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A completely randomized experiment allows us to estimate the causal effect by the difference in the averages of the outcome under the treatment and control. But, difference-in-means type estimators behave poorly if the potential outcomes have a heavy-tail, or contain a few extreme observations or outliers. We study an alternative estimator by Rosenbaum that estimates the causal effect by inverting a randomization test using ranks. We study the asymptotic properties of this estimator and develop a framework to compare the efficiencies of different estimators of the treatment effect in the setting of randomized experiments. In particular, we show that the Rosenbaum estimator has variance that is asymptotically, in the worst case, at most about 1.16 times the variance of the difference-in-means estimator, and can be much smaller when the potential outcomes are not light-tailed. We further derive a consistent estimator of the asymptotic standard error for the Rosenbaum estimator which immediately yields a readily computable confidence interval for the treatment effect, thereby alleviating the expensive numerical calculations needed to implement the original proposal of Rosenbaum. Further, we propose a regression adjusted version of the Rosenbaum estimator to incorporate additional covariate information in randomization inference. We prove gain in efficiency by this regression adjustment method under a linear regression model. Finally, we illustrate through simulations that, unlike the difference-in-means based estimators, either unadjusted or regression adjusted, these rank-based estimators are efficient and robust against heavy-tailed distributions, contamination, and various model misspecifications.

This paper deals with the grouped variable selection problem. A widely used strategy is to equip the loss function with a sparsity-promoting penalty. Existing methods include the group Lasso, group SCAD, and group MCP. The group Lasso solves a convex optimization problem but is plagued by underestimation bias. The group SCAD and group MCP avoid the estimation bias but require solving a non-convex optimization problem that suffers from local optima. In this work, we propose an alternative method based on the generalized minimax concave (GMC) penalty, which is a folded concave penalty that can maintain the convexity of the objective function. We develop a new method for grouped variable selection in linear regression, the group GMC, that generalizes the strategy of the original GMC estimator. We present an efficient algorithm for computing the group GMC estimator. We also prove properties of the solution path to guide its numerical computation and tuning parameter selection in practice. We establish error bounds for both the group GMC and original GMC estimators. A rich set of simulation studies and a real data application indicate that the proposed group GMC approach outperforms existing methods in several different aspects under a wide array of scenarios.

The $k$-center problem is to choose a subset of size $k$ from a set of $n$ points such that the maximum distance from each point to its nearest center is minimized. Let $Q=\{Q_1,\ldots,Q_n\}$ be a set of polygons or segments in the region-based uncertainty model, in which each $Q_i$ is an uncertain point, where the exact locations of the points in $Q_i$ are unknown. The geometric objects segments and polygons can be models of a point set. We define the uncertain version of the $k$-center problem as a generalization in which the objective is to find $k$ points from $Q$ to cover the remaining regions of $Q$ with minimum or maximum radius of the cluster to cover at least one or all exact instances of each $Q_i$, respectively. We modify the region-based model to allow multiple points to be chosen from a region and call the resulting model the aggregated uncertainty model. All these problems contain the point version as a special case, so they are all NP-hard with a lower bound 1.822. We give approximation algorithms for uncertain $k$-center of a set of segments and polygons. We also have implemented some of our algorithms on a data-set to show our theoretical performance guarantees can be achieved in practice.

We present an hp-adaptive virtual element method (VEM) based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems. We introduce a reliable and efficient a posteriori error estimator, which is computed by solving an auxiliary global mixed problem. We show that the mixed VEM satisfies a discrete inf-sup condition, with inf-sup constant independent of the discretization parameters. Furthermore, we construct a stabilization for the mixed VEM, with explicit bounds in terms of the local degree of accuracy of the method. The theoretical results are supported by several numerical experiments, including a comparison with the residual a posteriori error estimator. The numerics exhibit the p-robustness of the proposed error estimator. In addition, we provide a first step towards the localized flux reconstruction in the virtual element framework, which leads to an additional reliable a posteriori error estimator that is computed by solving local (cheap-to-solve and parallelizable) mixed problems. We provide theoretical and numerical evidence that the proposed local error estimator suffers from a lack of efficiency.

The classical Ka\v{c}anov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we introduce a modified Ka\v{c}anov method, which allows for (adaptive) damping, and, thereby, to derive a new convergence analysis under more general assumptions and for a wider range of applications. For instance, in the specific context of quasilinear diffusion models, our new approach does no longer require a standard monotonicity condition on the nonlinear diffusion coefficient to hold. Moreover, we propose two different adaptive strategies for the practical selection of the damping parameters involved.

