Jinchi Lv

Jianqing Fan, Yingying Fan, Jinchi Lv et al.

Large-scale network inference with uncertainty quantification has important applications in natural, social, and medical sciences. The recent work of Fan, Fan, Han and Lv (2022) introduced a general framework of statistical inference on membership profiles in large networks (SIMPLE) for testing the sharp null hypothesis that a pair of given nodes share the same membership profiles. In real applications, there are often groups of nodes under investigation that may share similar membership profiles at the presence of relatively weaker signals than the setting considered in SIMPLE. To address these practical challenges, in this paper we propose a SIMPLE method with random coupling (SIMPLE-RC) for testing the non-sharp null hypothesis that a group of given nodes share similar (not necessarily identical) membership profiles under weaker signals. Utilizing the idea of random coupling, we construct our test as the maximum of the SIMPLE tests for subsampled node pairs from the group. Such technique reduces significantly the correlation among individual SIMPLE tests while largely maintaining the power, enabling delicate analysis on the asymptotic distributions of the SIMPLE-RC test. Our method and theory cover both the cases with and without node degree heterogeneity. These new theoretical developments are empowered by a second-order expansion of spiked eigenvectors under the $\ell_\infty$-norm, built upon our work for random matrices with weak spikes. Our theoretical results and the practical advantages of the newly suggested method are demonstrated through several simulation and real data examples.

Boxin Zhao, Mladen Kolar, Jinchi Lv

Multi-task learning is effective for related applications, but its performance can deteriorate when the target sample size is small. Transfer learning can borrow strength from related studies; yet, many existing methods rely on restrictive bounded-difference assumptions between the source and target models. We propose SMART, a spectral transfer method for multi-task linear regression that instead assumes spectral similarity: the target left and right singular subspaces lie within the corresponding source subspaces and are sparsely aligned with the source singular bases. Such an assumption is natural when studies share latent structures and enables transfer beyond the bounded-difference settings. SMART estimates the target coefficient matrix through structured regularization that incorporates spectral information from a source study. Importantly, it requires only a fitted source model rather than the raw source data, making it useful when data sharing is limited. Although the optimization problem is nonconvex, we develop a practical ADMM-based algorithm. We establish general, non-asymptotic error bounds and a minimax lower bound in the noiseless-source regime. Under additional regularity conditions, these results yield near-minimax Frobenius error rates up to logarithmic factors. Simulations confirm improved estimation accuracy and robustness to negative transfer, and analysis of multi-modal single-cell data demonstrates better predictive performance. The Python implementation of SMART, along with the code to reproduce all experiments in this paper, is publicly available at https://github.com/boxinz17/smart.

Tianxing Mei, Yingying Fan, Mingming Leng et al.

CART random forests are among the most widely used modern predictive methods, with well-documented empirical success. Yet, at the mechanistic level, the algorithm is often treated as a black box because of its complexity. In this paper, we develop a stochastic-control perspective on feature-subsampled CART random forests, named CART random opportunity-set allocation (CART-ROSA). At each node, the random subset of features is interpreted as a random feasible action set, and the CART split rule as a masked-action allocation policy. This policy induces a controlled stochastic process over informative split-count states, whose terminal law determines both single-tree error and cross-tree interaction terms in the forest mean squared error (MSE). Such representation opens the black box of CART-forests by separating two design levers: the informative-opportunity rate induced by feature subsampling, and the contraction strength from the within-mask split policy. We establish that the CART policy is locally stabilizing: it contracts imbalances in informative split allocations and concentrates terminal tree geometry. At the system level, however, it can be globally suboptimal for the forest objective. Specializing to the linear model, we derive the MSE risk expansion explicitly. Our results show how an operations-research perspective makes tractable a theoretical gap difficult to access from the standard algorithmic description of CART forests.

Chien-Ming Chi, Yingying Fan, Jinchi Lv

Quantifying the usefulness of individual features in random forests learning can greatly enhance its interpretability. Existing studies have shown that some popularly used feature importance measures for random forests suffer from the bias issue. In addition, there lack comprehensive size and power analyses for most of these existing methods. In this paper, we approach the problem via hypothesis testing, and suggest a framework of the self-normalized feature-residual correlation test (FACT) for evaluating the significance of a given feature in the random forests model with bias-resistance property, where our null hypothesis concerns whether the feature is conditionally independent of the response given all other features. Such an endeavor on random forests inference is empowered by some recent developments on high-dimensional random forests consistency. Under a fairly general high-dimensional nonparametric model setting with dependent features, we formally establish that FACT can provide theoretically justified feature importance test with controlled type I error and enjoy appealing power property. The theoretical results and finite-sample advantages of the newly suggested method are illustrated with several simulation examples and an economic forecasting application.

