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Driven by the outstanding performance of neural networks in the structured euclidean domain, recent years have seen a surge of interest in developing neural networks for graphs and data supported on graphs. The graph is leveraged at each layer of the neural network as a parameterization to capture detail at the node level with a reduced number of parameters and computational complexity. Following this rationale, this paper puts forth a general framework that unifies state-of-the-art graph neural networks (GNNs) through the concept of EdgeNet. An EdgeNet is a GNN architecture that allows different nodes to use different parameters to weigh the information of different neighbors. By extrapolating this strategy to more iterations between neighboring nodes, the EdgeNet learns edge- and neighbor-dependent weights to capture local detail. This is a general linear and local operation that a node can perform and encompasses under one formulation all existing graph convolutional neural networks (GCNNs) as well as graph attention networks (GATs). In writing different GNN architectures with a common language, EdgeNets highlight specific architecture advantages and limitations, while providing guidelines to improve their capacity without compromising their local implementation. For instance, we show that GCNNs have a parameter sharing structure that induces permutation equivariance. This can be an advantage or a limitation, depending on the application. In cases where it is a limitation, we propose hybrid approaches and provide insights to develop several other solutions that promote parameter sharing without enforcing permutation equivariance. Another interesting conclusion is the unification of GCNNs and GATs —approaches that have been so far perceived as separate. In particular, we show that GATs are GCNNs on a graph that is learned from the features. This particularization opens the doors to develop alternative attention mechanisms for improving discriminatory power. [ABSTRACT FROM AUTHOR]

Few-shot human-object interaction (FS-HOI) recognition aims at inferring new interactions between human actions and surrounding objects merely with a few available instances. It is beneficial to alleviate the long-tail and combinatorial explosion problems in human-object interaction (HOI). Nevertheless, the existing FS-HOI methods only focus on modeling the relationships between labeled samples and unlabeled samples in the Euclidean domain, which neglects the rich relational structures of the visual information among labeled samples and between human actions and objects. Accordingly, we tackle the few-shot HOI task in the non-Euclidean domain and present a graph-based model, namely, task-oriented high-order context graph network (THCG-Net). It contains a task attention module (TA-Module) and a high-order context graph module (HG-Module). In TA-Module, an attention mechanism is designed by utilizing task information to build a task-oriented space, in which the discriminative information for the current task (episode) is captured by embedding the visual features into the task-oriented space. The HG-Module is proposed to construct a task-level graph and takes the context information as high-order knowledge, which provides discriminative guidance for propagating visual information. It captures the discriminability among different categories while highlights the commonality of related categories adaptively, which effectively transfers knowledge to related categories. Extensive experimental results on two benchmark datasets, HICO-FS and TUHOI-FS, are provided. It demonstrates that our THCG-Net significantly outperforms the state-of-the-art approaches, which proves its impressive effectiveness in recognizing various human actions and surrounding objects in few-shot scenarios. [ABSTRACT FROM AUTHOR]

This article presents a graph convolutional network model (GCNM) for gait phase classification to control a lower limb exoskeleton. The GCNM can recognize four leg phases between the foot and ground, including heel strike, foot flat, heel off, and swing. The proposed model can solve the gait phase classification problem from non-Euclidean domain based on a graph mechanism for exoskeletons. Real-time performance of the gait data acquisition system was then verified on a lower limb exoskeleton. The system quantifies the time delay with an optoelectronic system, VICON BONITA 10. Comparison works with some existing state-of-the-art methods from the Euclidean domain, i.e., long short-term memory (LSTM) and deep convolutional neural network (DCNN), were also provided. Experimental works show that the proposed model has a significantly higher prediction accuracy and better robustness in gait phase classification for different people in level, uphill, and downhill environments. Moreover, the label rate is under ${\rm {10}}{\rm {\% }}$ , compared to LSTM and DCNN, which are usually at ${\rm {70}}$ – ${\rm {80}}{\rm {\% }}$. Finally, the GCNM's maximum accuracy of gait phase classification is ${\rm {97}}{\rm {.43\% }}$ ; thus, the effectiveness of the proposed model is verified. [ABSTRACT FROM AUTHOR]

Computing and visualizing features in fluid flow often depends on the observer, or reference frame, relative to which the input velocity field is given. A desired property of feature detectors is therefore that they are objective, meaning independent of the input reference frame. However, the standard definition of objectivity is only given for Euclidean domains and cannot be applied in curved spaces. We build on methods from mathematical physics and Riemannian geometry to generalize objectivity to curved spaces, using the powerful notion of symmetry groups as the basis for definition. From this, we develop a general mathematical framework for the objective computation of observer fields for curved spaces, relative to which other computed measures become objective. An important property of our framework is that it works intrinsically in 2D, instead of in the 3D ambient space. This enables a direct generalization of the 2D computation via optimization of observer fields in flat space to curved domains, without having to perform optimization in 3D. We specifically develop the case of unsteady 2D geophysical flows given on spheres, such as the Earth. Our observer fields in curved spaces then enable objective feature computation as well as the visualization of the time evolution of scalar and vector fields, such that the automatically computed reference frames follow moving structures like vortices in a way that makes them appear to be steady. [ABSTRACT FROM AUTHOR]

