This article considers experimental design based on the strategy of rerandomization to increase the efficiency in experiments. Two aspects of rerandomization are addressed. First, we propose a two-stage allocation sample scheme for randomization inference to the units in experiments that guarantees that the difference-in-mean estimator is an unbiased estimator of the sample average treatment effect for any experiment, conserves the exactness of randomization inference, and halves the time consumption of the rerandomization design. Second, we propose a rank-based covariate-balance measure which can take into account the estimated relative weight of each covariate. Several strategies for estimating these weights using pre-experimental data are proposed. Using Monte Carlo simulations, the proposed strategies are compared to complete randomization and Mahalanobis-based rerandomization. An empirical example is given where the power of a mean difference test of electricity consumption of 54 households is increased by 99%, in comparison to complete randomization, using one of the proposed designs based on high frequency longitudinal electricity consumption data. Supplementary materials for this article are available online.

Recently a computational-based experimental design strategy called rerandomization has been proposed as an alternative or complement to traditional blocked designs. The idea of rerandomization is to remove, from consideration, those allocations with large imbalances in observed covariates according to a balance criterion, and then randomize within the set of acceptable allocations. Based on the Mahalanobis distance criterion for balancing the covariates, we show that asymptotic inference to the population, from which the units in the sample are randomly drawn, is possible using only the set of best, or ‘optimal’, allocations. Finally, we show that for the optimal and near optimal designs, the quite complex asymptotic sampling distribution derived by Li et al. (2018), is well approximated by a normal distribution.

Blocking is commonly used in randomized experiments to increase efficiency of estimation. A generalization of blocking removes allocations with imbalance in covariate distributions between treated and control units, and then randomizes within the remaining set of allocations with balance. This idea of rerandomization was formalized by Morgan and Rubin (Annals of Statistics, 2012, 40, 1263–1282), who suggested using Mahalanobis distance between treated and control covariate means as the criterion for removing unbalanced allocations. Kallus (Journal of the Royal Statistical Society, Series B: Statistical Methodology, 2018, 80, 85–112) proposed reducing the set of balanced allocations to the minimum. Here we discuss the implication of such an ‘optimal’ rerandomization design for inferences to the units in the sample and to the population from which the units in the sample were randomly drawn. We argue that, in general, it is a bad idea to seek the optimal design for an inference because that inference typically only reflects uncertainty from the random sampling of units, which is usually hypothetical, and not the randomization of units to treatment versus control.

In deconvolution in R^d, d≥1, with mixing density p(∈\mathscr{P}) and kernel h, the mixture density f_p(∈\mathscr{F}_p) is estimated with MDE f_{\hat{p}n}, having upper L1-error rate, an, in probability or in risk; \hat{p}_n∈\mathscr{P}. In one application, \mathscr{P} consists of L1-separable densities in R with differences changing sign at most J times and h(x−y) Totally Positive. When h is known and p is \tilde{q}-smooth, vanishing outside a compact in R^d, plug-in upper bounds are provided for the L2-error rate of \hat{p}_n and its [s]-th mixed partial derivative \hat{p}^{(s)}_n, via ∥f_{\hat{p}_n}−f_p∥_1, with rates (log a^{−1}_n)^{−N_1} and a^{N_2}_n, respectively, for h super-smooth and smooth; \tilde{q}∈R^+, [s] ≤ \tilde{q}, d≥1, N_1>0, N_2>0. For a_n ∼ (logn)^ζ⋅(n^{−δ}), the former rate is optimal for any δ>0 and the latter misses the optimal by the factor (logn)^ξ when δ=.5; ζ>0, ξ>0. N_1 and N_2 appear in optimal rates and lower error and risk bounds in the deconvolution literature.

Double-blind randomized controlled trials are traditionally seen as the gold standard for causal inferences as the difference-in-means estimator is an unbiased estimator of the average treatment effect in the experiment. The fact that this estimator is unbiased over all possible randomizations does not, however, mean that any given estimate is close to the true treatment effect. Similarly, while predetermined covariates will be balanced between treatment and control groups on average, large imbalances may be observed in a given experiment and the researcher may therefore want to condition on such covariates using linear regression. This article studies the theoretical properties of both the difference-in-means and OLS estimators conditional on observed differences in covariates. By deriving the statistical properties of the conditional estimators, we can establish guidance for how to deal with covariate imbalances.

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.

We propose a deep learning-based method for object detection in UAV-borne thermal images that have the capability of observing scenes in both day and night. Compared with visible images, thermal images have lower requirements for illumination conditions, but they typically have blurred edges and low contrast. Using a boundary-aware salient object detection network, we extract the saliency maps of the thermal images to improve the distinguishability. Thermal images are augmented with the corresponding saliency maps through channel replacement and pixel-level weighted fusion methods. Considering the limited computing power of UAV platforms, a lightweight combinational neural network ComNet is used as the core object detection method. The YOLOv3 model trained on the original images is used as a benchmark and compared with the proposed method. In the experiments, we analyze the detection performances of the ComNet models with different image fusion schemes. The experimental results show that the average precisions (APs) for pedestrian and vehicle detection have been improved by 2%~5% compared with the benchmark without saliency map fusion and MobileNetv2. The detection speed is increased by over 50%, while the model size is reduced by 58%. The results demonstrate that the proposed method provides a compromise model, which has application potential in UAV-borne detection tasks.