Yingying Fan, Yuxuan Han, Jinchi Lv et al.

We study contextual dynamic pricing in a semiparametric scalar-index valuation model where the latent value is $v_t=μ_\ast(\mathsf c_t)+ξ_t$, with an unknown utility map $μ_\ast$ and an unknown additive noise distribution. The key decision object is the one-dimensional oracle price map $u\mapsto p^\ast(u)$ induced by the scalar index $u=μ_\ast(\mathsf c)$ and the noise tail. Under the $β$-Hölder smoothness of the tail function for $β\geq 2$ and a revenue-geometry condition that gives a unique, stable, interior maximizer, this oracle map is itself $(β-1)$-smooth. We exploit such structure through $\mathsf{ORBIT}$, a modular coarse-to-fine policy that takes a scalar pilot index as input, localizes a benchmark price in each active bin, and learns a local polynomial approximation of the oracle map inside a trust region via bandit convex optimization. For the baseline linear utility model $μ_\ast(\mathsf c)=\mathsf c^\topθ_\ast$, an adaptive elliptical exploration scheme constructs the required scalar pilot online without distributional assumptions on the contexts. The resulting policy achieves regret $\widetilde{O}\big(T^{\frac{2β-1}{4β-3}}+\sqrt{dT}\big)$. For fixed $d$, we establish a matching lower bound in the horizon dependence, unveiling that the nonparametric oracle-map learning term is minimax sharp. The same scalar-pilot interface also yields extensions to sparse high-dimensional linear utility and nonparametric Hölder utility.

Jianqing Fan, Yingying Fan, Jinchi Lv et al.

Laplacian matrices are commonly employed in many real applications, encoding the underlying latent structural information such as graphs and manifolds. The use of the normalization terms naturally gives rise to random matrices with dependency. It is well-known that dependency is a major bottleneck of new random matrix theory (RMT) developments. To this end, in this paper, we formally introduce a class of generalized (and regularized) Laplacian matrices, which contains the Laplacian matrix and the random adjacency matrix as a specific case, and suggest the new framework of the asymptotic theory of eigenvectors for latent embeddings with generalized Laplacian matrices (ATE-GL). Our new theory is empowered by the tool of generalized quadratic vector equation for dealing with RMT under dependency, and delicate high-order asymptotic expansions of the empirical spiked eigenvectors and eigenvalues based on local laws. The asymptotic normalities established for both spiked eigenvectors and eigenvalues will enable us to conduct precise inference and uncertainty quantification for applications involving the generalized Laplacian matrices with flexibility. We discuss some applications of the suggested ATE-GL framework and showcase its validity through some numerical examples.

Yingying Fan, Lan Gao, Jinchi Lv et al.

We propose a unified theoretical framework for studying the robustness of the model-X knockoffs framework by investigating the asymptotic false discovery rate (FDR) control of the practically implemented approximate knockoffs procedure. This procedure deviates from the model-X knockoffs framework by substituting the true covariate distribution with a user-specified distribution that can be learned using in-sample observations. By replacing the distributional exchangeability condition of the model-X knockoff variables with three conditions on the approximate knockoff statistics, we establish that the approximate knockoffs procedure achieves the asymptotic FDR control. Using our unified framework, we further prove that an arguably most popularly used knockoff variable generation method--the Gaussian knockoffs generator based on the first two moments matching--achieves the asymptotic FDR control when the two-moment-based knockoff statistics are employed in the knockoffs inference procedure. For the first time in the literature, our theoretical results justify formally the effectiveness and robustness of the Gaussian knockoffs generator. Simulation and real data examples are conducted to validate the theoretical findings.

