Definitive screening designs (DSDs) are widely used for studying quantitative factors. However, DSDs constructed from different conference matrices are not equally good. We show DSDs using Paley’s conference matrices guarantee desirable performance (either optimal or near-optimal) under several important criteria.

We consider the edge statistics of large dimensional deformed rectangular matrices of the form $Y_t=Y+\sqrt{t}X,$ where $Y$ is a $p \times n$ deterministic signal matrix whose rank is comparable to $n$, $X$ is a $p\times n$ random noise matrix with centered i.i.d. entries with variance $n^{-1}$, and $t>0$ gives the noise level. This model is referred to as the interference-plus-noise matrix in the study of massive multiple-input multiple-output (MIMO) system, which belongs to the category of the so-called signal-plus-noise model. For the case $t=1$, the spectral statistics of this model have been studied to a certain extent in the literature \cite{DOZIER20071099,DOZIER2007678,VLM2012}. In this paper, we study the singular value and singular vector statistics of $Y_t$ around the right-most edge of the singular value spectrum in the harder regime $n^{-2/3}\ll t \ll 1$. This regime is harder than the $t=1$ case, because on one hand, the edge behavior of the empirical spectral distribution (ESD) of $YY^\top$ has a strong effect on the edge statistics of $Y_tY_t^\top$ since $t\ll 1$ is ``small", while on the other hand, the edge statistics of $Y_t$ is also not merely a perturbation of those of $Y$ since $t\gg n^{-2/3}$ is ``large". Under certain regularity assumptions on $Y,$ we prove the edge universality, eigenvalues rigidity and eigenvector delocalization for the matrices $Y_tY_t^\top$ and $Y_t^\top Y_t$. These results can be used to estimate and infer the massive MIMO system. To prove the main results, we analyze the edge behavior of the asymptotic ESD for $Y_tY_t^\top$, and establish some sharp local laws on the resolvent of $Y_tY_t^\top$. These results can be of independent interest, and used as useful inputs for many other problems regarding the spectral statistics of $Y_t$.

Large dimensional Gram type matrices are common objects in high-dimensional statistics and machine learning. In this paper, we study the limiting distribution of the edge eigenvalues for a general class of high-dimensional Gram type random matrices, including separable sample covariance matrices, sparse sample covariance matrices, bipartite stochastic block model and random Gram matrices with general variance profiles. Specifically, we prove that under (almost) sharp moment conditions and certain tractable regularity assumptions, the edge eigenvalues, i.e., the largest few eigenvalues of non-spiked Gram type random matrices or the extremal bulk eigenvalues of spiked Gram type random matrices, satisfy the Tracy-Widom distribution asymptotically. Our results can be used to construct adaptive, accurate and powerful statistics for high-dimensional statistical inference. In particular, we propose data-dependent statistics to infer the number of signals under general noise structure, test the one-sided sphericity of separable matrix, and test the structure of bipartite stochastic block model. Numerical simulations show strong support of our proposed statistics. The core of our proof is to establish the edge universality and Tracy-Widom distribution for a rectangular Dyson Brownian motion with regular initial data. This is a general strategy to study the edge statistics for high-dimensional Gram type random matrices without exploring the specific independence structure of the target matrices. It has potential to be applied to more general random matrices that are beyond the ones considered in this paper.

