arXiv subjects

Huijuan Zhou

Publications and source records attributed to Huijuan Zhou.

10 recordsLinked to original sources

arXiv2026

A Modern Theory for High-dimensional Cox Regression Models

The proportional hazards model has been extensively used in many fields such as biomedicine to estimate and perform statistical significance testing on the effects of covariates influencing the survival time of patients. The classical theory of maximum partial-likelihood estimation (MPLE) is used by most software packages to produce inference, e.g., the coxph function in R and the PHREG procedure in SAS. In this paper, we investigate the asymptotic behavior of the MPLE in the regime in which the number of parameters p is of the same order as the number of samples n. The main results are (i) existence of the MPLE undergoes a sharp 'phase transition'; (ii) the classical MPLE theory leads to invalid inference in the high-dimensional regime. We show that the asymptotic behavior of the MPLE is governed by a new asymptotic theory. These findings are further corroborated through numerical studies. The main technical tool in our proofs is the Convex Gaussian Min-max Theorem (CGMT), which has not been previously used in the analysis of partial likelihood. Our results thus extend the scope of CGMT and shed new light on the use of CGMT for examining the existence of MPLE and non-separable objective functions.

math.ST↗

arXiv2024

Generalization Bounds and Model Complexity for Kolmogorov-Arnold Networks

Kolmogorov-Arnold Network (KAN) is a network structure recently proposed by Liu et al. (2024) that offers improved interpretability and a more parsimonious design in many science-oriented tasks compared to multi-layer perceptrons. This work provides a rigorous theoretical analysis of KAN by establishing generalization bounds for KAN equipped with activation functions that are either represented by linear combinations of basis functions or lying in a low-rank Reproducing Kernel Hilbert Space (RKHS). In the first case, the generalization bound accommodates various choices of basis functions in forming the activation functions in each layer of KAN and is adapted to different operator norms at each layer. For a particular choice of operator norms, the bound scales with the $l_1$ norm of the coefficient matrices and the Lipschitz constants for the activation functions, and it has no dependence on combinatorial parameters (e.g., number of nodes) outside of logarithmic factors. Moreover, our result does not require the boundedness assumption on the loss function and, hence, is applicable to a general class of regression-type loss functions. In the low-rank case, the generalization bound scales polynomially with the underlying ranks as well as the Lipschitz constants of the activation functions in each layer. These bounds are empirically investigated for KANs trained with stochastic gradient descent on simulated and real data sets. The numerical results demonstrate the practical relevance of these bounds.

cs.LG↗

arXiv2024

Subsampling-based Tests in Mediation Analysis

Testing for mediation effect poses a challenge since the null hypothesis (i.e., the absence of mediation effects) is composite, making most existing mediation tests quite conservative and often underpowered. In this work, we propose a subsampling-based procedure to construct a test statistic whose asymptotic null distribution is pivotal and remains the same regardless of the three null cases encountered in mediation analysis. The method, when combined with the popular Sobel test, leads to an accurate size control under the null. We further introduce a Cauchy combination test to construct p-values from different subsample splits, which reduces variability in the testing results and increases detection power. Through numerical studies, our approach has demonstrated a more accurate size and higher detection power than the competing classical and contemporary methods.

stat.ME↗

arXiv2022

Data-driven forward-inverse problems for the variable coefficients Hirota equation using deep learning method

Data-driven forward-inverse problems for the variable coefficients Hirota (VCH) equation are discussed in this paper. The main idea is to use the improved physics-informed neural networks (IPINN) algorithm with neuron-wise locally adaptive activation function, slope recovery term and parameter regularization to recover the data-driven solitons and high-order soliton of the VCH equation with initial-boundary conditions, as well as the data-driven parameters discovery for VCH equation with unknown parameters under noise of different intensity. Numerical results are shown to demonstrate two facts: (i) data-driven soliton solutions of the VCH equation are successfully learned by adjusting the network layers, neurons, the original training data, spatiotemporal regions and other parameters of the IPINN algorithm; (ii) the prediction parameter can be trained stably and accurately by introducing a parameter regularization strategy with an appropriate weight coefficients into the IPINN algorithm. The results achieved in this work verify the effectiveness of the IPINN algorithm in solving the forward-inverse problems of the variable coefficients equation.

nlin.PS↗

arXiv2022

High-order soliton solutions and their dynamics in the inhomogeneous variable coefficients Hirota equation

A series of new soliton solutions are presented for the inhomogeneous variable coefficient Hirota equation by using the Riemann Hilbert method and transformation relationship. First, through a standard dressing procedure, the N-soliton matrix associated with the simple zeros in the Riemann Hilbert problem for the Hirota equation is constructed. Then the N-soliton matrix of the inhomogeneous variable coefficient Hirota equation can be obtained by a special transformation relationship from the N-soliton matrix of the Hirota equation. Next, using the generalized Darboux transformation, the high-order soliton solutions corresponding to the elementary high-order zeros in the Riemann Hilbert problem for the Hirota equation can be derived. Similarly, employing the transformation relationship mentioned above can lead to the high-order soliton solutions of the inhomogeneous variable coefficient Hirota equation. In addition, the collision dynamics of Hirota and inhomogeneous variable coefficient Hirota equations are analyzed; the asymptotic behaviors for multi-solitons and long-term asymptotic estimates for the high-order one-soliton of the Hirota equation are concretely calculated. Most notably, by analyzing the dynamics of the multi-solitons and high-order solitons of the inhomogeneous variable coefficient Hirota equation, we discover numerous new waveforms such as heart-shaped periodic wave solutions, O-shaped periodic wave solutions etc. that have never been reported before, which are crucial in theory and practice.

