Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no rigorous analysis of the fini
te-size error in the Hartree-Fock theory (HF) and the second order M{o}ller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified method to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of [Lyness, 1976] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations [Xing, Li, Lin, 2021]. Our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.
Few-shot image classification is a challenging problem which aims to achieve the human level of recognition based only on a small number of images. Deep learning algorithms such as meta-learning, transfer learning, and metric learning have been emplo
yed recently and achieved the state-of-the-art performance. In this survey, we review representative deep metric learning methods for few-shot classification, and categorize them into three groups according to the major problems and novelties they focus on. We conclude this review with a discussion on current challenges and future trends in few-shot image classification.
The calculation of the MP2 correlation energy for extended systems can be viewed as a multi-dimensional integral in the thermodynamic limit, and the standard method for evaluating the MP2 energy can be viewed as a trapezoidal quadrature scheme. We de
monstrate that existing analysis neglects certain contributions due to the non-smoothness of the integrand, and may significantly underestimate finite-size errors. We propose a new staggered mesh method, which uses two staggered Monkhorst-Pack meshes for occupied and virtual orbitals, respectively, to compute the MP2 energy. The staggered mesh method circumvents a significant error source in the standard method, in which certain quadrature nodes are always placed on points where the integrand is discontinuous. One significant advantage of the proposed method is that there are no tunable parameters, and the additional numerical effort needed can be negligible compared to the standard MP2 calculation. Numerical results indicate that the staggered mesh method can be particularly advantageous for quasi-1D systems, as well as quasi-2D and 3D systems with certain symmetries.
Recently, graph neural networks (GNNs) have shown powerful ability to handle few-shot classification problem, which aims at classifying unseen samples when trained with limited labeled samples per class. GNN-based few-shot learning architectures most
ly replace traditional metric with a learnable GNN. In the GNN, the nodes are set as the samples embedding, and the relationship between two connected nodes can be obtained by a network, the input of which is the difference of their embedding features. We consider this method of measuring relation of samples only models the sample-to-sample relation, while neglects the specificity of different tasks. That is, this method of measuring relation does not take the task-level information into account. To this end, we propose a new relation measure method, namely the task-level relation module (TLRM), to explicitly model the task-level relation of one sample to all the others. The proposed module captures the relation representations between nodes by considering the sample-to-task instead of sample-to-sample embedding features. We conducted extensive experiments on four benchmark datasets: mini-ImageNet, tiered-ImageNet, CUB-$200$-$2011$, and CIFAR-FS. Experimental results demonstrate that the proposed module is effective for GNN-based few-shot learning.
Few-shot learning for fine-grained image classification has gained recent attention in computer vision. Among the approaches for few-shot learning, due to the simplicity and effectiveness, metric-based methods are favorably state-of-the-art on many t
asks. Most of the metric-based methods assume a single similarity measure and thus obtain a single feature space. However, if samples can simultaneously be well classified via two distinct similarity measures, the samples within a class can distribute more compactly in a smaller feature space, producing more discriminative feature maps. Motivated by this, we propose a so-called textit{Bi-Similarity Network} (textit{BSNet}) that consists of a single embedding module and a bi-similarity module of two similarity measures. After the support images and the query images pass through the convolution-based embedding module, the bi-similarity module learns feature maps according to two similarity measures of diverse characteristics. In this way, the model is enabled to learn more discriminative and less similarity-biased features from few shots of fine-grained images, such that the model generalization ability can be significantly improved. Through extensive experiments by slightly modifying established metric/similarity based networks, we show that the proposed approach produces a substantial improvement on several fine-grained image benchmark datasets. Codes are available at: https://github.com/spraise/BSNet
Despite achieving state-of-the-art performance, deep learning methods generally require a large amount of labeled data during training and may suffer from overfitting when the sample size is small. To ensure good generalizability of deep networks und
er small sample sizes, learning discriminative features is crucial. To this end, several loss functions have been proposed to encourage large intra-class compactness and inter-class separability. In this paper, we propose to enhance the discriminative power of features from a new perspective by introducing a novel neural network termed Relation-and-Margin learning Network (ReMarNet). Our method assembles two networks of different backbones so as to learn the features that can perform excellently in both of the aforementioned two classification mechanisms. Specifically, a relation network is used to learn the features that can support classification based on the similarity between a sample and a class prototype; at the meantime, a fully connected network with the cross entropy loss is used for classification via the decision boundary. Experiments on four image datasets demonstrate that our approach is effective in learning discriminative features from a small set of labeled samples and achieves competitive performance against state-of-the-art methods. Codes are available at https://github.com/liyunyu08/ReMarNet.
Few-shot meta-learning has been recently reviving with expectations to mimic humanitys fast adaption to new concepts based on prior knowledge. In this short communication, we give a concise review on recent representative methods in few-shot meta-lea
rning, which are categorized into four branches according to their technical characteristics. We conclude this review with some vital current challenges and future prospects in few-shot meta-learning.
In this paper, we construct an efficient numerical scheme for full-potential electronic structure calculations of periodic systems. In this scheme, the computational domain is decomposed into a set of atomic spheres and an interstitial region, and di
fferent basis functions are used in different regions: radial basis functions times spherical harmonics in the atomic spheres and plane waves in the interstitial region. These parts are then patched together by discontinuous Galerkin (DG) method. Our scheme has the same philosophy as the widely used (L)APW methods in materials science, but possesses systematically spectral convergence rate. We provide a rigorous a priori error analysis of the DG approximations for the linear eigenvalue problems, and present some numerical simulations in electronic structure calculations.