Tianxing Mei, Yingying Fan, Jinchi Lv

We offer theoretical and empirical insights into the impact of exogenous randomness on the effectiveness of random forests with tree-building rules independent of training data. We formally introduce the concept of exogenous randomness and identify two types of commonly existing randomness: Type I from feature subsampling, and Type II from tie-breaking in tree-building processes. We develop non-asymptotic expansions for the mean squared error (MSE) for both individual trees and forests and establish sufficient and necessary conditions for their consistency. In the special example of the linear regression model with independent features, our MSE expansions are more explicit, providing more understanding of the random forests' mechanisms. It also allows us to derive an upper bound on the MSE with explicit consistency rates for trees and forests. Guided by our theoretical findings, we conduct simulations to further explore how exogenous randomness enhances random forest performance. Our findings unveil that feature subsampling reduces both the bias and variance of random forests compared to individual trees, serving as an adaptive mechanism to balance bias and variance. Furthermore, our results reveal an intriguing phenomenon: the presence of noise features can act as a "blessing" in enhancing the performance of random forests thanks to feature subsampling.

Wenxuan Zuo, Zifan Zhu, Yuxuan Du et al.

High-dimensional longitudinal time series data is prevalent across various real-world applications. Many such applications can be modeled as regression problems with high-dimensional time series covariates. Deep learning has been a popular and powerful tool for fitting these regression models. Yet, the development of interpretable and reproducible deep-learning models is challenging and remains underexplored. This study introduces a novel method, Deep Learning Inference using Knockoffs for Time series data (DeepLINK-T), focusing on the selection of significant time series variables in regression while controlling the false discovery rate (FDR) at a predetermined level. DeepLINK-T combines deep learning with knockoff inference to control FDR in feature selection for time series models, accommodating a wide variety of feature distributions. It addresses dependencies across time and features by leveraging a time-varying latent factor structure in time series covariates. Three key ingredients for DeepLINK-T are 1) a Long Short-Term Memory (LSTM) autoencoder for generating time series knockoff variables, 2) an LSTM prediction network using both original and knockoff variables, and 3) the application of the knockoffs framework for variable selection with FDR control. Extensive simulation studies have been conducted to evaluate DeepLINK-T's performance, showing its capability to control FDR effectively while demonstrating superior feature selection power for high-dimensional longitudinal time series data compared to its non-time series counterpart. DeepLINK-T is further applied to three metagenomic data sets, validating its practical utility and effectiveness, and underscoring its potential in real-world applications.

Yingying Fan, Jingyuan Liu, Jinchi Lv et al.

We propose a new inference framework, named MOSAIC, for change-point detection in dynamic networks with the simultaneous low-rank and sparse-change structure. We establish the minimax rate of detection boundary, which relies on the sparsity of changes. We then develop an eigen-decomposition-based test with screened signals that approaches the minimax rate in theory, with only a minor logarithmic loss. For practical implementation of MOSAIC, we adjust the theoretical test by a novel residual-based technique, resulting in a pivotal statistic that converges to a standard normal distribution via the martingale central limit theorem under the null hypothesis and achieves full power under the alternative hypothesis. We also analyze the minimax rate of testing boundary for dynamic networks without the low-rank structure, which almost aligns with the results in high-dimensional mean-vector change-point inference. We showcase the effectiveness of MOSAIC and verify our theoretical results with several simulation examples and a real data application.

Yingying Fan, Jinchi Lv, Ao Sun et al.

Forecasting the Consumer Price Index (CPI) is an important yet challenging task in economics, where most existing approaches rely on low-frequency, survey-based data. With the recent advances of large language models (LLMs), there is growing potential to leverage high-frequency online text data for improved CPI prediction, an area still largely unexplored. This paper proposes LLM-CPI, an LLM-based approach for CPI prediction inference incorporating online text time series. We collect a large set of high-frequency online texts from a popularly used Chinese social network site and employ LLMs such as ChatGPT and the trained BERT models to construct continuous inflation labels for posts that are related to inflation. Online text embeddings are extracted via LDA and BERT. We develop a joint time series framework that combines monthly CPI data with LLM-generated daily CPI surrogates. The monthly model employs an ARX structure combining observed CPI data with text embeddings and macroeconomic variables, while the daily model uses a VARX structure built on LLM-generated CPI surrogates and text embeddings. We establish the asymptotic properties of the method and provide two forms of constructed prediction intervals. The finite-sample performance and practical advantages of LLM-CPI are demonstrated through both simulation and real data examples.

