arXiv subjects

Cun-Hui Zhang

Publications and source records attributed to Cun-Hui Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2020

Adaptive Estimation In High-Dimensional Additive Models With Multi-Resolution Group Lasso

In additive models with many nonparametric components, a number of regularized estimators have been proposed and proven to attain various error bounds under different combinations of sparsity and fixed smoothness conditions. Some of these error bounds match minimax rates in the corresponding settings. Some of the rate minimax methods are non-convex and computationally costly. From these perspectives, the existing solutions to the high-dimensional additive nonparametric regression problem are fragmented. In this paper, we propose a multi-resolution group Lasso (MR-GL) method in a unified approach to simultaneously achieve or improve existing error bounds and provide new ones without the knowledge of the level of sparsity or the degree of smoothness of the unknown functions. Such adaptive convergence rates are established when a prediction factor can be treated as a constant. Furthermore, we prove that the prediction factor, which can be bounded in terms of a restricted eigenvalue or a compatibility coefficient, can be indeed treated as a constant for random designs under a nearly optimal sample size condition.

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arXiv2020

Confidence intervals for multiple isotonic regression and other monotone models

We consider the problem of constructing pointwise confidence intervals in the multiple isotonic regression model. Recently, [HZ19] obtained a pointwise limit distribution theory for the so-called block max-min and min-max estimators [FLN17] in this model, but inference remains a difficult problem due to the nuisance parameter in the limit distribution that involves multiple unknown partial derivatives of the true regression function. In this paper, we show that this difficult nuisance parameter can be effectively eliminated by taking advantage of information beyond point estimates in the block max-min and min-max estimators. Formally, let $\hat{u}(x_0)$ (resp. $\hat{v}(x_0)$) be the maximizing lower-left (resp. minimizing upper-right) vertex in the block max-min (resp. min-max) estimator, and $\hat{f}_n$ be the average of the block max-min and min-max estimators. If all (first-order) partial derivatives of $f_0$ are non-vanishing at $x_0$, then the following pivotal limit distribution theory holds: $$ \sqrt{n_{\hat{u},\hat{v}}(x_0)}\big(\hat{f}_n(x_0)-f_0(x_0)\big)\rightsquigarrow σ\cdot \mathbb{L}_{1_d}. $$ Here $n_{\hat{u},\hat{v}}(x_0)$ is the number of design points in the block $[\hat{u}(x_0),\hat{v}(x_0)]$, $σ$ is the standard deviation of the errors, and $\mathbb{L}_{1_d}$ is a universal limit distribution free of nuisance parameters. This immediately yields confidence intervals for $f_0(x_0)$ with asymptotically exact confidence level and oracle length. Notably, the construction of the confidence intervals, even new in the univariate setting, requires no more efforts than performing an isotonic regression for once using the block max-min and min-max estimators, and can be easily adapted to other common monotone models. Extensive simulations are carried out to support our theory.

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arXiv2020

Extreme Eigenvalues of Nonlinear Correlation Matrices with Applications to Additive Models

The maximum correlation of functions of a pair of random variables is an important measure of stochastic dependence. It is known that this maximum nonlinear correlation is identical to the absolute value of the Pearson correlation for a pair of Gaussian random variables or a pair of finite sums of iid random variables. This paper extends these results to pairwise Gaussian vectors and processes, nested sums of iid random variables, and permutation symmetric functions of sub-groups of iid random variables. It also discusses applications to additive regression models.

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arXiv2020

Second order Stein: SURE for SURE and other applications in high-dimensional inference

