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Attention-based graph neural networks (GNNs), such as graph attention networks (GATs), have become popular neural architectures for processing graph-structured data and learning node embeddings. Despite their empirical success, these models rely on labeled data and the theoretical properties of these models have yet to be fully understood. In this work, we propose a novel attention-based node embedding framework for graphs. Our framework builds upon a hierarchical kernel for multisets of subgraphs around nodes (e.g. neighborhoods) and each kernel leverages the geometry of a smooth statistical manifold to compare pairs of multisets, by "projecting" the multisets onto the manifold. By explicitly computing node embeddings with a manifold of Gaussian mixtures, our method leads to a new attention mechanism for neighborhood aggregation. We provide theoretical insights into genralizability and expressivity of our embeddings, contributing to a deeper understanding of attention-based GNNs. We propose efficient unsupervised and supervised methods for learning the embeddings, with the unsupervised method not requiring any labeled data. Through experiments on several node classification benchmarks, we demonstrate that our proposed method outperforms existing attention-based graph models like GATs. Our code is available at //github.com/BorgwardtLab/fisher_information_embedding.

Point clouds arise from acquisition processes applied in various scenarios, such as reverse engineering, rapid prototyping, or cultural preservation. To run various simulations via, e.g., finite element methods, on the derived data, a mesh has to be created from it. In this paper, a meshing algorithm for point clouds is presented, which is based on a sphere covering of the underlying surface. The algorithm provides a mesh close to uniformity in terms of edge lengths and angles of its triangles. Additionally, theoretical results guarantee the output to be manifold, given suitable input and parameter choices. We present both the underlying theory, which provides suitable parameter bounds, as well as experiments showing that our algorithm can compete with widely used competitors in terms of quality of the output and timings.

Not everybody can be equipped with professional photography skills and sufficient shooting time, and there can be some tilts in the captured images occasionally. In this paper, we propose a new and practical task, named Rotation Correction, to automatically correct the tilt with high content fidelity in the condition that the rotated angle is unknown. This task can be easily integrated into image editing applications, allowing users to correct the rotated images without any manual operations. To this end, we leverage a neural network to predict the optical flows that can warp the tilted images to be perceptually horizontal. Nevertheless, the pixel-wise optical flow estimation from a single image is severely unstable, especially in large-angle tilted images. To enhance its robustness, we propose a simple but effective prediction strategy to form a robust elastic warp. Particularly, we first regress the mesh deformation that can be transformed into robust initial optical flows. Then we estimate residual optical flows to facilitate our network the flexibility of pixel-wise deformation, further correcting the details of the tilted images. To establish an evaluation benchmark and train the learning framework, a comprehensive rotation correction dataset is presented with a large diversity in scenes and rotated angles. Extensive experiments demonstrate that even in the absence of the angle prior, our algorithm can outperform other state-of-the-art solutions requiring this prior. The code and dataset are available at //github.com/nie-lang/RotationCorrection.

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). Using classical potential theory, the Laplace-Beltrami operator can be pre-/post-conditioned with an integral operator whose kernel is translation invariant, resulting in well-conditioned Fredholm integral equations of the second-kind. These equations have the standard~$1/r$ kernel from potential theory, and therefore the equations can be solved rapidly and accurately using a combination of fast multipole methods (FMMs) and high-order quadrature corrections. In this work we detail such a scheme, presenting two alternative integral formulations of the Laplace-Beltrami problem, each of whose solution can be obtained via FMM acceleration. We then present several applications of the solvers, focusing on the computation of what are known as harmonic vector fields, relevant for many applications in electromagnetics. A battery of numerical results are presented for each application, detailing the performance of the solver in various geometries.

Phase retrieval is the nonlinear inverse problem of recovering a true signal from its Fourier magnitude measurements. It arises in many applications such as astronomical imaging, X-Ray crystallography, microscopy, and more. The problem is highly ill-posed due to the phase-induced ambiguities and the large number of possible images that can fit to the given measurements. Thus, there's a rich history of enforcing structural priors to improve solutions including sparsity priors and deep-learning-based generative models. However, such priors are often limited in their representational capacity or generalizability to slightly different distributions. Recent advancements in using denoisers as regularizers for non-convex optimization algorithms have shown promising performance and generalization. We present a way of leveraging the prior implicitly learned by a denoiser to solve phase retrieval problems by incorporating it in a classical alternating minimization framework. Compared to performant denoising-based algorithms for phase retrieval, we showcase competitive performance with Fourier measurements on in-distribution images and notable improvement on out-of-distribution images.