Zemin Zheng, Xin Zhou, Jinchi Lv

Data reduction with uncertainty quantification plays a key role in various multi-task learning applications, where large numbers of responses and features are present. To this end, a general framework of high-dimensional manifold-based SOFAR inference (SOFARI) was introduced recently in Zheng, Zhou, Fan and Lv (2024) for interpretable multi-task learning inference focusing on the left factor vectors and singular values exploiting the latent singular value decomposition (SVD) structure. Yet, designing a valid inference procedure on the latent right factor vectors is not straightforward from that of the left ones and can be even more challenging due to asymmetry of left and right singular vectors in the response matrix. To tackle these issues, in this paper we suggest a new method of high-dimensional manifold-based SOFAR inference for latent responses (SOFARI-R), where two variants of SOFARI-R are introduced. The first variant deals with strongly orthogonal factors by coupling left singular vectors with the design matrix and then appropriately rescaling them to generate new Stiefel manifolds. The second variant handles the more general weakly orthogonal factors by employing the hard-thresholded SOFARI estimates and delicately incorporating approximation errors into the distribution. Both variants produce bias-corrected estimators for the latent right factor vectors that enjoy asymptotically normal distributions with justified asymptotic variance estimates. We demonstrate the effectiveness of the newly suggested method using extensive simulation studies and an economic application.

Wenqin Du, Rundong Ding, Yingying Fan et al.

The problem of evaluating the effectiveness of a treatment or policy commonly appears in causal inference applications under network interference. In this paper, we suggest the new method of high-dimensional network causal inference (HNCI) that provides both valid confidence interval on the average direct treatment effect on the treated (ADET) and valid confidence set for the neighborhood size for interference effect. We exploit the model setting in Belloni et al. (2022) and allow certain type of heterogeneity in node interference neighborhood sizes. We propose a linear regression formulation of potential outcomes, where the regression coefficients correspond to the underlying true interference function values of nodes and exhibit a latent homogeneous structure. Such a formulation allows us to leverage existing literature from linear regression and homogeneity pursuit to conduct valid statistical inferences with theoretical guarantees. The resulting confidence intervals for the ADET are formally justified through asymptotic normalities with estimable variances. We further provide the confidence set for the neighborhood size with theoretical guarantees exploiting the repro samples approach. The practical utilities of the newly suggested methods are demonstrated through simulation and real data examples.

Yingying Fan, Yuxuan Han, Jinchi Lv et al.

In this paper, we study the behavior of the Upper Confidence Bound-Variance (UCB-V) algorithm for the Multi-Armed Bandit (MAB) problems, a variant of the canonical Upper Confidence Bound (UCB) algorithm that incorporates variance estimates into its decision-making process. More precisely, we provide an asymptotic characterization of the arm-pulling rates for UCB-V, extending recent results for the canonical UCB in Kalvit and Zeevi (2021) and Khamaru and Zhang (2024). In an interesting contrast to the canonical UCB, our analysis reveals that the behavior of UCB-V can exhibit instability, meaning that the arm-pulling rates may not always be asymptotically deterministic. Besides the asymptotic characterization, we also provide non-asymptotic bounds for the arm-pulling rates in the high probability regime, offering insights into the regret analysis. As an application of this high probability result, we establish that UCB-V can achieve a more refined regret bound, previously unknown even for more complicate and advanced variance-aware online decision-making algorithms.

Xinze Du, Yingying Fan, Jinchi Lv et al.

This paper investigates the estimation and inference of the average treatment effect (ATE) using deep neural networks (DNNs) in the potential outcomes framework. Under some regularity conditions, the observed response can be formulated as the response of a mean regression problem with both the confounding variables and the treatment indicator as the independent variables. Using such formulation, we investigate two methods for ATE estimation and inference based on the estimated mean regression function via DNN regression using a specific network architecture. We show that both DNN estimates of ATE are consistent with dimension-free consistency rates under some assumptions on the underlying true mean regression model. Our model assumptions accommodate the potentially complicated dependence structure of the observed response on the covariates, including latent factors and nonlinear interactions between the treatment indicator and confounding variables. We also establish the asymptotic normality of our estimators based on the idea of sample splitting, ensuring precise inference and uncertainty quantification. Simulation studies and real data application justify our theoretical findings and support our DNN estimation and inference methods.

Jianqing Fan, Yingying Fan, Xiao Han et al.