Stein's formula states that a random variable of the form $z^\top f(z) - \text{div} f(z)$ is mean-zero for functions $f$ with integrable gradient. Here, $\text{div} f$ is the divergence of the function $f$ and $z$ is a standard normal vector. This paper aims to propose a Second Order Stein formula to characterize the variance of such random variables for all functions $f(z)$ with square integrable gradient, and to demonstrate the usefulness of this formula in various applications. In the Gaussian sequence model, a consequence of Stein's formula is Stein's Unbiased Risk Estimate (SURE), an unbiased estimate of the mean squared risk for almost any estimator $\hatμ$ of the unknown mean. A first application of the Second Order Stein formula is an Unbiased Risk Estimate for SURE itself (SURE for SURE): an unbiased estimate {providing} information about the squared distance between SURE and the squared estimation error of $\hatμ$. SURE for SURE has a simple form as a function of the data and is applicable to all $\hatμ$ with square integrable gradient, e.g. the Lasso and the Elastic Net. In addition to SURE for SURE, the following applications are developed: (1) Upper bounds on the risk of SURE when the estimation target is the mean squared error; (2) Confidence regions based on SURE; (3) Oracle inequalities satisfied by SURE-tuned estimates; (4) An upper bound on the variance of the size of the model selected by the Lasso; (5) Explicit expressions of SURE for SURE for the Lasso and the Elastic-Net; (6) In the linear model, a general semi-parametric scheme to de-bias a differentiable initial estimator for inference of a low-dimensional projection of the unknown $β$, with a characterization of the variance after de-biasing; and (7) An accuracy analysis of a Gaussian Monte Carlo scheme to approximate the divergence of functions $f: R^n\to R^n$.

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arXiv2020

Beyond Gaussian Approximation: Bootstrap for Maxima of Sums of Independent Random Vectors

The Bonferroni adjustment, or the union bound, is commonly used to study rate optimality properties of statistical methods in high-dimensional problems. However, in practice, the Bonferroni adjustment is overly conservative. The extreme value theory has been proven to provide more accurate multiplicity adjustments in a number of settings, but only on ad hoc basis. Recently, Gaussian approximation has been used to justify bootstrap adjustments in large scale simultaneous inference in some general settings when $n \gg (\log p)^7$, where $p$ is the multiplicity of the inference problem and $n$ is the sample size. The thrust of this theory is the validity of the Gaussian approximation for maxima of sums of independent random vectors in high-dimension. In this paper, we reduce the sample size requirement to $n \gg (\log p)^5$ for the consistency of the empirical bootstrap and the multiplier/wild bootstrap in the Kolmogorov-Smirnov distance, possibly in the regime where the Gaussian approximation is not available. New comparison and anti-concentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable.

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arXiv2020

Isotonic Regression in Multi-Dimensional Spaces and Graphs

In this paper we study minimax and adaptation rates in general isotonic regression. For uniform deterministic and random designs in $[0,1]^d$ with $d\ge 2$ and $N(0,1)$ noise, the minimax rate for the $\ell_2$ risk is known to be bounded from below by $n^{-1/d}$ when the unknown mean function $f$ is nondecreasing and its range is bounded by a constant, while the least squares estimator (LSE) is known to nearly achieve the minimax rate up to a factor $(\log n)^γ$ where $n$ is sample size, $γ= 4$ in the lattice design and $γ= \max\{9/2, (d^2+d+1)/2 \}$ in the random design. Moreover, the LSE is known to achieve the adaptation rate $(K/n)^{-2/d}\{1\vee \log(n/K)\}^{2γ}$ when $f$ is piecewise constant on $K$ hyperrectangles in a partition of $[0,1]^d$. Due to the minimax theorem, the LSE is identical on every design point to both the max-min and min-max estimators over all upper and lower sets containing the design point. This motivates our consideration of estimators which lie in-between the max-min and min-max estimators over possibly smaller classes of upper and lower sets, including a subclass of block estimators. Under a $q$-th moment condition on the noise, we develop $\ell_q$ risk bounds for such general estimators for isotonic regression on graphs. For uniform deterministic and random designs in $[0,1]^d$ with $d\ge 3$, our $\ell_2$ risk bound for the block estimator matches the minimax rate $n^{-1/d}$ when the range of $f$ is bounded and achieves the near parametric adaptation rate $(K/n)\{1\vee\log(n/K)\}^{d}$ when $f$ is $K$-piecewise constant. Furthermore, the block estimator possesses the following oracle property in variable selection: When $f$ depends on only a subset $S$ of variables, the $\ell_2$ risk of the block estimator automatically achieves up to a poly-logarithmic factor the minimax rate based on the oracular knowledge of $S$.

