The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system coupling (1) a standard finite element discretisation localised around the scatterer and (2) integral equations whose unknowns are traces of the solution on the boundaries of a finite number of overlapping half-planes contained in the domain. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence of this HSM formulation to the original scattering problem have been established only for complex wavenumbers. In the present paper we show, in the case of a homogeneous background, that the HSM formulation is equivalent to the original scattering problem also for real wavenumbers, and so is well-posed, provided the traces satisfy radiation conditions at infinity analogous to the standard Sommerfeld radiation condition. As a key component of our argument we show that, if the trace on the boundary of a half-plane satisfies our new radiation condition, then the corresponding solution to the half-plane Dirichlet problem satisfies the Sommerfeld radiation condition in a slightly smaller half-plane. We expect that this last result will be of independent interest, in particular in studies of rough surface scattering.
相關內容
Computing the {\em matching statistics} of a string $P[1..m]$ with respect to a text $T[1..n]$ is a fundamental problem which has application to genome sequence comparison. In this paper, we study the problem of computing the matching statistics upon highly repetitive texts. We design three different data structures that are similar to LZ-compressed indexes. The space costs of all of them can be measured by $\gamma$, the size of the smallest string attractor [STOC'2018] and $\delta$, a better measure of repetitiveness [LATIN'2020].
We employ a toolset -- dubbed Dr. Frankenstein -- to analyse the similarity of representations in deep neural networks. With this toolset, we aim to match the activations on given layers of two trained neural networks by joining them with a stitching layer. We demonstrate that the inner representations emerging in deep convolutional neural networks with the same architecture but different initializations can be matched with a surprisingly high degree of accuracy even with a single, affine stitching layer. We choose the stitching layer from several possible classes of linear transformations and investigate their performance and properties. The task of matching representations is closely related to notions of similarity. Using this toolset, we also provide a novel viewpoint on the current line of research regarding similarity indices of neural network representations: the perspective of the performance on a task.
Recent works leveraging Graph Neural Networks to approach graph matching tasks have shown promising results. Recent progress in learning discrete distributions poses new opportunities for learning graph matching models. In this work, we propose a new model, Stochastic Iterative Graph MAtching (SIGMA), to address the graph matching problem. Our model defines a distribution of matchings for a graph pair so the model can explore a wide range of possible matchings. We further introduce a novel multi-step matching procedure, which learns how to refine a graph pair's matching results incrementally. The model also includes dummy nodes so that the model does not have to find matchings for nodes without correspondence. We fit this model to data via scalable stochastic optimization. We conduct extensive experiments across synthetic graph datasets as well as biochemistry and computer vision applications. Across all tasks, our results show that SIGMA can produce significantly improved graph matching results compared to state-of-the-art models. Ablation studies verify that each of our components (stochastic training, iterative matching, and dummy nodes) offers noticeable improvement.
The ad-hoc retrieval task is to rank related documents given a query and a document collection. A series of deep learning based approaches have been proposed to solve such problem and gained lots of attention. However, we argue that they are inherently based on local word sequences, ignoring the subtle long-distance document-level word relationships. To solve the problem, we explicitly model the document-level word relationship through the graph structure, capturing the subtle information via graph neural networks. In addition, due to the complexity and scale of the document collections, it is considerable to explore the different grain-sized hierarchical matching signals at a more general level. Therefore, we propose a Graph-based Hierarchical Relevance Matching model (GHRM) for ad-hoc retrieval, by which we can capture the subtle and general hierarchical matching signals simultaneously. We validate the effects of GHRM over two representative ad-hoc retrieval benchmarks, the comprehensive experiments and results demonstrate its superiority over state-of-the-art methods.