Convolutional neural networks (CNNs) provide a dramatically powerful class of models, but are subject to traditional convolution that can merely aggregate permutation-ordered and dimension-equal local inputs. It causes that CNNs are allowed to only manage signals on Euclidean or grid-like domains (e.g., images), not ones on non-Euclidean or graph domains (e.g., traffic networks). To eliminate this limitation, we develop a local-aggregation function, a sharable nonlinear operation, to aggregate permutation-unordered and dimension-unequal local inputs on non-Euclidean domains. In the context of the function approximation theory, the local-aggregation function is parameterized with a group of orthonormal polynomials in an effective and efficient manner. By replacing the traditional convolution in CNNs with the parameterized local-aggregation function, Local-Aggregation Graph Networks (LAGNs) are readily established, which enable to fit nonlinear functions without activation functions and can be expediently trained with the standard back-propagation. Extensive experiments on various datasets strongly demonstrate the effectiveness and efficiency of LAGNs, leading to superior performance on numerous pattern recognition and machine learning tasks, including text categorization, molecular activity detection, taxi flow prediction, and image classification. [ABSTRACT FROM AUTHOR]

Image denoising technologies in a Euclidean domain have achieved good results and are becoming mature. However, in recent years, many real-world applications encountered in computer vision and geometric modeling involve image data defined in irregular domains modeled by huge graphs, which results in the problem on how to solve image denoising problems defined on graphs. In this paper, we propose a novel model for removing mixed or unknown noise in images on graphs. The objective is to minimize the sum of a weighted fidelity term and a sparse regularization term that additionally utilizes wavelet frame transform on graphs to retain feature details of images defined on graphs. Specifically, the weighted fidelity term with $\ell _{1}$ -norm and $\ell _{2}$ -norm is designed based on a analysis of the distribution of mixed noise. The augmented Lagrangian and accelerated proximal gradient methods are employed to achieve the optimal solution to the problem. Finally, some supporting numerical results and comparative analyses with other denoising algorithms are provided. It is noted that we investigate image denoising with unknown noise or a wide range of mixed noise, especially the mixture of Poisson, Gaussian, and impulse noise. Experimental results reported for synthetic and real images on graphs demonstrate that the proposed method is effective and efficient, and exhibits better performance for the removal of mixed or unknown noise in images on graphs than other denoising algorithms in the literature. The method can effectively remove mixed or unknown noise and retain feature details of images on graphs. It delivers a new avenue for denoising images in irregular domains. [ABSTRACT FROM AUTHOR]

Hand–eye calibration, which consists in identifying the rigid-body transformation between a camera mounted on the robot end-effector and the end-effector itself, is a fundamental problem in robot vision. Mathematically, this problem can be formulated as: solve for $\boldsymbol {X}$ in $\boldsymbol {A}\boldsymbol {X}= \boldsymbol {X}\boldsymbol {B}$. In this paper, we provide a rigorous derivation of the covariance of the solution $\boldsymbol {X}$ , when $\boldsymbol {A}$ and $\boldsymbol {B}$ are randomly perturbed matrices. This fine-grained information is critical for applications that require a high degree of perception precision. Our approach consists in applying covariance propagation methods in $\boldsymbol {S}\boldsymbol {E}(3)$. Experiments involving synthetic and real calibration data confirm that our approach can predict the covariance of the hand–eye transformation with excellent precision. [ABSTRACT FROM AUTHOR]

We study convex optimization problems that feature low-rank matrix solutions. In such scenarios, non-convex methods offer significant advantages over convex methods due to their lower space complexity, as well as practical faster convergence. Under mild assumptions, these methods feature global convergence guarantees. In this paper, we extend the results on this matter by following a different path. We derive a non-Euclidean optimization framework in the non-convex setting that takes nonlinear gradient steps on the factors. Our framework enables the possibility to further exploit the underlying problem structures, such as sparsity or low-rankness on the factorized domain, or better dimensional dependence of the smoothness parameters of the objectives. We prove that the non-Euclidean methods enjoy the same rigorous guarantees as their Euclidean counterparts under appropriate assumptions. Numerical evidence with Fourier ptychography and FastText applications, using real data, shows that our approach can enhance solution quality, as well as convergence speed over the standard non-convex approaches. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study a new model of Imperfect Maintenance in Reliability. A model of arithmetic reduction of intensity is assumed on the interarrival times of failures on a system subject to recurrent failures. Based on the observation of the recurrent failures of a single repairable system and assuming that a perfect repair is operated after $N$ failures, we introduce estimators of the parameters (Euclidean and functional) of this semiparametric model and we prove their asymptotic normality. Statistical inference is also carried out using this model. Finally, a simulation study is performed to learn the behavior of these estimators on samples of small or moderate size. We end this work with an application on a real dataset. [ABSTRACT FROM AUTHOR]

