arXiv subjects

Zuofeng Shang

Publications and source records attributed to Zuofeng Shang.

At least 19 recordsLinked to original sources

arXiv2026

A Consistent Feature Screening Approach for Tensor Responses with Applications to Genome-Wide Facial Shape Association

As data collecting technologies advance, data structures are getting more and more complex, from single vectors to multi-dimensional tensors. This article is motivated by a variable selection problem to detect important genes from an ultrahigh dimensional pool that are associated with human facial shape variations. We propose a data-driven trimmed feature screening method based on a tensor ridge regression model (TrimTenRidge) through setting thresholds on the tensor coefficients to perform a feature screening procedure. Unlike existing approaches, the TrimTenRidge does not require any sparse structures. In addition, it not only detects important predictors but also locates specific regions/components of the tensor response that are associated with each of the selected predictors. We prove the theoretical selection consistency and also assess its empirical performance through various simulation settings. The approach copes with ultra-high dimensional predictors and tensor responses simultaneously and contributes to the literature from theoretical, methodological, and five applicational aspects. We further apply the TrimTenRidge approach to genome-wide human facial shape data, from which the entire facial shapes form a $2,342\times 7,160\times 3$ tensor, and we successfully detect several novel genetic loci and also confirm some existing findings that are associated to facial shape.

stat.ME↗

arXiv2026

Memory as a Markov Matrix: Sample Efficient Knowledge Expansion via Token-to-Dictionary Mapping

Continual incorporation of new knowledge is essential for the long-term evolution of large language models (LLMs). Existing approaches typically rely on parameter-update algorithms to mitigate catastrophic forgetting, yet they suffer from fundamental limitations: 1) forgetting is unavoidable as the amount of newly injected knowledge grows; and 2) model updates are often irreversible. As modern LLMs become increasingly expressive, it is natural to question whether large-scale weight updates are necessary for acquiring a small amount of new knowledge. In this work, we propose a principled framework that models autoregressive language generation as a Markov process over tokens, where model memory is represented by a Markov transition matrix. Under this formulation, incorporating new knowledge/tokens corresponds to extending the state space, and preserving existing transitions guarantees retention of previously learned knowledge. We then prove a sample complexity bound for incorporating new tokens via a token-to-dictionary mapping strategy. In particular, for learning the transition behavior of each new token, the required number of samples scales linearly with the number of existing tokens it is mapped to. To realize this mapping, we propose an embedding-tuning algorithm that requires minimal parameter updates and induces zero forgetting. Experimental results further demonstrate the effectiveness of our method and validate our theoretical findings.

cs.LG↗

arXiv2025

Matrix Autoregressive Model with Vector Time Series Covariates for Spatio-Temporal Data

We develop a new methodology for forecasting matrix-valued time series with historical matrix data and auxiliary vector time series data. We focus on a time series of matrices defined on a static 2-D spatial grid and an auxiliary time series of non-spatial vectors. The proposed model, Matrix AutoRegression with Auxiliary Covariates (MARAC), contains an autoregressive component for the historical matrix predictors and an additive component that maps the auxiliary vector predictors to a matrix response via tensor-vector product. The autoregressive component adopts a bi-linear transformation framework following Chen et al. (2021), significantly reducing the number of parameters. The auxiliary component posits that the tensor coefficient, which maps non-spatial predictors to a spatial response, contains slices of spatially smooth matrix coefficients that are discrete evaluations of smooth functions on a spatial grid from a Reproducing Kernel Hilbert Space (RKHS). We propose to estimate the model parameters under a penalized maximum likelihood estimation framework coupled with an alternating minimization algorithm. We establish the joint asymptotics of the autoregressive and tensor parameters under fixed and high-dimensional regimes. Extensive simulations and a geophysical application for forecasting the global Total Electron Content (TEC) are conducted to validate the performance of MARAC.

stat.ME↗

arXiv2025

Bootstrap Nonparametric Inference under Data Integration

We propose multiplier bootstrap procedures for nonparametric inference and uncertainty quantification of the target mean function, based on a novel framework of integrating target and source data. We begin with the relatively easier covariate shift scenario with equal target and source mean functions and propose estimation and inferential procedures through a straightforward combination of all target and source datasets. We next consider the more general and flexible distribution shift scenario with arbitrary target and source mean functions, and propose a two-step inferential procedure. First, we estimate the target-to-source differences based on separate portions of the target and source data. Second, the remaining source data are adjusted by these differences and combined with the remaining target data to perform the multiplier bootstrap procedure. Our method enables local and global inference on the target mean function without using asymptotic distributions. To justify our approach, we derive an optimal convergence rate for the nonparametric estimator and establish bootstrap consistency to estimate the asymptotic distribution of the nonparametric estimator. The proof of global bootstrap consistency involves a central limit theorem for quadratic forms with dependent variables under a conditional probability measure. Our method applies to arbitrary source and target datasets, provided that the data sizes meet a specific quantitative relationship. Simulation studies and real data analysis are provided to examine the performance of our approach.