We propose a scalable Gromov-Wasserstein learning (S-GWL) method and establish a novel and theoretically-supported paradigm for large-scale graph analysis. The proposed method is based on the fact that Gromov-Wasserstein discrepancy is a pseudometric on graphs. Given two graphs, the optimal transport associated with their Gromov-Wasserstein discrepancy provides the correspondence between their nodes and achieves graph matching. When one of the graphs has isolated but self-connected nodes ($i.e.$, a disconnected graph), the optimal transport indicates the clustering structure of the other graph and achieves graph partitioning. Using this concept, we extend our method to multi-graph partitioning and matching by learning a Gromov-Wasserstein barycenter graph for multiple observed graphs; the barycenter graph plays the role of the disconnected graph, and since it is learned, so is the clustering. Our method combines a recursive $K$-partition mechanism with a regularized proximal gradient algorithm, whose time complexity is $\mathcal{O}(K(E+V)\log_K V)$ for graphs with $V$ nodes and $E$ edges. To our knowledge, our method is the first attempt to make Gromov-Wasserstein discrepancy applicable to large-scale graph analysis and unify graph partitioning and matching into the same framework. It outperforms state-of-the-art graph partitioning and matching methods, achieving a trade-off between accuracy and efficiency.
For many computer vision applications such as image captioning, visual question answering, and person search, learning discriminative feature representations at both image and text level is an essential yet challenging problem. Its challenges originate from the large word variance in the text domain as well as the difficulty of accurately measuring the distance between the features of the two modalities. Most prior work focuses on the latter challenge, by introducing loss functions that help the network learn better feature representations but fail to account for the complexity of the textual input. With that in mind, we introduce TIMAM: a Text-Image Modality Adversarial Matching approach that learns modality-invariant feature representations using adversarial and cross-modal matching objectives. In addition, we demonstrate that BERT, a publicly-available language model that extracts word embeddings, can successfully be applied in the text-to-image matching domain. The proposed approach achieves state-of-the-art cross-modal matching performance on four widely-used publicly-available datasets resulting in absolute improvements ranging from 2% to 5% in terms of rank-1 accuracy.
This paper seeks to model human language by the mathematical framework of quantum physics. With the well-designed mathematical formulations in quantum physics, this framework unifies different linguistic units in a single complex-valued vector space, e.g. words as particles in quantum states and sentences as mixed systems. A complex-valued network is built to implement this framework for semantic matching. With well-constrained complex-valued components, the network admits interpretations to explicit physical meanings. The proposed complex-valued network for matching (CNM) achieves comparable performances to strong CNN and RNN baselines on two benchmarking question answering (QA) datasets.
In text classification, the problem of overfitting arises due to the high dimensionality, making regularization essential. Although classic regularizers provide sparsity, they fail to return highly accurate models. On the contrary, state-of-the-art group-lasso regularizers provide better results at the expense of low sparsity. In this paper, we apply a greedy variable selection algorithm, called Orthogonal Matching Pursuit, for the text classification task. We also extend standard group OMP by introducing overlapping group OMP to handle overlapping groups of features. Empirical analysis verifies that both OMP and overlapping GOMP constitute powerful regularizers, able to produce effective and super-sparse models. Code and data are available at: //www.dropbox.com/sh/7w7hjns71ol0xrz/AAC\string_G0\string_0DlcGkq6tQb2zqAaca\string?dl\string=0 .
This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.
Learning from a few examples remains a key challenge in machine learning. Despite recent advances in important domains such as vision and language, the standard supervised deep learning paradigm does not offer a satisfactory solution for learning new concepts rapidly from little data. In this work, we employ ideas from metric learning based on deep neural features and from recent advances that augment neural networks with external memories. Our framework learns a network that maps a small labelled support set and an unlabelled example to its label, obviating the need for fine-tuning to adapt to new class types. We then define one-shot learning problems on vision (using Omniglot, ImageNet) and language tasks. Our algorithm improves one-shot accuracy on ImageNet from 87.6% to 93.2% and from 88.0% to 93.8% on Omniglot compared to competing approaches. We also demonstrate the usefulness of the same model on language modeling by introducing a one-shot task on the Penn Treebank.
小貼士
登錄享
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191