Network data is prevalent in many contemporary big data applications in which a common interest is to unveil important latent links between different pairs of nodes. Yet a simple fundamental question of how to precisely quantify the statistical uncertainty associated with the identification of latent links still remains largely unexplored. In this paper, we propose the method of statistical inference on membership profiles in large networks (SIMPLE) in the setting of degree-corrected mixed membership model, where the null hypothesis assumes that the pair of nodes share the same profile of community memberships. In the simpler case of no degree heterogeneity, the model reduces to the mixed membership model for which an alternative more robust test is also proposed. Both tests are of the Hotelling-type statistics based on the rows of empirical eigenvectors or their ratios, whose asymptotic covariance matrices are very challenging to derive and estimate. Nevertheless, their analytical expressions are unveiled and the unknown covariance matrices are consistently estimated. Under some mild regularity conditions, we establish the exact limiting distributions of the two forms of SIMPLE test statistics under the null hypothesis and contiguous alternative hypothesis. They are the chi-square distributions and the noncentral chi-square distributions, respectively, with degrees of freedom depending on whether the degrees are corrected or not. We also address the important issue of estimating the unknown number of communities and establish the asymptotic properties of the associated test statistics. The advantages and practical utility of our new procedures in terms of both size and power are demonstrated through several simulation examples and real network applications.

Yingying Fan, Jinchi Lv, Mahrad Sharifvaghefi et al.

Interpretability and stability are two important features that are desired in many contemporary big data applications arising in economics and finance. While the former is enjoyed to some extent by many existing forecasting approaches, the latter in the sense of controlling the fraction of wrongly discovered features which can enhance greatly the interpretability is still largely underdeveloped in the econometric settings. To this end, in this paper we exploit the general framework of model-X knockoffs introduced recently in Candès, Fan, Janson and Lv (2018), which is nonconventional for reproducible large-scale inference in that the framework is completely free of the use of p-values for significance testing, and suggest a new method of intertwined probabilistic factors decoupling (IPAD) for stable interpretable forecasting with knockoffs inference in high-dimensional models. The recipe of the method is constructing the knockoff variables by assuming a latent factor model that is exploited widely in economics and finance for the association structure of covariates. Our method and work are distinct from the existing literature in that we estimate the covariate distribution from data instead of assuming that it is known when constructing the knockoff variables, our procedure does not require any sample splitting, we provide theoretical justifications on the asymptotic false discovery rate control, and the theory for the power analysis is also established. Several simulation examples and the real data analysis further demonstrate that the newly suggested method has appealing finite-sample performance with desired interpretability and stability compared to some popularly used forecasting methods.

Yang Young Lu, Yingying Fan, Jinchi Lv et al.

Deep learning has become increasingly popular in both supervised and unsupervised machine learning thanks to its outstanding empirical performance. However, because of their intrinsic complexity, most deep learning methods are largely treated as black box tools with little interpretability. Even though recent attempts have been made to facilitate the interpretability of deep neural networks (DNNs), existing methods are susceptible to noise and lack of robustness. Therefore, scientists are justifiably cautious about the reproducibility of the discoveries, which is often related to the interpretability of the underlying statistical models. In this paper, we describe a method to increase the interpretability and reproducibility of DNNs by incorporating the idea of feature selection with controlled error rate. By designing a new DNN architecture and integrating it with the recently proposed knockoffs framework, we perform feature selection with a controlled error rate, while maintaining high power. This new method, DeepPINK (Deep feature selection using Paired-Input Nonlinear Knockoffs), is applied to both simulated and real data sets to demonstrate its empirical utility.

Emre Demirkaya, Yingying Fan, Lan Gao et al.

The weighted nearest neighbors (WNN) estimator has been popularly used as a flexible and easy-to-implement nonparametric tool for mean regression estimation. The bagging technique is an elegant way to form WNN estimators with weights automatically generated to the nearest neighbors; we name the resulting estimator as the distributional nearest neighbors (DNN) for easy reference. Yet, there is a lack of distributional results for such estimator, limiting its application to statistical inference. Moreover, when the mean regression function has higher-order smoothness, DNN does not achieve the optimal nonparametric convergence rate, mainly because of the bias issue. In this work, we provide an in-depth technical analysis of the DNN, based on which we suggest a bias reduction approach for the DNN estimator by linearly combining two DNN estimators with different subsampling scales, resulting in the novel two-scale DNN (TDNN) estimator. The two-scale DNN estimator has an equivalent representation of WNN with weights admitting explicit forms and some being negative. We prove that, thanks to the use of negative weights, the two-scale DNN estimator enjoys the optimal nonparametric rate of convergence in estimating the regression function under the fourth-order smoothness condition. We further go beyond estimation and establish that the DNN and two-scale DNN are both asymptotically normal as the subsampling scales and sample size diverge to infinity. For the practical implementation, we also provide variance estimators and a distribution estimator using the jackknife and bootstrap techniques for the two-scale DNN. These estimators can be exploited for constructing valid confidence intervals for nonparametric inference of the regression function. The theoretical results and appealing finite-sample performance of the suggested two-scale DNN method are illustrated with several numerical examples.