stat.ME↗

arXiv2024

Variational Nonparametric Inference in Functional Stochastic Block Model

We propose a functional stochastic block model whose vertices involve functional data information. This new model extends the classic stochastic block model with vector-valued nodal information, and finds applications in real-world networks whose nodal information could be functional curves. Examples include international trade data in which a network vertex (country) is associated with the annual or quarterly GDP over certain time period, and MyFitnessPal data in which a network vertex (MyFitnessPal user) is associated with daily calorie information measured over certain time period. Two statistical tasks will be jointly executed. First, we will detect community structures of the network vertices assisted by the functional nodal information. Second, we propose computationally efficient variational test to examine the significance of the functional nodal information. We show that the community detection algorithms achieve weak and strong consistency, and the variational test is asymptotically chi-square with diverging degrees of freedom. As a byproduct, we propose pointwise confidence intervals for the slop function of the functional nodal information. Our methods are examined through both simulated and real datasets.

stat.ME↗

arXiv2023

Scalable inference in functional linear regression with streaming data

Traditional static functional data analysis is facing new challenges due to streaming data, where data constantly flow in. A major challenge is that storing such an ever-increasing amount of data in memory is nearly impossible. In addition, existing inferential tools in online learning are mainly developed for finite-dimensional problems, while inference methods for functional data are focused on the batch learning setting. In this paper, we tackle these issues by developing functional stochastic gradient descent algorithms and proposing an online bootstrap resampling procedure to systematically study the inference problem for functional linear regression. In particular, the proposed estimation and inference procedures use only one pass over the data; thus they are easy to implement and suitable to the situation where data arrive in a streaming manner. Furthermore, we establish the convergence rate as well as the asymptotic distribution of the proposed estimator. Meanwhile, the proposed perturbed estimator from the bootstrap procedure is shown to enjoy the same theoretical properties, which provide the theoretical justification for our online inference tool. As far as we know, this is the first inference result on the functional linear regression model with streaming data. Simulation studies are conducted to investigate the finite-sample performance of the proposed procedure. An application is illustrated with the Beijing multi-site air-quality data.

stat.ME↗

arXiv2023

Scalable Statistical Inference in Non-parametric Least Squares

Stochastic approximation (SA) is a powerful and scalable computational method for iteratively estimating the solution of optimization problems in the presence of randomness, particularly well-suited for large-scale and streaming data settings. In this work, we propose a theoretical framework for stochastic approximation (SA) applied to non-parametric least squares in reproducing kernel Hilbert spaces (RKHS), enabling online statistical inference in non-parametric regression models. We achieve this by constructing asymptotically valid pointwise (and simultaneous) confidence intervals (bands) for local (and global) inference of the nonlinear regression function, via employing an online multiplier bootstrap approach to functional stochastic gradient descent (SGD) algorithm in the RKHS. Our main theoretical contributions consist of a unified framework for characterizing the non-asymptotic behavior of the functional SGD estimator and demonstrating the consistency of the multiplier bootstrap method. The proof techniques involve the development of a higher-order expansion of the functional SGD estimator under the supremum norm metric and the Gaussian approximation of suprema of weighted and non-identically distributed empirical processes. Our theory specifically reveals an interesting relationship between the tuning of step sizes in SGD for estimation and the accuracy of uncertainty quantification.

math.ST↗

arXiv2023

Calibrating multi-dimensional complex ODE from noisy data via deep neural networks

Ordinary differential equations (ODEs) are widely used to model complex dynamics that arises in biology, chemistry, engineering, finance, physics, etc. Calibration of a complicated ODE system using noisy data is generally very difficult. In this work, we propose a two-stage nonparametric approach to address this problem. We first extract the de-noised data and their higher order derivatives using boundary kernel method, and then feed them into a sparsely connected deep neural network with ReLU activation function. Our method is able to recover the ODE system without being subject to the curse of dimensionality and complicated ODE structure. When the ODE possesses a general modular structure, with each modular component involving only a few input variables, and the network architecture is properly chosen, our method is proven to be consistent. Theoretical properties are corroborated by an extensive simulation study that demonstrates the validity and effectiveness of the proposed method. Finally, we use our method to simultaneously characterize the growth rate of Covid-19 infection cases from 50 states of the USA.