The European Union has proposed the Artificial Intelligence Act which introduces a proportional risk-based approach to AI regulation including detailed requirements for transparency and explainability. Many of these requirements may be addressed in practice by the field of explainable AI (XAI), however, there are fundamental differences between XAI and the Act regarding what transparency and explainability are. These basic definitions should be aligned to assure that regulation continually translates into appropriate technical practices. To facilitate this alignment, we first give an overview of how XAI and European regulation view basic definitions of transparency with a particular focus on the AI Act and the related General Data Protection Regulation (GDPR). We then present a comparison of XAI and regulatory approaches to identify the main points that would improve alignment between the fields: clarification of the scope of transparency, the legal status of XAI, oversight issues in conformity assessments, and dataset-related transparency.

Knowledge graph embedding models learn the representations of entities and relations in the knowledge graphs for predicting missing links (relations) between entities. Their effectiveness are deeply affected by the ability of modeling and inferring different relation patterns such as symmetry, asymmetry, inversion, composition and transitivity. Although existing models are already able to model many of these relations patterns, transitivity, a very common relation pattern, is still not been fully supported. In this paper, we first theoretically show that the transitive relations can be modeled with projections. We then propose the Rot-Pro model which combines the projection and relational rotation together. We prove that Rot-Pro can infer all the above relation patterns. Experimental results show that the proposed Rot-Pro model effectively learns the transitivity pattern and achieves the state-of-the-art results on the link prediction task in the datasets containing transitive relations.

Data in Knowledge Graphs often represents part of the current state of the real world. Thus, to stay up-to-date the graph data needs to be updated frequently. To utilize information from Knowledge Graphs, many state-of-the-art machine learning approaches use embedding techniques. These techniques typically compute an embedding, i.e., vector representations of the nodes as input for the main machine learning algorithm. If a graph update occurs later on -- specifically when nodes are added or removed -- the training has to be done all over again. This is undesirable, because of the time it takes and also because downstream models which were trained with these embeddings have to be retrained if they change significantly. In this paper, we investigate embedding updates that do not require full retraining and evaluate them in combination with various embedding models on real dynamic Knowledge Graphs covering multiple use cases. We study approaches that place newly appearing nodes optimally according to local information, but notice that this does not work well. However, we find that if we continue the training of the old embedding, interleaved with epochs during which we only optimize for the added and removed parts, we obtain good results in terms of typical metrics used in link prediction. This performance is obtained much faster than with a complete retraining and hence makes it possible to maintain embeddings for dynamic Knowledge Graphs.

Learning latent representations of nodes in graphs is an important and ubiquitous task with widespread applications such as link prediction, node classification, and graph visualization. Previous methods on graph representation learning mainly focus on static graphs, however, many real-world graphs are dynamic and evolve over time. In this paper, we present Dynamic Self-Attention Network (DySAT), a novel neural architecture that operates on dynamic graphs and learns node representations that capture both structural properties and temporal evolutionary patterns. Specifically, DySAT computes node representations by jointly employing self-attention layers along two dimensions: structural neighborhood and temporal dynamics. We conduct link prediction experiments on two classes of graphs: communication networks and bipartite rating networks. Our experimental results show that DySAT has a significant performance gain over several different state-of-the-art graph embedding baselines.

Multi-view networks are ubiquitous in real-world applications. In order to extract knowledge or business value, it is of interest to transform such networks into representations that are easily machine-actionable. Meanwhile, network embedding has emerged as an effective approach to generate distributed network representations. Therefore, we are motivated to study the problem of multi-view network embedding, with a focus on the characteristics that are specific and important in embedding this type of networks. In our practice of embedding real-world multi-view networks, we identify two such characteristics, which we refer to as preservation and collaboration. We then explore the feasibility of achieving better embedding quality by simultaneously modeling preservation and collaboration, and propose the mvn2vec algorithms. With experiments on a series of synthetic datasets, an internal Snapchat dataset, and two public datasets, we further confirm the presence and importance of preservation and collaboration. These experiments also demonstrate that better embedding can be obtained by simultaneously modeling the two characteristics, while not over-complicating the model or requiring additional supervision.