Lei Zhang

Qinchen Song, Lei Zhang, Min Tang

When solving the time-dependent radiative transport equation (RTE), implicit time discretization is often employed for its robustness and stability. This results in a sequence of steady-state RTEs with identical cross-sections but varying source terms, whose repeated solution is computationally costly. To address this, we first apply the adaptive tailored finite point scheme (TFPS) for spatial discretization. This scheme exploits prior knowledge of the background media's optical properties to adaptively compress the angular domain, constructing a compressed linear system. A key feature is its ability to reconstruct the layer structure after compression, faithfully capturing the variance at the layer. We then use the Recursive Skeleton Method (RSM) to obtain an explicit multilevel decomposition of the inverse discrete operator, which is reused for all steady-state solutions. Numerical experiments show that our framework achieves high accuracy and significant efficiency across diverse scenarios.

Ying Sun, Hongwei Yong, Lei Zhang

Due to the effectiveness of second-order algorithms in solving classical optimization problems, designing second-order optimizers to train deep neural networks (DNNs) has attracted much research interest in recent years. However, because of the very high dimension of intermediate features in DNNs, it is difficult to directly compute and store the Hessian matrix for network optimization. Most of the previous second-order methods approximate the Hessian information imprecisely, resulting in unstable performance. In this work, we propose a compound optimizer, which is a combination of a second-order optimizer with a precise partial Hessian matrix for updating channel-wise parameters and the first-order stochastic gradient descent (SGD) optimizer for updating the other parameters. We show that the associated Hessian matrices of channel-wise parameters are diagonal and can be extracted directly and precisely from Hessian-free methods. The proposed method, namely SGD with Partial Hessian (SGD-PH), inherits the advantages of both first-order and second-order optimizers. Compared with first-order optimizers, it adopts a certain amount of information from the Hessian matrix to assist optimization, while compared with the existing second-order optimizers, it keeps the good generalization performance of first-order optimizers. Experiments on image classification tasks demonstrate the effectiveness of our proposed optimizer SGD-PH. The code is publicly available at \url{https://github.com/myingysun/SGDPH}.

Qinshuo Liu, Zixin Wang, Xi-An Li et al.

Semiparametric statistics play a pivotal role in a wide range of domains, including but not limited to missing data, causal inference, and transfer learning, to name a few. In many settings, semiparametric theory leads to (nearly) statistically optimal procedures that yet involve numerically solving Fredholm integral equations of the second kind. Traditional numerical methods, such as polynomial or spline approximations, are difficult to scale to multi-dimensional problems. Alternatively, statisticians may choose to approximate the original integral equations by ones with closed-form solutions, resulting in computationally more efficient, but statistically suboptimal or even incorrect procedures. To bridge this gap, we propose a novel framework by formulating the semiparametric estimation problem as a bi-level optimization problem; and then we develop a scalable algorithm called Deep Neural-Nets Assisted Semiparametric Estimation (DNA-SE) by leveraging the universal approximation property of Deep Neural-Nets (DNN) to streamline semiparametric procedures. Through extensive numerical experiments and a real data analysis, we demonstrate the numerical and statistical advantages of $\dnase$ over traditional methods. To the best of our knowledge, we are the first to bring DNN into semiparametric statistics as a numerical solver of integral equations in our proposed general framework.