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arXiv2019

Limit distribution theory for block estimators in multiple isotonic regression

We study limit distributions for the tuning-free max-min block estimator originally proposed in [FLN17] in the problem of multiple isotonic regression, under both fixed lattice design and random design settings. We show that, if the regression function $f_0$ admits vanishing derivatives up to order $α_k$ along the $k$-th dimension ($k=1,\ldots,d$) at a fixed point $x_0 \in (0,1)^d$, and the errors have variance $σ^2$, then the max-min block estimator $\hat{f}_n$ satisfies \begin{align*} (n_\ast/σ^2)^{\frac{1}{2+\sum_{k \in \mathcal{D}_\ast} α_k^{-1}}}\big(\hat{f}_n(x_0)-f_0(x_0)\big)\rightsquigarrow \mathbb{C}(f_0,x_0). \end{align*} Here $\mathcal{D}_\ast, n_\ast$, depending on $\{α_k\}$ and the design points, are the set of all `effective dimensions' and the size of `effective samples' that drive the asymptotic limiting distribution, respectively. If furthermore either $\{α_k\}$ are relative primes to each other or all mixed derivatives of $f_0$ of certain critical order vanish at $x_0$, then the limiting distribution can be represented as $\mathbb{C}(f_0,x_0) =_d K(f_0,x_0) \cdot \mathbb{D}_α$, where $K(f_0,x_0)$ is a constant depending on the local structure of the regression function $f_0$ at $x_0$, and $\mathbb{D}_α$ is a non-standard limiting distribution generalizing the well-known Chernoff distribution in univariate problems. The above limit theorem is also shown to be optimal both in terms of the local rate of convergence and the dependence on the unknown regression function whenever such dependence is explicit (i.e. $K(f_0,x_0)$), for the full range of $\{α_k\}$ in a local asymptotic minimax sense.

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arXiv2019

On Estimation of Isotonic Piecewise Constant Signals

Consider a sequence of real data points $X_1,\ldots, X_n$ with underlying means $θ^*_1,\dots,θ^*_n$. This paper starts from studying the setting that $θ^*_i$ is both piecewise constant and monotone as a function of the index $i$. For this, we establish the exact minimax rate of estimating such monotone functions, and thus give a non-trivial answer to an open problem in the shape-constrained analysis literature. The minimax rate involves an interesting iterated logarithmic dependence on the dimension, a phenomenon that is revealed through characterizing the interplay between the isotonic shape constraint and model selection complexity. We then develop a penalized least-squares procedure for estimating the vector $θ^*=(θ^*_1,\dots,θ^*_n)^T$. This estimator is shown to achieve the derived minimax rate adaptively. For the proposed estimator, we further allow the model to be misspecified and derive oracle inequalities with the optimal rates, and show there exists a computationally efficient algorithm to compute the exact solution.

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arXiv2018

Statistically Optimal and Computationally Efficient Low Rank Tensor Completion from Noisy Entries

In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of noisy tensor completion. Much of the attention has been focused on the fundamental computational challenges often associated with problems involving higher order tensors, yet very little is known about their statistical performance. To fill in this void, in this article, we characterize the fundamental statistical limits of noisy tensor completion by establishing minimax optimal rates of convergence for estimating a $k$th order low rank tensor under the general $\ell_p$ ($1\le p\le 2$) norm which suggest significant room for improvement over the existing approaches. Furthermore, we propose a polynomial-time computable estimating procedure based upon power iteration and a second-order spectral initialization that achieves the optimal rates of convergence. Our method is fairly easy to implement and numerical experiments are presented to further demonstrate the practical merits of our estimator.

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arXiv2017

Sorted Concave Penalized Regression

The Lasso is biased. Concave penalized least squares estimation (PLSE) takes advantage of signal strength to reduce this bias, leading to sharper error bounds in prediction, coefficient estimation and variable selection. For prediction and estimation, the bias of the Lasso can be also reduced by taking a smaller penalty level than what selection consistency requires, but such smaller penalty level depends on the sparsity of the true coefficient vector. The sorted L1 penalized estimation (Slope) was proposed for adaptation to such smaller penalty levels. However, the advantages of concave PLSE and Slope do not subsume each other. We propose sorted concave penalized estimation to combine the advantages of concave and sorted penalizations. We prove that sorted concave penalties adaptively choose the smaller penalty level and at the same time benefits from signal strength, especially when a significant proportion of signals are stronger than the corresponding adaptively selected penalty levels. A local convex approximation, which extends the local linear and quadratic approximations to sorted concave penalties, is developed to facilitate the computation of sorted concave PLSE and proven to possess desired prediction and estimation error bounds. We carry out a unified treatment of penalty functions in a general optimization setting, including the penalty levels and concavity of the above mentioned sorted penalties and mixed penalties motivated by Bayesian considerations. Our analysis of prediction and estimation errors requires the restricted eigenvalue condition on the design, not beyond, and provides selection consistency under a required minimum signal strength condition in addition. Thus, our results also sharpens existing results on concave PLSE by removing the upper sparse eigenvalue component of the sparse Riesz condition.

