Shuting Shen

We derive novel anti-concentration bounds for the difference between the maximal values of two Gaussian random vectors across various settings. Our bounds are dimension-free, scaling with the dimension of the Gaussian vectors only through the smaller expected maximum of the Gaussian subvectors. In addition, our bounds hold under the degenerate covariance structures, which previous results do not cover. In addition, we show that our conditions are sharp under the homogeneous component-wise variance setting, while we only impose some mild assumptions on the covariance structures under the heterogeneous variance setting. We apply the new anti-concentration bounds to derive the central limit theorem for the maximizers of discrete empirical processes. Finally, we back up our theoretical findings with comprehensive numerical studies.

Principal component analysis (PCA) is one of the most popular methods for dimension reduction. In light of rapidly increasing large-scale data in federated ecosystems, where data cannot leave individual warehouses, such as banks and healthcare systems, the traditional PCA method is often not applicable due to privacy protection considerations and large computational burden. Fast PCA algorithms have been proposed to lower the computational cost for large-scale data, but they cannot handle federated data. Distributed PCA algorithms have been developed to handle federated data by applying traditional PCA to data at each site and aggregating site-specific PCA results. However, they are not computationally efficient and not scalable when data at each site are large with many samples and variables, such as biobanks. In this paper, we propose the FAst DIstributed (FADI) PCA method that performs PCA analysis of large federated data with high computational efficiency and low statistical error without the need of sharing the data across sites. Specifically, FADI applies fast PCA to site-specific data using multiple random sketches and aggregates the fast PCA results across sites. We perform a non-asymptotic theoretical study to show that under some regularity conditions, FADI enjoys the same error rate as the traditional full sample PCA and a significantly smaller order of computational burden compared to the existing methods. We perform extensive simulation studies to compare the finite sample performance of FADI with the existing algorithms, and show that FADI substantially outperforms the other methods in computational efficiency without sacrificing statistical accuracy. We apply FADI to the analysis of the 1000 Genomes data to study the population structure.

Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.

In this paper, we propose an inferential framework testing the general community combinatorial properties of the stochastic block model. We aim to test the hypothesis on whether a certain community property is satisfied and provide p-values to assess the statistical uncertainty. For instance, we test whether a given set of nodes belong to the same community. We present a general framework applicable to all symmetric community properties. To ease the challenges caused by the combinatorial nature of communities properties, we develop a novel shadowing bootstrap testing method. Utilizing the symmetry, our method can find a shadowing representative of the true assignment and the number of assignments to be tested in the alternative can be largely reduced. In theory, we introduce a combi- natorial distance between two community classes and show a combinatorial-probabilistic trade-off phenomenon. Our test is honest as long as the product of the combinatorial distance between two communities and the probabilistic distance between two assignment probabilities is sufficiently large. Besides, we show that such trade-off also exists in the information-theoretic lower bound of the community property test. We also implement numerical experiments on both the synthetic data and the protein interaction application to show the validity of our method.