We study the conformal capacity by using novel computational algorithms based on implementations of the fast multipole method, and analytic techniques. Especially, we apply domain functionals to study the capacities of condensers $(G,E)$ where $G$ is a simply connected domain in the complex plane and $E$ is a compact subset of $G$. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of $E$. Our computational experiments demonstrate, for instance, sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.
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For an image with multiple scene texts, different people may be interested in different text information. Current text-aware image captioning models are not able to generate distinctive captions according to various information needs. To explore how to generate personalized text-aware captions, we define a new challenging task, namely Question-controlled Text-aware Image Captioning (Qc-TextCap). With questions as control signals, this task requires models to understand questions, find related scene texts and describe them together with objects fluently in human language. Based on two existing text-aware captioning datasets, we automatically construct two datasets, ControlTextCaps and ControlVizWiz to support the task. We propose a novel Geometry and Question Aware Model (GQAM). GQAM first applies a Geometry-informed Visual Encoder to fuse region-level object features and region-level scene text features with considering spatial relationships. Then, we design a Question-guided Encoder to select the most relevant visual features for each question. Finally, GQAM generates a personalized text-aware caption with a Multimodal Decoder. Our model achieves better captioning performance and question answering ability than carefully designed baselines on both two datasets. With questions as control signals, our model generates more informative and diverse captions than the state-of-the-art text-aware captioning model. Our code and datasets are publicly available at //github.com/HAWLYQ/Qc-TextCap.
Graph neural networks (GNNs) are typically applied to static graphs that are assumed to be known upfront. This static input structure is often informed purely by insight of the machine learning practitioner, and might not be optimal for the actual task the GNN is solving. In absence of reliable domain expertise, one might resort to inferring the latent graph structure, which is often difficult due to the vast search space of possible graphs. Here we introduce Pointer Graph Networks (PGNs) which augment sets or graphs with additional inferred edges for improved model expressivity. PGNs allow each node to dynamically point to another node, followed by message passing over these pointers. The sparsity of this adaptable graph structure makes learning tractable while still being sufficiently expressive to simulate complex algorithms. Critically, the pointing mechanism is directly supervised to model long-term sequences of operations on classical data structures, incorporating useful structural inductive biases from theoretical computer science. Qualitatively, we demonstrate that PGNs can learn parallelisable variants of pointer-based data structures, namely disjoint set unions and link/cut trees. PGNs generalise out-of-distribution to 5x larger test inputs on dynamic graph connectivity tasks, outperforming unrestricted GNNs and Deep Sets.
Knowledge graph (KG) embeddings learn low-dimensional representations of entities and relations to predict missing facts. KGs often exhibit hierarchical and logical patterns which must be preserved in the embedding space. For hierarchical data, hyperbolic embedding methods have shown promise for high-fidelity and parsimonious representations. However, existing hyperbolic embedding methods do not account for the rich logical patterns in KGs. In this work, we introduce a class of hyperbolic KG embedding models that simultaneously capture hierarchical and logical patterns. Our approach combines hyperbolic reflections and rotations with attention to model complex relational patterns. Experimental results on standard KG benchmarks show that our method improves over previous Euclidean- and hyperbolic-based efforts by up to 6.1% in mean reciprocal rank (MRR) in low dimensions. Furthermore, we observe that different geometric transformations capture different types of relations while attention-based transformations generalize to multiple relations. In high dimensions, our approach yields new state-of-the-art MRRs of 49.6% on WN18RR and 57.7% on YAGO3-10.
Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.
