Zhiqiang Cai

Zhiqiang Cai, Tong Ding, Min Liu et al.

In this paper, we propose a structure-guided Gauss-Newton (SgGN) method for solving least squares problems using a shallow ReLU neural network. The method effectively takes advantage of both the least squares structure and the neural network structure of the objective function. By categorizing the weights and biases of the hidden and output layers of the network as nonlinear and linear parameters, respectively, the method iterates back and forth between the nonlinear and linear parameters. The nonlinear parameters are updated by a damped Gauss-Newton method and the linear ones are updated by a linear solver. Moreover, at the Gauss-Newton step, a special form of the Gauss-Newton matrix is derived for the shallow ReLU neural network and is used for efficient iterations. It is shown that the corresponding mass and Gauss-Newton matrices in the respective linear and nonlinear steps are symmetric and positive definite under reasonable assumptions. Thus, the SgGN method naturally produces an effective search direction without the need of additional techniques like shifting in the Levenberg-Marquardt method to achieve invertibility of the Gauss-Newton matrix. The convergence and accuracy of the method are demonstrated numerically for several challenging function approximation problems, especially those with discontinuities or sharp transition layers that pose significant challenges for commonly used training algorithms in machine learning.

Jialin Chen, Zhiqiang Cai, Ke Xu et al.

Considering the noise level limit, one crucial aspect for quantum machine learning is to design a high-performing variational quantum circuit architecture with small number of quantum gates. As the classical neural architecture search (NAS), quantum architecture search methods (QAS) employ methods like reinforcement learning, evolutionary algorithms and supernet optimiza-tion to improve the search efficiency. In this paper, we propose a novel qubit-wise architec-ture search (QWAS) method, which progres-sively search one-qubit configuration per stage, and combine with Monte Carlo Tree Search al-gorithm to find good quantum architectures by partitioning the search space into several good and bad subregions. The numerical experimental results indicate that our proposed method can balance the exploration and exploitation of cir-cuit performance and size in some real-world tasks, such as MNIST, Fashion and MOSI. As far as we know, QWAS achieves the state-of-art re-sults of all tasks in the terms of accuracy and circuit size.

Zhiqiang Cai, Jingshuang Chen, Min Liu

We introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the method outperforms mesh-based numerical methods in terms of the number of degrees of freedom. This paper studies the LSNN method for scalar nonlinear hyperbolic conservation law. The method is a discretization of an equivalent least-squares (LS) formulation in the set of neural network functions with the ReLU activation function. Evaluation of the LS functional is done by using numerical integration and conservative finite volume scheme. Numerical results of some test problems show that the method is capable of approximating the discontinuous interface of the underlying problem automatically through the free breaking lines of the ReLU neural network. Moreover, the method does not exhibit the common Gibbs phenomena along the discontinuous interface.

Tai Wang, Xiangen Hu, Keith Shubeck et al.

The relationship between reading and writing (RRW) is one of the major themes in learning science. One of its obstacles is that it is difficult to define or measure the latent background knowledge of the individual. However, in an academic research setting, scholars are required to explicitly list their background knowledge in the citation sections of their manuscripts. This unique opportunity was taken advantage of to observe RRW, especially in the published academic commentary scenario. RRW was visualized under a proposed topic process model by using a state of the art version of latent Dirichlet allocation (LDA). The empirical study showed that the academic commentary is modulated both by its target paper and the author's background knowledge. Although this conclusion was obtained in a unique environment, we suggest its implications can also shed light on other similar interesting areas, such as dialog and conversation, group discussion, and social media.