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We investigate the asymptotic distributions of coordinates of regression M-estimates in the moderate p/n regime, where the number of covariates p grows proportionally with the sample size n. Under appropriate regularity conditions, we establish the coordinate-wise asymptotic normality of regression M-estimates assuming a fixed-design matrix. Our proof is based on the second-order Poincare inequality (Chatterjee, 2009) and leave-one-out analysis (El Karoui et al., 2011). Some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices. We also show a counterexample, namely the ANOVA-type design, to emphasize that the technical assumptions are not just artifacts of the proof. Finally, the numerical experiments confirm and complement our theoretical results.

We consider the fundamental problem of solving quadratic systems of m equations in n variables. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach. There are several key distinguishing features, most notably, a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e. in time proportional to reading the data and as soon as the ratio m/n between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems and prove that our algorithms achieve a statistical accuracy, which is nearly un-improvable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size---hence the title of this paper. For instance, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least-squares problem of the same size.

Community detection is a fundamental problem in social network analysis. It is desirable to have fast community detection algorithm that applicable to large-scale networks we see today and also have reliable community detection results. This paper proposes Spectral Clustering On Ratios-of-eigenvectors (SCORE) as such an algorithm. The algorithm is fast in computation, is applicable to real large networks, and is competitive in real data performance when applied to some representative data sets in this area. We study the algorithm with the Degree-Corrected Block Model (DCBM). Compare to most works in this area that focus on the much narrower Block Models, our analysis is more difficult and requires new techniques. The novelty of our algorithm lies in that it uses entry-wise ratios between leading eigenvectors of the adjacency to reduce a seemingly hard problem to a problem that is very much tractable.

Cryo-electron microscopy (cryo-EM) has recently emerged as a powerful tool for obtaining three-dimensional (3D) structures of biological macromolecules in native states. A minimum cryo-EM image data set for deriving a meaningful reconstruction is comprised of thousands of randomly orientated projections of identical particles photographed with a small number of electrons. The computation of 3D structure from 2D projections requires clustering, which aims to enhance the signal to noise ratio in each view by grouping similarly oriented images. Nevertheless, the prevailing clustering techniques are often compromised by three characteristics of cryo-EM data: high noise content, high dimensionality and large number of clusters. Moreover, since clustering requires registering images of similar orientation into the same pixel coordinates by 2D alignment, it is desired that the clustering algorithm can label misaligned images as outliers. Herein, we introduce a clustering algorithm γ-SUP to model the data with a q-Gaussian mixture and adopt the minimum γ-divergence for estimation, and then use a self-updating procedure to obtain the numerical solution. We apply γ-SUP to the cryo-EM images of two benchmark macromolecules, RNA polymerase II and ribosome. In the former case, simulated images were chosen to decouple clustering from alignment to demonstrate γ-SUP is more robust to misalignment outliers than the existing clustering methods used in the cryo-EM community. In the latter case, the clustering of real cryo-EM data by our γ-SUP method eliminates noise in many views to reveal true structure features of ribosome at the projection level.

We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments, and dependence structure. Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given. Some previous results about specific BLM distributions are improved. In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters. Besides, we point out that Slepian’s inequality also holds true for BLM distributions.

This paper is about the propagation of the singularities in the solutions to the Cauchy problem of the spatially inhomogeneous Boltzmann equation with angular cutoff assumption. It is motivated by the work of BoudinDesvillettes on the propagation of singularities in solutions near vacuum. It shows that for the solution near a global Maxwellian, singularities in the initial data propagate like the free transportation. Precisely, the solution is the sum of two parts in which one keeps the singularities of the initial data and the other one is regular with locally bounded derivatives of fractional order in some Sobolev space. In addition, the dependence of the regularity on the cross-section is also given.

In this paper, we provide the O() corrections to the hydrodynamic model derived by Degond and Motsch from a kinetic version of the model by Vicsek and co-authors describing flocking biological agents. The parameter stands for the ratio of the microscopic to the macroscopic scales. The O() corrected model involves diffusion terms in both the mass and velocity equations as well as terms which are quadratic functions of the first-order derivatives of the density and velocity. The derivation method is based on the standard ChapmanEnskog theory, but is significantly more complex than usual due to both the non-isotropy of the fluid and the lack of momentum conservation.

The approach combines second and fourth order statistics to perform BSS of instantaneous mixtures. It applies for any number of receivers if they are as many as sources. It is a batch algorithm that uses non-Gaussianity and stationarity of source signals. It is linear algebra based direct method, reliable and robust, though large dimensions of sources may slow down the computation significantly. It is however limited to instantaneous mixtures.