Graph neural network (GNN) has shown superior performance in dealing with graphs, which has attracted considerable research attention recently. However, most of the existing GNN models are primarily designed for graphs in Euclidean spaces. Recent research has proven that the graph data exhibits non-Euclidean latent anatomy. Unfortunately, there was rarely study of GNN in non-Euclidean settings so far. To bridge this gap, in this paper, we study the GNN with attention mechanism in hyperbolic spaces at the first attempt. The research of hyperbolic GNN has some unique challenges: since the hyperbolic spaces are not vector spaces, the vector operations (e.g., vector addition, subtraction, and scalar multiplication) cannot be carried. To tackle this problem, we employ the gyrovector spaces, which provide an elegant algebraic formalism for hyperbolic geometry, to transform the features in a graph; and then we propose the hyperbolic proximity based attention mechanism to aggregate the features. Moreover, as mathematical operations in hyperbolic spaces could be more complicated than those in Euclidean spaces, we further devise a novel acceleration strategy using logarithmic and exponential mappings to improve the efficiency of our proposed model. The comprehensive experimental results on four real-world datasets demonstrate the performance of our proposed hyperbolic graph attention network model, by comparisons with other state-of-the-art baseline methods.
Learning embeddings of entities and relations existing in knowledge bases allows the discovery of hidden patterns in data. In this work, we examine the geometrical space's contribution to the task of knowledge base completion. We focus on the family of translational models, whose performance has been lagging, and propose a model, dubbed HyperKG, which exploits the hyperbolic space in order to better reflect the topological properties of knowledge bases. We investigate the type of regularities that our model can capture and we show that it is a prominent candidate for effectively representing a subset of Datalog rules. We empirically show, using a variety of link prediction datasets, that hyperbolic space allows to narrow down significantly the performance gap between translational and bilinear models.
In this paper, we introduce an embedding model, named CapsE, exploring a capsule network to model relationship triples (subject, relation, object). Our CapsE represents each triple as a 3-column matrix where each column vector represents the embedding of an element in the triple. This 3-column matrix is then fed to a convolution layer where multiple filters are operated to generate different feature maps. These feature maps are reconstructed into corresponding capsules which are then routed to another capsule to produce a continuous vector. The length of this vector is used to measure the plausibility score of the triple. Our proposed CapsE obtains better performance than previous state-of-the-art embedding models for knowledge graph completion on two benchmark datasets WN18RR and FB15k-237, and outperforms strong search personalization baselines on SEARCH17.
Most attention-based image captioning models attend to the image once per word. However, attending once per word is rigid and is easy to miss some information. Attending more times can adjust the attention position, find the missing information back and avoid generating the wrong word. In this paper, we show that attending more times per word can gain improvements in the image captioning task. We propose a flexible two-LSTM merge model to make it convenient to encode more attentions than words. Our captioning model uses two LSTMs to encode the word sequence and the attention sequence respectively. The information of the two LSTMs and the image feature are combined to predict the next word. Experiments on the MSCOCO caption dataset show that our method outperforms the state-of-the-art. Using bottom up features and self-critical training method, our method gets BLEU-4, METEOR, ROUGE-L and CIDEr scores of 0.381, 0.283, 0.580 and 1.261 on the Karpathy test split.
This paper presents an empirical exploration of the use of capsule networks for text classification. While it has been shown that capsule networks are effective for image classification, their validity in the domain of text has not been explored. In this paper, we show that capsule networks indeed have the potential for text classification and that they have several advantages over convolutional neural networks. We further suggest a simple routing method that effectively reduces the computational complexity of dynamic routing. We utilized seven benchmark datasets to demonstrate that capsule networks, along with the proposed routing method provide comparable results.
We introduce hyperbolic attention networks to endow neural networks with enough capacity to match the complexity of data with hierarchical and power-law structure. A few recent approaches have successfully demonstrated the benefits of imposing hyperbolic geometry on the parameters of shallow networks. We extend this line of work by imposing hyperbolic geometry on the activations of neural networks. This allows us to exploit hyperbolic geometry to reason about embeddings produced by deep networks. We achieve this by re-expressing the ubiquitous mechanism of soft attention in terms of operations defined for hyperboloid and Klein models. Our method shows improvements in terms of generalization on neural machine translation, learning on graphs and visual question answering tasks while keeping the neural representations compact.
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