasts.

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