arXiv subjects

Zhenyuan Liu

Publications and source records attributed to Zhenyuan Liu.

14 recordsLinked to original sources

arXiv2026

Liouville Type Theorem for the Steady Fractional Compressible MHD Equations in $\mathbb{R}^{3}$

This paper is concerned with the Liouville-type problem for the stationary fractional compressible magnetohydrodynamics (MHD) equations. The main difficulty comes from the nonlocal fractional Laplace operator $(-Δ)^s$. To overcome it, we combine the Caffarelli-Silvestre extension technique with truncation arguments. Under suitable regularity and decay conditions, we prove that the only solution is trivial, establishing a Liouville-type theorem.

math.AP↗

arXiv2026

Liouville Type Theorem for the Fractional MHD and Hall-MHD equations in $\mathbb{R}^{3}

In this paper, we are mainly concerned with the Liouville type problem for the stationary fractional magnetohydrodynamics(MHD) and stationary fractional Hall-MHD equations. In addition, we present the results of the Navier-Stokes equation as a byproduct. The key point is to use the Caffarelli-Sivestre extension to overcome the difficulty caused by the non-local operator $(-\triangle)^{s}$ and combined with Yuan and Xiao's method (J. Math. Anal. Appl. 491 (2020) 124343).

math.AP↗

arXiv2024

BiGS: Bidirectional Gaussian Primitives for Relightable 3D Gaussian Splatting

We present Bidirectional Gaussian Primitives, an image-based novel view synthesis technique designed to represent and render 3D objects with surface and volumetric materials under dynamic illumination. Our approach integrates light intrinsic decomposition into the Gaussian splatting framework, enabling real-time relighting of 3D objects. To unify surface and volumetric material within a cohesive appearance model, we adopt a light- and view-dependent scattering representation via bidirectional spherical harmonics. Our model does not use a specific surface normal-related reflectance function, making it more compatible with volumetric representations like Gaussian splatting, where the normals are undefined. We demonstrate our method by reconstructing and rendering objects with complex materials. Using One-Light-At-a-Time (OLAT) data as input, we can reproduce photorealistic appearances under novel lighting conditions in real time.

cs.CV↗

arXiv2024

Shape-Constrained Distributional Optimization via Importance-Weighted Sample Average Approximation

Shape-constrained optimization arises in a wide range of problems including distributionally robust optimization (DRO) that has surging popularity in recent years. In the DRO literature, these problems are usually solved via reduction into moment-constrained problems using the Choquet representation. While powerful, such an approach could face tractability challenges arising from the geometries and the compatibility between the shape and the objective function and moment constraints. In this paper, we propose an alternative methodology to solve shape-constrained optimization problems by integrating sample average approximation with importance sampling, the latter used to convert the distributional optimization into an optimization problem over the likelihood ratio with respect to a sampling distribution. We demonstrate how our approach, which relies on finite-dimensional linear programs, can handle a range of shape-constrained problems beyond the reach of previous Choquet-based reformulations, and entails vanishing and quantifiable optimality gaps. Moreover, our theoretical analyses based on strong duality and empirical processes reveal the critical role of shape constraints in guaranteeing desirable consistency and convergence rates.

math.OC↗

arXiv2024

Quantifying Distributional Input Uncertainty via Inflated Kolmogorov-Smirnov Confidence Band

In stochastic simulation, input uncertainty refers to the propagation of the statistical noise in calibrating input models to impact output accuracy, in addition to the Monte Carlo simulation noise. The vast majority of the input uncertainty literature focuses on estimating target output quantities that are real-valued. However, outputs of simulation models are random and real-valued targets essentially serve only as summary statistics. To provide a more holistic assessment, we study the input uncertainty problem from a distributional view, namely we construct confidence bands for the entire output distribution function. Our approach utilizes a novel test statistic whose asymptotic consists of the supremum of the sum of a Brownian bridge and a suitable mean-zero Gaussian process, which generalizes the Kolmogorov-Smirnov statistic to account for input uncertainty. Regarding implementation, we also demonstrate how to use subsampling to efficiently estimate the covariance function of the Gaussian process, thereby leading to an implementable estimation of the quantile of the test statistic and a statistically valid confidence band. Numerical results demonstrate how our new confidence bands provide valid coverage for output distributions under input uncertainty that is not achievable by conventional approaches.