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arXiv2017

Penalized Estimation in Additive Regression with High-Dimensional Data

Additive regression provides an extension of linear regression by modeling the signal of a response as a sum of functions of covariates of relatively low complexity. We study penalized estimation in high-dimensional nonparametric additive regression where functional semi-norms are used to induce smoothness of component functions and the empirical $L_2$ norm is used to induce sparsity. The functional semi-norms can be of Sobolev or bounded variation types and are allowed to be different amongst individual component functions. We establish new oracle inequalities for the predictive performance of such methods under three simple technical conditions: a sub-gaussian condition on the noise, a compatibility condition on the design and the functional classes under consideration, and an entropy condition on the functional classes. For random designs, the sample compatibility condition can be replaced by its population version under an additional condition to ensure suitable convergence of empirical norms. In homogeneous settings where the complexities of the component functions are of the same order, our results provide a spectrum of explicit convergence rates, from the so-called slow rate without requiring the compatibility condition to the fast rate under the hard sparsity or certain $L_q$ sparsity to allow many small components in the true regression function. These results significantly broadens and sharpens existing ones in the literature.

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arXiv2017

Group-Linear Empirical Bayes Estimates for a Heteroscedastic Normal Mean

The problem of estimating the mean of a normal vector with known but unequal variances introduces substantial difficulties that impair the adequacy of traditional empirical Bayes estimators. By taking a different approach, that treats the known variances as part of the random observations, we restore symmetry and thus the effectiveness of such methods. We suggest a group-linear empirical Bayes estimator, which collects observations with similar variances and applies a spherically symmetric estimator to each group separately. The proposed estimator is motivated by a new oracle rule which is stronger than the best linear rule, and thus provides a more ambitious benchmark than that considered in previous literature. Our estimator asymptotically achieves the new oracle risk (under appropriate conditions) and at the same time is minimax. The group-linear estimator is particularly advantageous in situations where the true means and observed variances are empirically dependent. To demonstrate the merits of the proposed methods in real applications, we analyze the baseball data used in Brown (2008), where the group-linear methods achieved the prediction error of the best nonparametric estimates that have been applied to the dataset, and significantly lower error than other parametric and semi-parametric empirical Bayes estimators.

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arXiv2016

Theory of the GMM Kernel

We develop some theoretical results for a robust similarity measure named "generalized min-max" (GMM). This similarity has direct applications in machine learning as a positive definite kernel and can be efficiently computed via probabilistic hashing. Owing to the discrete nature, the hashed values can also be used for efficient near neighbor search. We prove the theoretical limit of GMM and the consistency result, assuming that the data follow an elliptical distribution, which is a very general family of distributions and includes the multivariate $t$-distribution as a special case. The consistency result holds as long as the data have bounded first moment (an assumption which essentially holds for datasets commonly encountered in practice). Furthermore, we establish the asymptotic normality of GMM. Compared to the "cosine" similarity which is routinely adopted in current practice in statistics and machine learning, the consistency of GMM requires much weaker conditions. Interestingly, when the data follow the $t$-distribution with $ν$ degrees of freedom, GMM typically provides a better measure of similarity than "cosine" roughly when $ν<8$ (which is already very close to normal). These theoretical results will help explain the recent success of GMM in learning tasks.

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arXiv2016

High-dimensional simultaneous inference with the bootstrap

We propose a residual and wild bootstrap methodology for individual and simultaneous inference in high-dimensional linear models with possibly non-Gaussian and heteroscedastic errors. We establish asymptotic consistency for simultaneous inference for parameters in groups $G$, where $p \gg n$, $s_0 = o(n^{1/2}/\{\log(p) \log(|G|)^{1/2}\})$ and $\log(|G|) = o(n^{1/7})$, with $p$ the number of variables, $n$ the sample size and $s_0$ denoting the sparsity. The theory is complemented by many empirical results. Our proposed procedures are implemented in the R-package hdi.