Emre Demirkaya, Yang Feng, Pallavi Basu et al.

Model selection is crucial to high-dimensional learning and inference for contemporary big data applications in pinpointing the best set of covariates among a sequence of candidate interpretable models. Most existing work assumes implicitly that the models are correctly specified or have fixed dimensionality. Yet both features of model misspecification and high dimensionality are prevalent in practice. In this paper, we exploit the framework of model selection principles in misspecified models originated in Lv and Liu (2014) and investigate the asymptotic expansion of Bayesian principle of model selection in the setting of high-dimensional misspecified models. With a natural choice of prior probabilities that encourages interpretability and incorporates Kullback-Leibler divergence, we suggest the high-dimensional generalized Bayesian information criterion with prior probability (HGBIC_p) for large-scale model selection with misspecification. Our new information criterion characterizes the impacts of both model misspecification and high dimensionality on model selection. We further establish the consistency of covariance contrast matrix estimation and the model selection consistency of HGBIC_p in ultra-high dimensions under some mild regularity conditions. The advantages of our new method are supported by numerical studies.

Zemin Zheng, Jinchi Lv, Wei Lin

As a popular tool for producing meaningful and interpretable models, large-scale sparse learning works efficiently when the underlying structures are indeed or close to sparse. However, naively applying the existing regularization methods can result in misleading outcomes due to model misspecification. In particular, the direct sparsity assumption on coefficient vectors has been questioned in real applications. Therefore, we consider nonsparse learning with the conditional sparsity structure that the coefficient vector becomes sparse after taking out the impacts of certain unobservable latent variables. A new methodology of nonsparse learning with latent variables (NSL) is proposed to simultaneously recover the significant observable predictors and latent factors as well as their effects. We explore a common latent family incorporating population principal components and derive the convergence rates of both sample principal components and their score vectors that hold for a wide class of distributions. With the properly estimated latent variables, properties including model selection consistency and oracle inequalities under various prediction and estimation losses are established for the proposed methodology. Our new methodology and results are evidenced by simulation and real data examples.

Yingying Fan, Emre Demirkaya, Gaorong Li et al.

Power and reproducibility are key to enabling refined scientific discoveries in contemporary big data applications with general high-dimensional nonlinear models. In this paper, we provide theoretical foundations on the power and robustness for the model-free knockoffs procedure introduced recently in Candès, Fan, Janson and Lv (2016) in high-dimensional setting when the covariate distribution is characterized by Gaussian graphical model. We establish that under mild regularity conditions, the power of the oracle knockoffs procedure with known covariate distribution in high-dimensional linear models is asymptotically one as sample size goes to infinity. When moving away from the ideal case, we suggest the modified model-free knockoffs method called graphical nonlinear knockoffs (RANK) to accommodate the unknown covariate distribution. We provide theoretical justifications on the robustness of our modified procedure by showing that the false discovery rate (FDR) is asymptotically controlled at the target level and the power is asymptotically one with the estimated covariate distribution. To the best of our knowledge, this is the first formal theoretical result on the power for the knockoffs procedure. Simulation results demonstrate that compared to existing approaches, our method performs competitively in both FDR control and power. A real data set is analyzed to further assess the performance of the suggested knockoffs procedure.

Yoshimasa Uematsu, Yingying Fan, Kun Chen et al.

Many modern big data applications feature large scale in both numbers of responses and predictors. Better statistical efficiency and scientific insights can be enabled by understanding the large-scale response-predictor association network structures via layers of sparse latent factors ranked by importance. Yet sparsity and orthogonality have been two largely incompatible goals. To accommodate both features, in this paper we suggest the method of sparse orthogonal factor regression (SOFAR) via the sparse singular value decomposition with orthogonality constrained optimization to learn the underlying association networks, with broad applications to both unsupervised and supervised learning tasks such as biclustering with sparse singular value decomposition, sparse principal component analysis, sparse factor analysis, and spare vector autoregression analysis. Exploiting the framework of convexity-assisted nonconvex optimization, we derive nonasymptotic error bounds for the suggested procedure characterizing the theoretical advantages. The statistical guarantees are powered by an efficient SOFAR algorithm with convergence property. Both computational and theoretical advantages of our procedure are demonstrated with several simulation and real data examples.