stat.ML↗

arXiv2023

Nonparametric Inference under B-bits Quantization

Statistical inference based on lossy or incomplete samples is often needed in research areas such as signal/image processing, medical image storage, remote sensing, signal transmission. In this paper, we propose a nonparametric testing procedure based on samples quantized to $B$ bits through a computationally efficient algorithm. Under mild technical conditions, we establish the asymptotic properties of the proposed test statistic and investigate how the testing power changes as $B$ increases. In particular, we show that if $B$ exceeds a certain threshold, the proposed nonparametric testing procedure achieves the classical minimax rate of testing (Shang and Cheng, 2015) for spline models. We further extend our theoretical investigations to a nonparametric linearity test and an adaptive nonparametric test, expanding the applicability of the proposed methods. Extensive simulation studies {together with a real-data analysis} are used to demonstrate the validity and effectiveness of the proposed tests.

math.ST↗

arXiv2023

Empirical likelihood test for community structure in networks

Network data, characterized by interconnected nodes and edges, is pervasive in various domains and has gained significant popularity in recent years. In network data analysis, testing the presence of community structure in a network is one of the important research tasks. Existing tests are mainly developed for unweighted networks. In this paper, we study the problem of testing the existence of community structure in general (either weighted or unweighted) networks. We propose two new tests: the Weighted Signed-Triangle (WST) test and the empirical likelihood (EL) test. Both tests can be applied to weighted or unweighted networks and outperform existing tests for small networks. The EL test may outperform the WST test for small networks.

stat.ME↗

arXiv2023

Distributed Adaptive Nearest Neighbor Classifier: Algorithm and Theory

When data is of an extraordinarily large size or physically stored in different locations, the distributed nearest neighbor (NN) classifier is an attractive tool for classification. We propose a novel distributed adaptive NN classifier for which the number of nearest neighbors is a tuning parameter stochastically chosen by a data-driven criterion. An early stopping rule is proposed when searching for the optimal tuning parameter, which not only speeds up the computation but also improves the finite sample performance of the proposed Algorithm. Convergence rate of excess risk of the distributed adaptive NN classifier is investigated under various sub-sample size compositions. In particular, we show that when the sub-sample sizes are sufficiently large, the proposed classifier achieves the nearly optimal convergence rate. Effectiveness of the proposed approach is demonstrated through simulation studies as well as an empirical application to a real-world dataset.

stat.ML↗

arXiv2023

Minimax optimal high-dimensional classification using deep neural networks

High-dimensional classification is a fundamentally important research problem in high-dimensional data analysis. In this paper, we derive a nonasymptotic rate for the minimax excess misclassification risk when feature dimension exponentially diverges with the sample size and the Bayes classifier possesses a complicated modular structure. We also show that classifiers based on deep neural networks can attain the above rate, hence, are minimax optimal.

math.ST↗

arXiv2023

Statistical Inference with Stochastic Gradient Methods under $\phi$-mixing Data

Stochastic gradient descent (SGD) is a scalable and memory-efficient optimization algorithm for large datasets and stream data, which has drawn a great deal of attention and popularity. The applications of SGD-based estimators to statistical inference such as interval estimation have also achieved great success. However, most of the related works are based on i.i.d. observations or Markov chains. When the observations come from a mixing time series, how to conduct valid statistical inference remains unexplored. As a matter of fact, the general correlation among observations imposes a challenge on interval estimation. Most existing methods may ignore this correlation and lead to invalid confidence intervals. In this paper, we propose a mini-batch SGD estimator for statistical inference when the data is $\phi$-mixing. The confidence intervals are constructed using an associated mini-batch bootstrap SGD procedure. Using ``independent block'' trick from \cite{yu1994rates}, we show that the proposed estimator is asymptotically normal, and its limiting distribution can be effectively approximated by the bootstrap procedure. The proposed method is memory-efficient and easy to implement in practice. Simulation studies on synthetic data and an application to a real-world dataset confirm our theory.

stat.ME↗

arXiv2022

Solar Flare Index Prediction Using SDO/HMI Vector Magnetic Data Products with Statistical and Machine Learning Methods

