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This survey reviews hyperbolic graph embedding models, and evaluate them on anomaly detection, highlighting their advantages over Euclidean methods in capturing complex structures. Evaluating models like \textit{HGCAE},…

Criticality and symmetry, studied by the renormalization groups, lie at the heart of modern physics theories of matters and complex systems. However, surveying these properties with massive experimental data is bottlenecked by the…

Temporal knowledge graph (TKG) reasoning predicts future events based on historical data, but it's challenging due to the complex semantic and hierarchical information involved. Existing Euclidean models excel at capturing semantics but…

Hyperbolic representation learning has been widely used to extract implicit hierarchies within data, and recently it has found its way to the open-world classification task of Generalized Category Discovery (GCD). However, prior hyperbolic…

Many high-dimensional and large-volume data sets of practical relevance have hierarchical structures induced by trees, graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional…

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…

When high-dimensional non-uniformly hyperbolic chaotic systems undergo dynamical perturbations, their long-time statistics are generally observed to respond differentiably with respect to the perturbation. Although important in…

Subgroup discovery is a descriptive and exploratory data mining technique to identify subgroups in a population that exhibit interesting behavior with respect to a variable of interest. Subgroup discovery has numerous applications in…

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of…

To comprehend complex systems with multiple states, it is imperative to reveal the identity of these states by system outputs. Nevertheless, the mathematical models describing these systems often exhibit nonlinearity so that render the…

Hamiltonian Learning is a process of recovering system Hamiltonian from measurements, which is a fundamental problem in quantum information processing. In this study, we investigate the problem of learning the symmetric Hamiltonian from its…

Research in modern data-driven dynamical systems is typically focused on the three key challenges of high dimensionality, unknown dynamics, and nonlinearity. The dynamic mode decomposition (DMD) has emerged as a cornerstone for modeling…

For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure.…

The engineering design process often relies on mathematical modeling that can describe the underlying dynamic behavior. In this work, we present a data-driven methodology for modeling the dynamics of nonlinear systems. To simplify this…

Hyperbolic deep learning has become a growing research direction in computer vision due to the unique properties afforded by the alternate embedding space. The negative curvature and exponentially growing distance metric provide a natural…

Hyperbolic space is a natural setting for mining and visualizing data with hierarchical structure. In order to compute a hyperbolic embedding from comparison or similarity information, one has to solve a hyperbolic distance geometry…

In this paper, a novel learning paradigm is presented to automatically identify groups of informative and correlated features from very high dimensions. Specifically, we explicitly incorporate correlation measures as constraints and then…

We extend our two-scale neural-network method for scalar singularly perturbed problems with one small parameter to dynamical systems with multiple small parameters. To accommodate multiple small parameters, we use a single effective scale…

Matrix geometric means between two positive definite matrices can be defined from distinct perspectives - as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain…

Label inventories for fine-grained entity typing have grown in size and complexity. Nonetheless, they exhibit a hierarchical structure. Hyperbolic spaces offer a mathematically appealing approach for learning hierarchical representations of…