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Recent research has shown that alignment between the structure of graph data and the geometry of an embedding space is crucial for learning high-quality representations of the data. The uniform geometry of Euclidean and hyperbolic spaces…

Probabilistic Manifold Decomposition (PMD)\cite{doi:10.1137/25M1738863}, developed in our earlier work, provides a nonlinear model reduction by embedding high-dimensional dynamics onto low-dimensional probabilistic manifolds. The PMD has…

Object detection, for the most part, has been formulated in the euclidean space, where euclidean or spherical geodesic distances measure the similarity of an image region to an object class prototype. In this work, we study whether a…

Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…

In the framework of solid mechanics, the task of deriving material parameters from experimental data has recently re-emerged with the progress in full-field measurement capabilities and the renewed advances of machine learning. In this…

In many real-world applications of regression, conditional probability estimation, and uncertainty quantification, exploiting symmetries rooted in physics or geometry can dramatically improve generalization and sample efficiency. While…

A complex system comprises multiple interacting entities whose interdependencies form a unified whole, exhibiting emergent behaviours not present in individual components. Examples include the human brain, living cells, soft matter, Earth's…

Medical studies frequently require to extract the relationship between each covariate and the outcome with statistical confidence measures. To do this, simple parametric models are frequently used (e.g. coefficients of linear regression)…

We present a geometric neural network-based tracking controller for systems evolving on matrix Lie groups under unknown dynamics, actuator faults, and bounded disturbances. Leveraging the left-invariance of the tangent bundle of matrix Lie…

We propose a data-driven Machine-Learning Symmetry Discovery (MLSD) framework for identifying continuous symmetry generators and their Lie-algebraic structure directly from phase-space trajectory data expressed in canonical coordinates.…

Foundation models pre-trained on massive datasets, including large language models (LLMs), vision-language models (VLMs), and large multimodal models, have demonstrated remarkable success in diverse downstream tasks. However, recent studies…

One of the strategies to detect the pose and shape of unknown objects is their geometric modeling, consisting on fitting known geometric entities. Classical geometric modeling fits simple shapes such as spheres or cylinders, but often those…

This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of SL(2,R) group. We describe here geometries of…

In this note we show how to find the stable model of a one-parameter family of elliptic surfaces with sections. More specifically, we perform the log Minimal Model Program in an explicit manner by means of toric geometry, in each such one…

Learning the representation of data with hierarchical structures in the hyperbolic space attracts increasing attention in recent years. Due to the constant negative curvature, the hyperbolic space resembles tree metrics and captures the…

We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.

Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are…

We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that differ from one another by a low-rank variation in the state matrix. Usual approaches for parametric model reduction…

In science, we are often interested in obtaining a generative model of the underlying system dynamics from observed time series. While powerful methods for dynamical systems reconstruction (DSR) exist when data come from a single domain,…

We introduce group crosscoders, an extension of crosscoders that systematically discover and analyse symmetrical features in neural networks. While neural networks often develop equivariant representations without explicit architectural…