In this paper, we prove a CLT for the sample canonical correlation coefficients between two high-dimensional random vectors with finite rank correlations. More precisely, consider two random vectors $\wt{\bx}=\mathbf x + A \mathbf z $ and $\wt{\by}=\mathbf y + B \mathbf z $, where $\mathbf x \in \R^p$, $\mathbf y \in \R^q$ and $\mathbf z\in \R^r$ are independent random vectors with i.i.d.\;entries of mean zero and variance one, and $A \in \R^{p\times r}$ and $B\in \R^{q\times r}$ are two arbitrary deterministic matrices. Given $n$ samples of $\wt{\bx}$ and $\wt{\by}$, we stack them into two matrices $\cal X= X+AZ$ and $\cal Y= Y+BZ$, where $X\in \R^{p\times n}$, $Y\in \R^{q\times n}$ and $Z\in \R^{r\times n}$ are random matrices with i.i.d.\;entries of mean zero and variance one. Let $\wt\lambda_1 \ge \wt\lambda_2\ge \cdots \ge \wt\lambda_{r}$ be the largest $r$ eigenvalues of the sample canonical correlation (SCC) matrix $\cal C_{\cal X\cal Y}=(\cal X\cal X^\top)^{-1/2}\cal X\cal Y^\top (\cal Y\cal Y^\top)^{-1}\cal Y \cal X^\top (\cal X\cal X^\top)^{-1/2}$, and let $t_1\ge t_2 \ge \cdots\ge t_r$ be the squares of the population canonical correlation coefficients between $\wt{\bx}$ and $\wt{\by}$. Under certain moment assumptions, we show that there exists a threshold $t_c \in(0, 1)$ such that if $t_i>t_c$, then \smash{$\sqrt{n} (\wt\lambda_i-\theta_i)$} converges weakly to a centered normal distribution, where $\theta_i $ is a fixed outlier location determined by $t_i$. Our proof uses a self-adjoint linearization of the SCC matrix and a sharp local law on the inverse of the linearized matrix.

Consider two random vectors $\wt{\bx} \in \R^p$ and $\wt{\by} \in \mathbb R^q$ of the forms $\wt{\bx}=A \mathbf z + \bC_1^{1/2}\mathbf x $ and $\wt{\by}=B \mathbf z + \bC_2^{1/2}\mathbf y $, where $\mathbf x \in \R^p$, $\mathbf y \in \R^q$ and $\mathbf z\in \R^r$ are independent random vectors with i.i.d. entries of zero mean and unit variance, $\bC_1$ and $\bC_2$ are $p \times p$ and $q\times q$ deterministic population covariance matrices, and $A$ and $B$ are $p \times r$ and $q\times r$ deterministic factor loading matrices. With $n$ independent observations of $(\wt{\mathbf x},\wt{\mathbf y})$, we study the sample canonical correlations between $\wt{\bx} $ and $\wt{\by}$. We consider the high-dimensional setting with finite rank correlations, that is, ${p}/{n}\to c_1$ and ${q}/{n}\to c_2$ as $n\to \infty$ for some constants $c_1\in (0,1)$ and $c_2\in (0,1-c_1)$, and $r$ is a fixed integer. Let $t_1\ge t_2 \ge \cdots\ge t_r\ge 0$ be the squares of the nontrivial population canonical correlation coefficients between $\wt {\bx}$ and $\wt{\by}$, and let $\wt\lambda_1 \ge \wt\lambda_2\ge \cdots \ge \wt\lambda_{p\wedge q}\ge 0$ be the squares of the sample canonical correlation coefficients. If the entries of $\mathbf x$, $\mathbf y$ and $\mathbf z$ are i.i.d. Gaussian, then the following dichotomy has been shown in \cite{CCA} for a fixed threshold $t_c \in(0, 1)$: for any $1\le i \le r$, if $t_i < t_c$, then $\wt\lambda_i$ converges to the right-edge $\lambda_+$ of the limiting eigenvalue spectrum of the sample canonical correlation matrix, and moreover, $n^{2/3}(\wt\lambda_i-\lambda_+)$ converges weakly to the Tracy-Widom law; if $t_i>t_c$, then $\wt\lambda_i$ converges to a deterministic limit $\theta_i \in (\lambda_+, 1)$ that is determined by $c_1$, $c_2$ and $t_i$. In this paper, we prove that these results hold universally under the sharp fourth moment conditions on the entries of $\mathbf x$, $\mathbf y$ and $\mathbf z$. Moreover, we prove the results in full generality, in the sense that they also hold for near-degenerate $t_i$'s and for $t_i$'s that are close to the threshold $t_c$. Finally, we also provide almost sharp convergence rates for the sample canonical correlation coefficients under a general $a$-th moment assumption.