Wen-Xin Zhou

Matrix completion via max-norm constrained optimization

Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly when the underlying sampling scheme is non-uniform. In this paper we propose and analyze a max-norm constrained empirical risk minimization method for noisy matrix completion under a general sampling model. The optimal rate of convergence is established under the Frobenius norm loss in the context of approximately low-rank matrix reconstruction. It is shown that the max-norm constrained method is minimax rate-optimal and yields a unified and robust approximate recovery guarantee, with respect to the sampling distributions. The computational effectiveness of this method is also discussed, based on first-order algorithms for solving convex optimizations involving max-norm regularization.

Are discoveries spurious? Distributions of maximum spurious correlations and their applications

Over the last two decades, many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool. Can the discoveries by such data mining approaches be spurious due to high dimensionality and limited sample size? Can our fundamental assumptions on exogeneity of covariates needed for such variable selection be validated with the data? To answer these questions, we need to derive the distributions of the maximum spurious correlations given certain number of predictors, namely, the distribution of the correlation of a response variable Y with the best s linear combinations of p covariates X, even when X and Y are independent. When the covariance matrix of X possesses the restricted eigenvalue property, we derive such distributions for both finite s and diverging s, using Gaussian approximation and empirical process techniques. However, such a distribution depends on the unknown covariance matrix of X. Hence, we use the multiplier bootstrap procedure to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach. The results are further extended to the situation where residuals are from regularized fits. Our approach is then applied to construct the upper confidence limit for the maximum spurious correlation and testing exogeneity of covariates. The former provides a baseline for guarding against false discoveries due to data mining and the latter tests whether our fundamental assumptions for high-dimensional model selection are statistically valid. Our techniques and results are illustrated by both numerical examples and real data analysis.

Nonparametric covariate-adjusted regression

We consider nonparametric estimation of a regression curve when the data are observed with multiplicative distortion which depends on an observed confounding variable. We suggest several estimators, ranging from a relatively simple one that relies on restrictive assumptions usually made in the literature, to a sophisticated piecewise approach that involves reconstructing a smooth curve from an estimator of a constant multiple of its absolute value, and which can be applied in much more general scenarios. We show that, although our nonparametric estimators are constructed from predictors of the unobserved undistorted data, they have the same first order asymptotic properties as the standard estimators that could be computed if the undistorted data were available. We illustrate the good numerical performance of our methods on both simulated and real datasets.

Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices

Let

Max-norm optimization for robust matrix recovery

\mathbf {x}_1,\ldots,\mathbf {x}_n

Nonparametric and Parametric Estimators of Prevalence From Group Testing Data With Aggregated Covariates

be a random sample from a

Two-sample smooth tests for the equality of distributions

p

A new perspective on robust

-dimensional population distribution, where

M

p=p_n\to\infty

-estimation: Finite sample theory and applications to dependence-adjusted multiple testing

and