stat.ME↗

arXiv2023

Propagation of Input Tail Uncertainty in Rare-Event Estimation: A Light versus Heavy Tail Dichotomy

We consider the estimation of small probabilities or other risk quantities associated with rare but catastrophic events. In the model-based literature, much of the focus has been devoted to efficient Monte Carlo computation or analytical approximation assuming the model is accurately specified. In this paper, we study a distinct direction on the propagation of model uncertainty and how it impacts the reliability of rare-event estimates. Specifically, we consider the basic setup of the exceedance of i.i.d. sum, and investigate how the lack of tail information of each input summand can affect the output probability. We argue that heavy-tailed problems are much more vulnerable to input uncertainty than light-tailed problems, reasoned through their large deviations behaviors and numerical evidence. We also investigate some approaches to quantify model errors in this problem using a combination of the bootstrap and extreme value theory, showing some positive outcomes but also uncovering some statistical challenges.

math.ST↗

arXiv2023

Smoothed $f$-Divergence Distributionally Robust Optimization

In data-driven optimization, sample average approximation (SAA) is known to suffer from the so-called optimizer's curse that causes an over-optimistic evaluation of the solution performance. We argue that a special type of distributionallly robust optimization (DRO) formulation offers theoretical advantages in correcting for this optimizer's curse compared to simple ``margin'' adjustments to SAA and other DRO approaches: It attains a statistical bound on the out-of-sample performance, for a wide class of objective functions and distributions, that is nearly tightest in terms of exponential decay rate. This DRO uses an ambiguity set based on a Kullback Leibler (KL) divergence smoothed by the Wasserstein or Lévy-Prokhorov (LP) distance via a suitable distance optimization. Computationally, we also show that such a DRO, and its generalized versions using smoothed $f$-divergence, are not harder than DRO problems based on $f$-divergence or Wasserstein distances, rendering our DRO formulations both statistically optimal and computationally viable.

math.OC↗

arXiv2023

Hierarchical-level rain image generative model based on GAN

Autonomous vehicles are exposed to various weather during operation, which is likely to trigger the performance limitations of the perception system, leading to the safety of the intended functionality (SOTIF) problems. To efficiently generate data for testing the performance of visual perception algorithms under various weather conditions, a hierarchical-level rain image generative model, rain conditional CycleGAN (RCCycleGAN), is constructed. RCCycleGAN is based on the generative adversarial network (GAN) and can generate images of light, medium, and heavy rain. Different rain intensities are introduced as labels in conditional GAN (CGAN). Meanwhile, the model structure is optimized and the training strategy is adjusted to alleviate the problem of mode collapse. In addition, natural rain images of different intensities are collected and processed for model training and validation. Compared with the two baseline models, CycleGAN and DerainCycleGAN, the peak signal-to-noise ratio (PSNR) of RCCycleGAN on the test dataset is improved by 2.58 dB and 0.74 dB, and the structural similarity (SSIM) is improved by 18% and 8%, respectively. The ablation experiments are also carried out to validate the effectiveness of the model tuning.

cs.CV↗

arXiv2023

Bootstrap in High Dimension with Low Computation

The bootstrap is a popular data-driven method to quantify statistical uncertainty, but for modern high-dimensional problems, it could suffer from huge computational costs due to the need to repeatedly generate resamples and refit models. We study the use of bootstraps in high-dimensional environments with a small number of resamples. In particular, we show that with a recent "cheap" bootstrap perspective, using a number of resamples as small as one could attain valid coverage even when the dimension grows closely with the sample size, thus strongly supporting the implementability of the bootstrap for large-scale problems. We validate our theoretical results and compare the performance of our approach with other benchmarks via a range of experiments.

stat.ME↗

arXiv2022

Gate-Level Side-Channel Leakage Assessment with Architecture Correlation Analysis

While side-channel leakage is traditionally evaluated from a fabricated chip, it is more time-efficient and cost-effective to do so during the design phase of the chip. We present a methodology to rank the gates of a design according to their contribution to the side-channel leakage of the chip. The methodology relies on logic synthesis, logic simulation, gate-level power estimation, and gate leakage assessment to compute a ranking. The ranking metric can be defined as a specific test by correlating gate-level activity with a leakage model, or else as a non-specific test by evaluating gate-level activity in response to distinct test vector groups. Our results show that only a minority of the gates in a design contribute most of the side-channel leakage. We demonstrate this property for several designs, including a hardware AES coprocessor and a cryptographic hardware/software interface in a five-stage pipelined RISC processor.

