A prototype knockoff filter for group selection with FDR control

In many applications, we need to study a linear regression model that consists of a response variable and a large number of potential explanatory variables, and determine which variables are truly associated with the response. In Foygel Barber & Candès (2015, Ann. Statist., 43, 2055–2085), the authors introduced a new variable selection procedure called the knockoff filter to control the false discovery rate (FDR) and proved that this method achieves exact FDR control. In this paper, we propose a prototype knockoff filter for group selection by extending the Reid–Tibshirani (2016, Biostatistics, 17, 364–376) prototype method. Our prototype knockoff filter improves the computational efficiency and statistical power of the Reid–Tibshirani prototype method when it is applied for group selection. In some cases when the group features are spanned by one or a few hidden factors, we demonstrate that the Principal Component Analysis (PCA) prototype knockoff filter outperforms the Dai–Foygel Barber (2016, 33rd International Conference on Machine Learning (ICML 2016)) group knockoff filter. We present several numerical experiments to compare our prototype knockoff filter with the Reid–Tibshirani prototype method and the group knockoff filter. We have also conducted some analysis of the knockoff filter. Our analysis reveals that some knockoff path method statistics, including the Lasso path statistic, may lead to loss of power for certain design matrices and a specially designed response even if their signal strengths are still relatively strong

On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equation

We present a novel method of analysis and prove finite time self-similar blowup of the original De Gregorio model [DG90,DG96] for smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model [OSW08] for the entire range of parameter on R or S¹ for Hölder continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions and take into account cancellation among various nonlinear terms to extract the inviscid damping effect from the linearized operator around the approximate self-similar profile. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations

Finite time blowup of 2D Boussinesq and 3D Euler equations with C1,αC^{1,\alpha} velocity and boundary

Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou \cite{luo2013potentially-1,luo2013potentially-2} and the recent breakthrough by Elgindi \cite{elgindi2019finite} on the singularity formation of the 3D Euler equation without swirl with $C^{1,\alpha}$ initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with $C^{1,\alpha}$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in \cite{luo2013potentially-1,luo2013potentially-2} share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $C^{1,\alpha}$ initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in \cite{elgindi2019finite}. We also use some strategy proposed in our recent joint work with Huang in \cite{chen2019finite} and adopt several methods of analysis in \cite{elgindi2019finite} to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with $C^{1,\alpha}$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time.Comment: Revised introduction. Added motivations and discussions in Sections 5, 9. 82 Page

Reasoning over Hierarchical Question Decomposition Tree for Explainable Question Answering

Explainable question answering (XQA) aims to answer a given question and provide an explanation why the answer is selected. Existing XQA methods focus on reasoning on a single knowledge source, e.g., structured knowledge bases, unstructured corpora, etc. However, integrating information from heterogeneous knowledge sources is essential to answer complex questions. In this paper, we propose to leverage question decomposing for heterogeneous knowledge integration, by breaking down a complex question into simpler ones, and selecting the appropriate knowledge source for each sub-question. To facilitate reasoning, we propose a novel two-stage XQA framework, Reasoning over Hierarchical Question Decomposition Tree (RoHT). First, we build the Hierarchical Question Decomposition Tree (HQDT) to understand the semantics of a complex question; then, we conduct probabilistic reasoning over HQDT from root to leaves recursively, to aggregate heterogeneous knowledge at different tree levels and search for a best solution considering the decomposing and answering probabilities. The experiments on complex QA datasets KQA Pro and Musique show that our framework outperforms SOTA methods significantly, demonstrating the effectiveness of leveraging question decomposing for knowledge integration and our RoHT framework.Comment: has been accepted by ACL202