We study the problem of density estimation for a random vector ${\boldsymbol X}$ in $\mathbb R^d$ with probability density $f(\boldsymbol x)$. For a spanning tree $T$ defined on the vertex set $\{1,\dots ,d\}$, the tree density $f_{T}$ is a product of bivariate conditional densities. The optimal spanning tree $T^*$ is the spanning tree $T$, for which the Kullback-Leibler divergence of $f$ and $f_{T}$ is the smallest. From i.i.d. data we identify the optimal tree $T^*$ and computationally efficiently construct a tree density estimate $f_n$ such that, without any regularity conditions on the density $f$, one has that $\lim_{n\to \infty} \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x=0$ a.s. For Lipschitz continuous $f$ with bounded support, $\mathbb E\{ \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x\}=O(n^{-1/4})$.

The centrality in a network is a popular metric for agents' network positions and is often used in regression models to model the network effect on an outcome variable of interest. In empirical studies, researchers often adopt a two-stage procedure to first estimate the centrality and then infer the network effect using the estimated centrality. Despite its prevalent adoption, this two-stage procedure lacks theoretical backing and can fail in both estimation and inference. We, therefore, propose a unified framework, under which we prove the shortcomings of the two-stage in centrality estimation and the undesirable consequences in the regression. We then propose a novel supervised network centrality estimation (SuperCENT) methodology that simultaneously yields superior estimations of the centrality and the network effect and provides valid and narrower confidence intervals than those from the two-stage. We showcase the superiority of SuperCENT in predicting the currency risk premium based on the global trade network.

Heatmap-based methods dominate in the field of human pose estimation by modelling the output distribution through likelihood heatmaps. In contrast, regression-based methods are more efficient but suffer from inferior performance. In this work, we explore maximum likelihood estimation (MLE) to develop an efficient and effective regression-based methods. From the perspective of MLE, adopting different regression losses is making different assumptions about the output density function. A density function closer to the true distribution leads to a better regression performance. In light of this, we propose a novel regression paradigm with Residual Log-likelihood Estimation (RLE) to capture the underlying output distribution. Concretely, RLE learns the change of the distribution instead of the unreferenced underlying distribution to facilitate the training process. With the proposed reparameterization design, our method is compatible with off-the-shelf flow models. The proposed method is effective, efficient and flexible. We show its potential in various human pose estimation tasks with comprehensive experiments. Compared to the conventional regression paradigm, regression with RLE bring 12.4 mAP improvement on MSCOCO without any test-time overhead. Moreover, for the first time, especially on multi-person pose estimation, our regression method is superior to the heatmap-based methods. Our code is available at //github.com/Jeff-sjtu/res-loglikelihood-regression

Knowledge graph embedding aims at modeling entities and relations with low-dimensional vectors. Most previous methods require that all entities should be seen during training, which is unpractical for real-world knowledge graphs with new entities emerging on a daily basis. Recent efforts on this issue suggest training a neighborhood aggregator in conjunction with the conventional entity and relation embeddings, which may help embed new entities inductively via their existing neighbors. However, their neighborhood aggregators neglect the unordered and unequal natures of an entity's neighbors. To this end, we summarize the desired properties that may lead to effective neighborhood aggregators. We also introduce a novel aggregator, namely, Logic Attention Network (LAN), which addresses the properties by aggregating neighbors with both rules- and network-based attention weights. By comparing with conventional aggregators on two knowledge graph completion tasks, we experimentally validate LAN's superiority in terms of the desired properties.

Non-maximum suppression (NMS) is essential for state-of-the-art object detectors to localize object from a set of candidate locations. However, accurate candidate location sometimes is not associated with a high classification score, which leads to object localization failure during NMS. In this paper, we introduce a novel bounding box regression loss for learning bounding box transformation and localization variance together. The resulting localization variance exhibits a strong connection to localization accuracy, which is then utilized in our new non-maximum suppression method to improve localization accuracy for object detection. On MS-COCO, we boost the AP of VGG-16 faster R-CNN from 23.6% to 29.1% with a single model and nearly no additional computational overhead. More importantly, our method is able to improve the AP of ResNet-50 FPN fast R-CNN from 36.8% to 37.8%, which achieves state-of-the-art bounding box refinement result.