Zhao Ren, Yongjian Kang, Yingying Fan et al.

Heterogeneity is often natural in many contemporary applications involving massive data. While posing new challenges to effective learning, it can play a crucial role in powering meaningful scientific discoveries through the understanding of important differences among subpopulations of interest. In this paper, we exploit multiple networks with Gaussian graphs to encode the connectivity patterns of a large number of features on the subpopulations. To uncover the heterogeneity of these structures across subpopulations, we suggest a new framework of tuning-free heterogeneity pursuit (THP) via large-scale inference, where the number of networks is allowed to diverge. In particular, two new tests, the chi-based test and the linear functional-based test, are introduced and their asymptotic null distributions are established. Under mild regularity conditions, we establish that both tests are optimal in achieving the testable region boundary and the sample size requirement for the latter test is minimal. Both theoretical guarantees and the tuning-free feature stem from efficient multiple-network estimation by our newly suggested approach of heterogeneous group square-root Lasso (HGSL) for high-dimensional multi-response regression with heterogeneous noises. To solve this convex program, we further introduce a tuning-free algorithm that is scalable and enjoys provable convergence to the global optimum. Both computational and theoretical advantages of our procedure are elucidated through simulation and real data examples.

Yingying Fan, Yinfei Kong, Daoji Li et al.

Understanding how features interact with each other is of paramount importance in many scientific discoveries and contemporary applications. Yet interaction identification becomes challenging even for a moderate number of covariates. In this paper, we suggest an efficient and flexible procedure, called the interaction pursuit (IP), for interaction identification in ultra-high dimensions. The suggested method first reduces the number of interactions and main effects to a moderate scale by a new feature screening approach, and then selects important interactions and main effects in the reduced feature space using regularization methods. Compared to existing approaches, our method screens interactions separately from main effects and thus can be more effective in interaction screening. Under a fairly general framework, we establish that for both interactions and main effects, the method enjoys the sure screening property in screening and oracle inequalities in selection. Our method and theoretical results are supported by several simulation and real data examples.

Yingying Fan, Jinchi Lv

Two important goals of high-dimensional modeling are prediction and variable selection. In this article, we consider regularization with combined $L_1$ and concave penalties, and study the sampling properties of the global optimum of the suggested method in ultra-high dimensional settings. The $L_1$-penalty provides the minimum regularization needed for removing noise variables in order to achieve oracle prediction risk, while concave penalty imposes additional regularization to control model sparsity. In the linear model setting, we prove that the global optimum of our method enjoys the same oracle inequalities as the lasso estimator and admits an explicit bound on the false sign rate, which can be asymptotically vanishing. Moreover, we establish oracle risk inequalities for the method and the sampling properties of computable solutions. Numerical studies suggest that our method yields more stable estimates than using a concave penalty alone.

Yinfei Kong, Daoji Li, Yingying Fan et al.

Feature interactions can contribute to a large proportion of variation in many prediction models. In the era of big data, the coexistence of high dimensionality in both responses and covariates poses unprecedented challenges in identifying important interactions. In this paper, we suggest a two-stage interaction identification method, called the interaction pursuit via distance correlation (IPDC), in the setting of high-dimensional multi-response interaction models that exploits feature screening applied to transformed variables with distance correlation followed by feature selection. Such a procedure is computationally efficient, generally applicable beyond the heredity assumption, and effective even when the number of responses diverges with the sample size. Under mild regularity conditions, we show that this method enjoys nice theoretical properties including the sure screening property, support union recovery, and oracle inequalities in prediction and estimation for both interactions and main effects. The advantages of our method are supported by several simulation studies and real data analysis.

Yingying Fan, Jinchi Lv

Large-scale precision matrix estimation is of fundamental importance yet challenging in many contemporary applications for recovering Gaussian graphical models. In this paper, we suggest a new approach of innovated scalable efficient estimation (ISEE) for estimating large precision matrix. Motivated by the innovated transformation, we convert the original problem into that of large covariance matrix estimation. The suggested method combines the strengths of recent advances in high-dimensional sparse modeling and large covariance matrix estimation. Compared to existing approaches, our method is scalable and can deal with much larger precision matrices with simple tuning. Under mild regularity conditions, we establish that this procedure can recover the underlying graphical structure with significant probability and provide efficient estimation of link strengths. Both computational and theoretical advantages of the procedure are evidenced through simulation and real data examples.