Visual observations of dynamic phenomena, such as human actions, are often represented as sequences of smoothly-varying features. In cases where the feature spaces can be structured as Riemannian manifolds, the corresponding representations become trajectories on manifolds. Analysis of these trajectories is challenging due to non-linearity of underlying spaces and high-dimensionality of trajectories. In vision problems, given the nature of physical systems involved, these phenomena are better characterized on a low-dimensional manifold compared to the space of Riemannian trajectories. For instance, if one does not impose physical constraints of the human body, in data involving human action analysis, the resulting representation space will have highly redundant features. Learning an effective, low-dimensional embedding for action representations will have a huge impact in the areas of search and retrieval, visualization, learning, and recognition. Traditional manifold learning addresses this problem for static points in the euclidean space, but its extension to Riemannian trajectories is non-trivial and remains unexplored. The difficulty lies in inherent non-linearity of the domain and temporal variability of actions that can distort any traditional metric between trajectories. To overcome these issues, we use the framework based on transported square-root velocity fields (TSRVF); this framework has several desirable properties, including a rate-invariant metric and vector space representations. We propose to learn an embedding such that each action trajectory is mapped to a single point in a low-dimensional euclidean space, and the trajectories that differ only in temporal rates map to the same point. We utilize the TSRVF representation, and accompanying statistical summaries of Riemannian trajectories, to extend existing coding methods such as PCA, KSVD and Label Consistent KSVD to Riemannian trajectories or more generally to Riemannian functions. We show that such coding efficiently captures trajectories in applications such as action recognition, stroke rehabilitation, visual speech recognition, clustering and diverse sequence sampling. Using this framework, we obtain state-of-the-art recognition results, while reducing the dimensionality/complexity by a factor of $100-250\times$ . Since these mappings and codes are invertible, they can also be used to interactively visualize Riemannian trajectories and synthesize actions. [ABSTRACT FROM AUTHOR]

Tree-structured data usually contain both topological and geometrical information, and are necessarily considered on manifold instead of euclidean space for appropriate data parameterization and analysis. In this study, we propose a novel tree-structured data parameterization, called Topology-Attribute matrix (T-A matrix), so the data clustering task can be conducted on matrix manifold. We incorporate the structure constraints embedded in data into the non-negative matrix factorization method to determine meta-trees from the T-A matrix, and the signature vector of each single tree can then be extracted by meta-tree decomposition. The meta-tree space turns out to be a cone space, in which we explore the distance metric and implement the clustering algorithm based on the concepts like Fréchet mean. Finally, the T-A matrix based clustering (TAMBAC) framework is evaluated and compared using both simulated data and real retinal images to illustrate its efficiency and accuracy. [ABSTRACT FROM AUTHOR]

In response to their growing use in the real world, this article presents two algorithms for direct parameterization of quadrilateral meshes. The proposed algorithms are angle-preserving mappings, with one mapping a topological disk surface onto a Euclidean plane and one mapping a topological sphere surface onto a unit sphere. Specifically, for topological disk surfaces, the authors devise a discrete conformal energy function to flatten the quadrilateral meshes with a length-preserving boundary condition. For topological sphere surfaces, a derived Tuette energy function is applied to the initialization of parameterization for a mesh, and then the final spherical parametrization result is obtained by minimizing a devised harmonic energy function. Experimental results demonstrate the efficiency of the proposed methods. [ABSTRACT FROM PUBLISHER]

We consider the problem of retrieving the database points nearest to a given hyperplane query without exhaustively scanning the entire database. For this problem, we propose two hashing-based solutions. Our first approach maps the data to 2-bit binary keys that are locality sensitive for the angle between the hyperplane normal and a database point. Our second approach embeds the data into a vector space where the euclidean norm reflects the desired distance between the original points and hyperplane query. Both use hashing to retrieve near points in sublinear time. Our first method's preprocessing stage is more efficient, while the second has stronger accuracy guarantees. We apply both to pool-based active learning: Taking the current hyperplane classifier as a query, our algorithm identifies those points (approximately) satisfying the well-known minimal distance-to-hyperplane selection criterion. We empirically demonstrate our methods' tradeoffs and show that they make it practical to perform active selection with millions of unlabeled points. [ABSTRACT FROM PUBLISHER]