Solar flares, especially the M- and X-class flares, are often associated with coronal mass ejections (CMEs). They are the most important sources of space weather effects, that can severely impact the near-Earth environment. Thus it is essential to forecast flares (especially the M-and X-class ones) to mitigate their destructive and hazardous consequences. Here, we introduce several statistical and Machine Learning approaches to the prediction of the AR's Flare Index (FI) that quantifies the flare productivity of an AR by taking into account the numbers of different class flares within a certain time interval. Specifically, our sample includes 563 ARs appeared on solar disk from May 2010 to Dec 2017. The 25 magnetic parameters, provided by the Space-weather HMI Active Region Patches (SHARP) from Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO), characterize coronal magnetic energy stored in ARs by proxy and are used as the predictors. We investigate the relationship between these SHARP parameters and the FI of ARs with a machine-learning algorithm (spline regression) and the resampling method (Synthetic Minority Over-Sampling Technique for Regression with Gaussian Noise, short by SMOGN). Based on the established relationship, we are able to predict the value of FIs for a given AR within the next 1-day period. Compared with other 4 popular machine learning algorithms, our methods improve the accuracy of FI prediction, especially for large FI. In addition, we sort the importance of SHARP parameters by Borda Count method calculated from the ranks that are rendered by 9 different machine learning methods.

astro-ph.SR↗

arXiv2022

Minimax Optimal Deep Neural Network Classifiers Under Smooth Decision Boundary

Deep learning has gained huge empirical successes in large-scale classification problems. In contrast, there is a lack of statistical understanding about deep learning methods, particularly in the minimax optimality perspective. For instance, in the classical smooth decision boundary setting, existing deep neural network (DNN) approaches are rate-suboptimal, and it remains elusive how to construct minimax optimal DNN classifiers. Moreover, it is interesting to explore whether DNN classifiers can circumvent the curse of dimensionality in handling high-dimensional data. The contributions of this paper are two-fold. First, based on a localized margin framework, we discover the source of suboptimality of existing DNN approaches. Motivated by this, we propose a new deep learning classifier using a divide-and-conquer technique: DNN classifiers are constructed on each local region and then aggregated to a global one. We further propose a localized version of the classical Tsybakov's noise condition, under which statistical optimality of our new classifier is established. Second, we show that DNN classifiers can adapt to low-dimensional data structures and circumvent the curse of dimensionality in the sense that the minimax rate only depends on the effective dimension, potentially much smaller than the actual data dimension. Numerical experiments are conducted on simulated data to corroborate our theoretical results.

math.ST↗

arXiv2022

Deep Neural Network Classifier for Multi-dimensional Functional Data

We propose a new approach, called as functional deep neural network (FDNN), for classifying multi-dimensional functional data. Specifically, a deep neural network is trained based on the principle components of the training data which shall be used to predict the class label of a future data function. Unlike the popular functional discriminant analysis approaches which rely on Gaussian assumption, the proposed FDNN approach applies to general non-Gaussian multi-dimensional functional data. Moreover, when the log density ratio possesses a locally connected functional modular structure, we show that FDNN achieves minimax optimality. The superiority of our approach is demonstrated through both simulated and real-world datasets.

stat.ML↗

arXiv2022

Information-theoretic Limits for Testing Community Structures in Weighted Networks

Community detection refers to the problem of clustering the nodes of a network into groups. Existing inferential methods for community structure mainly focus on unweighted (binary) networks. Many real-world networks are nonetheless weighted and a common practice is to dichotomize a weighted network to an unweighted one which is known to result in information loss. Literature on hypothesis testing in the latter situation is still missing. In this paper, we study the problem of testing the existence of community structure in weighted networks. Our contributions are threefold: (a). We use the (possibly infinite-dimensional) exponential family to model the weights and derive the sharp information-theoretic limit for the existence of consistent test. Within the limit, any test is inconsistent; and beyond the limit, we propose a useful consistent test. (b). Based on the information-theoretic limits, we provide the first formal way to quantify the loss of information incurred by dichotomizing weighted graphs into unweighted graphs in the context of hypothesis testing. (c). We propose several new and practically useful test statistics. Simulation study show that the proposed tests have good performance. Finally, we apply the proposed tests to an animal social network.

math.ST↗

arXiv2022

Online Statistical Inference for Parameters Estimation with Linear-Equality Constraints

Stochastic gradient descent (SGD) and projected stochastic gradient descent (PSGD) are scalable algorithms to compute model parameters in unconstrained and constrained optimization problems. In comparison with SGD, PSGD forces its iterative values into the constrained parameter space via projection. From a statistical point of view, this paper studies the limiting distribution of PSGD-based estimate when the true parameters satisfy some linear-equality constraints. Our theoretical findings reveal the role of projection played in the uncertainty of the PSGD-based estimate. As a byproduct, we propose an online hypothesis testing procedure to test the linear-equality constraints. Simulation studies on synthetic data and an application to a real-world dataset confirm our theory.

stat.ML↗