nlin.SI↗

arXiv2022

LinDA: linear models for differential abundance analysis of microbiome compositional data

Differential abundance analysis is at the core of statistical analysis of microbiome data. The compositional nature of microbiome sequencing data makes false positive control challenging. Here, we show that the compositional effects can be addressed by a simple, yet highly flexible and scalable, approach. The proposed method, LinDA, only requires fitting linear regression models on the centered log-ratio transformed data, and correcting the bias due to compositional effects. We show that LinDA enjoys asymptotic FDR control and can be extended to mixed-effect models for correlated microbiome data. Using simulations and real examples, we demonstrate the effectiveness of LinDA.

stat.ME↗

arXiv2021

Complex excitations for the derivative nonlinear Schrödinger equation

The Darboux transformation (DT) formulae for the derivative nonlinear Schrödinger (DNLS) equation are expressed in concise forms, from which the multi-solitons, n-periodic solutions, higher-order hybrid-pattern solitons and some mixed solutions are obtained. These complex excitations can be constructed thanks to more general semi-degenerate DTs. Even the non-degenerate N-fold DT with a zero seed can generate complicated n-periodic solutions. It is proved that the solution q[N] at the origin depends only on the summation of the spectral parameters. We find the maximum amplitudes of several classes of the wave solutions are determined by the summation. Many interesting phenomena are discovered from these new solutions. For instance, the interactions between n-periodic waves produce peaks with different amplitudes and sizes; A soliton on a single periodic wave background shares a similar feature as a breather due to the interference of the periodic background. In addition, the results are extended to the reverse-space-time DNLS equation.

nlin.SI↗

arXiv2020

Covariate Adaptive Family-wise Error Rate Control for Genome-Wide Association Studies

The family-wise error rate (FWER) has been widely used in genome-wide association studies. With the increasing availability of functional genomics data, it is possible to increase the detection power by leveraging these genomic functional annotations. Previous efforts to accommodate covariates in multiple testing focus on the false discovery rate control while covariate-adaptive FWER-controlling procedures remain under-developed. Here we propose a novel covariate-adaptive FWER-controlling procedure that incorporates external covariates which are potentially informative of either the statistical power or the prior null probability. An efficient algorithm is developed to implement the proposed method. We prove its asymptotic validity and obtain the rate of convergence through a perturbation-type argument. Our numerical studies show that the new procedure is more powerful than competing methods and maintains robustness across different settings. We apply the proposed approach to the UK Biobank data and analyze 27 traits with 9 million single-nucleotide polymorphisms tested for associations. Seventy-five genomic annotations are used as covariates. Our approach detects more genome-wide significant loci than other methods in 21 out of the 27 traits.

stat.ME↗

arXiv2020

High-order soliton matrix for an extended nonlinear Schrödinger equation

The extended nonlinear Schrödinger (ENLS) equation with third-order term and fourth-order term which describes the wave propagation in the optical fibers is more accurate than the NLS equation. A study of high-order soliton matrix is presented for an ENLS equation in the framework of the Riemann-Hilbert problem (RHP). Through a standard dressing procedure and the generalized Darboux transformation (gDT), soliton matrix for simple zeros and elementary high-order zeros in the RHP for the ENLS equation are constructed. Then the N-soliton solutions and high-order soliton solutions for the ENLS equation can be determined. Moreover, collision dynamics along with the asymptotic behavior for the two-solitons and long-time asymptotic estimations for the high-order one-soliton are concretely analyzed. For the given spectral parameters, we can control the propagation direction, velocity, width and other physical quantities of solitons by adjusting the free parameters of ENLS equation.

nlin.SI↗

arXiv2017

Rogue waves of the Frobenius nonlinear Schrödinger equation

In this paper, by considering the potential application in two mode nonlinear waves in nonlinear fibers under a certain case, we define a coupled nonlinear Schrödinger equation(called Frobenius NLS equation) including its Lax pair. Afterwards, we construct the Darboux transformations of the Frobenius NLS equation. New solutions can be obtained from known seed solutions by the Dardoux transformations. Soliton solutions are generated from trivial seed solutions. Also we derive breather solutions $q,r$ of the Frobenius NLS equation obtained from periodic seed solutions. Interesting enough, we find the amplitudes of $r$ vary in size in different areas with period-like fluctuations in the background. This is very different from the solution $q$ of the single-component classical nonlinear Schrödinger equation. Then, the rogue waves of the Frobenius NLS equation are given explicitly by a Taylor series expansion about the breather solutions $q,r$. Also the graph of rogue wave solution $r$ shows that the rogue wave has fluctuations around the peak. The reason of this phenomenon should be in the dynamical dependence of $r$ on $q$ which is independent on $r$.

nlin.SI↗