Yinfei Kong, Zemin Zheng, Jinchi Lv

The Dantzig selector has received popularity for many applications such as compressed sensing and sparse modeling, thanks to its computational efficiency as a linear programming problem and its nice sampling properties. Existing results show that it can recover sparse signals mimicking the accuracy of the ideal procedure, up to a logarithmic factor of the dimensionality. Such a factor has been shown to hold for many regularization methods. An important question is whether this factor can be reduced to a logarithmic factor of the sample size in ultra-high dimensions under mild regularity conditions. To provide an affirmative answer, in this paper we suggest the constrained Dantzig selector, which has more flexible constraints and parameter space. We prove that the suggested method can achieve convergence rates within a logarithmic factor of the sample size of the oracle rates and improved sparsity, under a fairly weak assumption on the signal strength. Such improvement is significant in ultra-high dimensions. This method can be implemented efficiently through sequential linear programming. Numerical studies confirm that the sample size needed for a certain level of accuracy in these problems can be much reduced.

Yingying Fan, Jinchi Lv

High-dimensional data analysis has motivated a spectrum of regularization methods for variable selection and sparse modeling, with two popular classes of convex ones and concave ones. A long debate has been on whether one class dominates the other, an important question both in theory and to practitioners. In this paper, we characterize the asymptotic equivalence of regularization methods, with general penalty functions, in a thresholded parameter space under the generalized linear model setting, where the dimensionality can grow up to exponentially with the sample size. To assess their performance, we establish the oracle inequalities, as in Bickel, Ritov and Tsybakov (2009), of the global minimizer for these methods under various prediction and variable selection losses. These results reveal an interesting phase transition phenomenon. For polynomially growing dimensionality, the $L_1$-regularization method of Lasso and concave methods are asymptotically equivalent, having the same convergence rates in the oracle inequalities. For exponentially growing dimensionality, concave methods are asymptotically equivalent but have faster convergence rates than the Lasso. We also establish a stronger property of the oracle risk inequalities of the regularization methods, as well as the sampling properties of computable solutions. Our new theoretical results are illustrated and justified by simulation and real data examples.

Zemin Zheng, Yingying Fan, Jinchi Lv

High-dimensional sparse modeling via regularization provides a powerful tool for analyzing large-scale data sets and obtaining meaningful, interpretable models. The use of nonconvex penalty functions shows advantage in selecting important features in high dimensions, but the global optimality of such methods still demands more understanding. In this paper, we consider sparse regression with hard-thresholding penalty, which we show to give rise to thresholded regression. This approach is motivated by its close connection with the $L_0$-regularization, which can be unrealistic to implement in practice but of appealing sampling properties, and its computational advantage. Under some mild regularity conditions allowing possibly exponentially growing dimensionality, we establish the oracle inequalities of the resulting regularized estimator, as the global minimizer, under various prediction and variable selection losses, as well as the oracle risk inequalities of the hard-thresholded estimator followed by a further $L_2$-regularization. The risk properties exhibit interesting shrinkage effects under both estimation and prediction losses. We identify the optimal choice of the ridge parameter, which is shown to have simultaneous advantages to both the $L_2$-loss and prediction loss. These new results and phenomena are evidenced by simulation and real data examples.

Pallavi Basu, Yang Feng, Jinchi Lv

Model selection is indispensable to high-dimensional sparse modeling in selecting the best set of covariates among a sequence of candidate models. Most existing work assumes implicitly that the model is correctly specified or of fixed dimensions. Yet model misspecification and high dimensionality are common in real applications. In this paper, we investigate two classical Kullback-Leibler divergence and Bayesian principles of model selection in the setting of high-dimensional misspecified models. Asymptotic expansions of these principles reveal that the effect of model misspecification is crucial and should be taken into account, leading to the generalized AIC and generalized BIC in high dimensions. With a natural choice of prior probabilities, we suggest the generalized BIC with prior probability which involves a logarithmic factor of the dimensionality in penalizing model complexity. We further establish the consistency of the covariance contrast matrix estimator in a general setting. Our results and new method are supported by numerical studies.