Classification of Data Generated by Gaussian Mixture Models Using Deep ReLU Networks

This paper studies the binary classification of unbounded data from ${\mathbb R}^d$ generated under Gaussian Mixture Models (GMMs) using deep ReLU neural networks. We obtain \unicode{x2013} for the first time \unicode{x2013} non-asymptotic upper bounds and convergence rates of the excess risk (excess misclassification error) for the classification without restrictions on model parameters. The convergence rates we derive do not depend on dimension $d$, demonstrating that deep ReLU networks can overcome the curse of dimensionality in classification. While the majority of existing generalization analysis of classification algorithms relies on a bounded domain, we consider an unbounded domain by leveraging the analyticity and fast decay of Gaussian distributions. To facilitate our analysis, we give a novel approximation error bound for general analytic functions using ReLU networks, which may be of independent interest. Gaussian distributions can be adopted nicely to model data arising in applications, e.g., speeches, images, and texts; our results provide a theoretical verification of the observed efficiency of deep neural networks in practical classification problems

Learning Ability of Interpolating Deep Convolutional Neural Networks

It is frequently observed that overparameterized neural networks generalize well. Regarding such phenomena, existing theoretical work mainly devotes to linear settings or fully-connected neural networks. This paper studies the learning ability of an important family of deep neural networks, deep convolutional neural networks (DCNNs), under both underparameterized and overparameterized settings. We establish the first learning rates of underparameterized DCNNs without parameter or function variable structure restrictions presented in the literature. We also show that by adding well-defined layers to a non-interpolating DCNN, we can obtain some interpolating DCNNs that maintain the good learning rates of the non-interpolating DCNN. This result is achieved by a novel network deepening scheme designed for DCNNs. Our work provides theoretical verification of how overfitted DCNNs generalize well

QAOA with fewer qubits: a coupling framework to solve larger-scale Max-Cut problem

Maximum cut (Max-Cut) problem is one of the most important combinatorial optimization problems because of its various applications in real life, and recently Quantum Approximate Optimization Algorithm (QAOA) has been widely employed to solve it. However, as the size of the problem increases, the number of qubits required will become larger. With the aim of saving qubits, we propose a coupling framework for designing QAOA circuits to solve larger-scale Max-Cut problem. This framework relies on a classical algorithm that approximately solves a certain variant of Max-Cut, and we derive an approximation guarantee theoretically, assuming the approximation ratio of the classical algorithm and QAOA. Furthermore we design a heuristic approach that fits in our framework and perform sufficient numerical experiments, where we solve Max-Cut on various $24$-vertex Erd\H{o}s-R\'enyi graphs. Our framework only consumes $18$ qubits and achieves $0.9950$ approximation ratio on average, which outperforms the previous methods showing $0.9778$ (quantum algorithm using the same number of qubits) and $0.9643$ (classical algorithm). The experimental results indicate our well-designed quantum-classical coupling framework gives satisfactory approximation ratio while reduces the qubit cost, which sheds light on more potential computing power of NISQ devices

The Transfiguration of the Woman’s Body: A Study of Holy Bible and The Woman’s Bible

This article employs the method of critical discourse analysis to study the changing portrayal of the woman’s body. Our analysis shows the ways in which the woman’s body is constructed as silent, instrumentalized, sinful, and unclean in Holy Bible under the dominance of male discourses. Then we examined how the woman’s body is reconstructed as an independent and glorified entity in The Woman’s Bible, along with the subversion of male discourses by feminist renditions thereof. The study has implications for women who are fighting for their bodily rights

1,3-Dioxo-2,3-dihydro-1H-isoindol-2-yl 2,3,4-tri-O-acetyl-β-d-xyloside

The title compound, C19H19NO10, was obtained from the reaction of α-d-1-bromo-2,3,4-tri-O-acetylxylose with N-hy­droxy­phthalimide in the presence of potassium carbonate. The asymmetric unit contains two independent mol­ecules, in which the O—CH—O—N torsion angles are 73.0 (4) and 65.0 (4)°. The hexa­pyranosyl rings adopt chair conformations and the substituent groups are in equatorial positions. In the crystal, mol­ecules are linked by nonclassical C—H⋯O hydrogen bonds