cs.CR↗

arXiv2022

Leverage the Average: Averaged Sampling in Pre-Silicon Side-Channel Leakage Assessment

Pre-silicon side-channel leakage assessment is a useful tool to identify hardware vulnerabilities at design time, but it requires many high-resolution power traces and increases the power simulation cost of the design. By downsampling and averaging these high-resolution traces, we show that the power simulation cost can be considerably reduced without significant loss of side-channel leakage assessment quality. We introduce a theoretical basis for our claims. Our results demonstrate up to 6.5-fold power-simulation speed improvement on a gate-level side-channel leakage assessment of a RISC-V SoC. Furthermore, we clarify the conditions under which the averaged sampling technique can be successfully used.

cs.AR↗

arXiv2022

Test Against High-Dimensional Uncertainties: Accelerated Evaluation of Autonomous Vehicles with Deep Importance Sampling

Evaluating the performance of autonomous vehicles (AV) and their complex subsystems to high precision under naturalistic circumstances remains a challenge, especially when failure or dangerous cases are rare. Rarity does not only require an enormous sample size for a naive method to achieve high confidence estimation, but it also causes dangerous underestimation of the true failure rate and it is extremely hard to detect. Meanwhile, the state-of-the-art approach that comes with a correctness guarantee can only compute an upper bound for the failure rate under certain conditions, which could limit its practical uses. In this work, we present Deep Importance Sampling (Deep IS) framework that utilizes a deep neural network to obtain an efficient IS that is on par with the state-of-the-art, capable of reducing the required sample size 43 times smaller than the naive sampling method to achieve 10% relative error and while producing an estimate that is much less conservative. Our high-dimensional experiment estimating the misclassification rate of one of the state-of-the-art traffic sign classifiers further reveals that this efficiency still holds true even when the target is very small, achieving over 600 times efficiency boost. This highlights the potential of Deep IS in providing a precise estimate even against high-dimensional uncertainties.

cs.LG↗

arXiv2021

Orthounimodal Distributionally Robust Optimization: Representation, Computation and Multivariate Extreme Event Applications

This paper studies a basic notion of distributional shape known as orthounimodality (OU) and its use in shape-constrained distributionally robust optimization (DRO). As a key motivation, we argue how such type of DRO is well-suited to tackle multivariate extreme event estimation by giving statistically valid confidence bounds on target extremal probabilities. In particular, we explain how DRO can be used as a nonparametric alternative to conventional extreme value theory that extrapolates tails based on theoretical limiting distributions, which could face challenges in bias-variance control and other technical complications. We also explain how OU resolves the challenges in interpretability and robustness faced by existing distributional shape notions used in the DRO literature. Methodologically, we characterize the extreme points of the OU distribution class in terms of what we call OU sets and build a corresponding Choquet representation, which subsequently allows us to reduce OU-DRO into moment problems over infinite-dimensional random variables. We then develop, in the bivariate setting, a geometric approach to reduce such moment problems into finite dimension via a specially constructed variational problem designed to eliminate suboptimal solutions. Numerical results illustrate how our approach gives rise to valid and competitive confidence bounds for extremal probabilities.

math.OC↗

arXiv2018

Diagnostics of Data-Driven Models: Uncertainty Quantification of PM7 Semi-Empirical Quantum Chemical Method

We report an evaluation of a semi-empirical quantum chemical method PM7 from the perspective of uncertainty quantification. Specifically, we apply Bound-to-Bound Data Collaboration, an uncertainty quantification framework, to characterize a) variability of PM7 model parameter values consistent with the uncertainty in the training data, and b) uncertainty propagation from the training data to the model predictions. Experimental heats of formation of a homologous series of linear alkanes are used as the property of interest. The training data are chemically accurate, i.e., they have very low uncertainty by the standards of computational chemistry. The analysis does not find evidence of PM7 consistency with the entire data set considered as no single set of parameter values is found that captures the experimental uncertainties of all training data. Nevertheless, PM7 is found to be consistent for subsets of the training data. In such cases, uncertainty propagation from the chemically accurate training data to the predicted values preserves error within bounds of chemical accuracy if predictions are made for the molecules of comparable size. Otherwise, the error grows linearly with the relative size of the molecules.

physics.chem-ph↗