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arXiv2016

Incoherent Tensor Norms and Their Applications in Higher Order Tensor Completion

In this paper, we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion. We introduce a class of tensor norms by allowing for different levels of coherence, which allows us to leverage the incoherence of a tensor. In particular, we show that a $k$th order tensor of rank $r$ and dimension $d\times\cdots\times d$ can be recovered perfectly from as few as $O((r^{(k-1)/2}d^{3/2}+r^{k-1}d)(\log(d))^2)$ uniformly sampled entries through an appropriate incoherent nuclear norm minimization. Our results demonstrate some key differences between completing a matrix and a higher order tensor: They not only point to potential room for improvement over the usual nuclear norm minimization but also highlight the importance of explicitly accounting for incoherence, when dealing with higher order tensors.

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arXiv2016

The Benefit of Group Sparsity in Group Inference with De-biased Scaled Group Lasso

We study confidence regions and approximate chi-squared tests for variable groups in high-dimensional linear regression. When the size of the group is small, low-dimensional projection estimators for individual coefficients can be directly used to construct efficient confidence regions and p-values for the group. However, the existing analyses of low-dimensional projection estimators do not directly carry through for chi-squared-based inference of a large group of variables without inflating the sample size by a factor of the group size. We propose to de-bias a scaled group Lasso for chi-squared-based statistical inference for potentially very large groups of variables. We prove that the proposed methods capture the benefit of group sparsity under proper conditions, for statistical inference of the noise level and variable groups, large and small. Such benefit is especially strong when the group size is large.

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arXiv2015

Lasso adjustments of treatment effect estimates in randomized experiments

We provide a principled way for investigators to analyze randomized experiments when the number of covariates is large. Investigators often use linear multivariate regression to analyze randomized experiments instead of simply reporting the difference of means between treatment and control groups. Their aim is to reduce the variance of the estimated treatment effect by adjusting for covariates. If there are a large number of covariates relative to the number of observations, regression may perform poorly because of overfitting. In such cases, the Lasso may be helpful. We study the resulting Lasso-based treatment effect estimator under the Neyman-Rubin model of randomized experiments. We present theoretical conditions that guarantee that the estimator is more efficient than the simple difference-of-means estimator, and we provide a conservative estimator of the asymptotic variance, which can yield tighter confidence intervals than the difference-of-means estimator. Simulation and data examples show that Lasso-based adjustment can be advantageous even when the number of covariates is less than the number of observations. Specifically, a variant using Lasso for selection and OLS for estimation performs particularly well, and it chooses a smoothing parameter based on combined performance of Lasso and OLS.

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arXiv2015

Asymptotic normality and optimalities in estimation of large Gaussian graphical models

The Gaussian graphical model, a popular paradigm for studying relationship among variables in a wide range of applications, has attracted great attention in recent years. This paper considers a fundamental question: When is it possible to estimate low-dimensional parameters at parametric square-root rate in a large Gaussian graphical model? A novel regression approach is proposed to obtain asymptotically efficient estimation of each entry of a precision matrix under a sparseness condition relative to the sample size. When the precision matrix is not sufficiently sparse, or equivalently the sample size is not sufficiently large, a lower bound is established to show that it is no longer possible to achieve the parametric rate in the estimation of each entry. This lower bound result, which provides an answer to the delicate sample size question, is established with a novel construction of a subset of sparse precision matrices in an application of Le Cam's lemma. Moreover, the proposed estimator is proven to have optimal convergence rate when the parametric rate cannot be achieved, under a minimal sample requirement. The proposed estimator is applied to test the presence of an edge in the Gaussian graphical model or to recover the support of the entire model, to obtain adaptive rate-optimal estimation of the entire precision matrix as measured by the matrix $\ell_q$ operator norm and to make inference in latent variables in the graphical model. All of this is achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum. This significantly relaxes the commonly imposed uniform signal strength condition on the precision matrix, irrepresentability condition on the Hessian tensor operator of the covariance matrix or the $\ell_1$ constraint on the precision matrix. Numerical results confirm our theoretical findings. The ROC curve of the proposed algorithm, Asymptotic Normal Thresholding (ANT), for support recovery significantly outperforms that of the popular GLasso algorithm.

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