Related papers: Learning on hexagonal structures and Monge-Amp\`er…

It is shown that applying manifold learning techniques to Poincar\'e sections of high-dimensional, chaotic dynamical systems can uncover their low-dimensional topological organization. Manifold learning provides a low-dimensional embedding…

We analyze the performance of a class of manifold-learning algorithms that find their output by minimizing a quadratic form under some normalization constraints. This class consists of Locally Linear Embedding (LLE), Laplacian Eigenmap,…

The contractive auto-encoder learns a representation of the input data that captures the local manifold structure around each data point, through the leading singular vectors of the Jacobian of the transformation from input to…

Energy-based models provide a natural bridge between statistical physics and machine learning by representing data through structured energy landscapes. Boltzmann machines are a particularly compelling class of such models for capturing…

Statistical learning methods show great promise in providing an accurate prediction of materials and molecular properties, while minimizing the need for computationally demanding electronic structure calculations. The accuracy and…

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a…

Orthogonal statistical learning and double machine learning have emerged as general frameworks for two-stage statistical prediction in the presence of a nuisance component. We establish non-asymptotic bounds on the excess risk of orthogonal…

Learning the structure of dependencies among multiple random variables is a problem of considerable theoretical and practical interest. Within the context of Bayesian Networks, a practical and surprisingly successful solution to this…

The manifold hypothesis presumes that high-dimensional data lies on or near a low-dimensional manifold. While the utility of encoding geometric structure has been demonstrated empirically, rigorous analysis of its impact on the learnability…

We investigate the landscape of generalized geometries that can be derived from Monge-Amp\`ere structures. Instead of following the approaches of Banos, Roubtsov, Kosmann-Schwarzbach, and others, we take a new path inspired by the results…

We present an improved algorithm for {\em quasi-properly} learning convex polyhedra in the realizable PAC setting from data with a margin. Our learning algorithm constructs a consistent polyhedron as an intersection of about $t \log t$…

Moir\'e patterns made of two-dimensional (2D) materials represent highly tunable electronic Hamiltonians, allowing a wide range of quantum phases to emerge in a single material. Current modeling techniques for moir\'e electrons requires…

The Restricted Boltzmann Machine (RBM) is one of the simplest generative neural networks capable of learning input distributions. Despite its simplicity, the analysis of its performance in learning from the training data is only well…

The study of topological bandstructures is an active area of research in condensed matter physics and beyond. Here, we combine recent progress in this field with developments in machine-learning, another rising topic of interest.…

We prove that Moore machines can be exactly realized as state-space models (SSMs), establishing a formal correspondence between symbolic automata and these continuous machine learning architectures. These Moore-SSMs preserve both the…

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning…

Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure,…

Optimal transportation of raw material from suppliers to customers is an issue arising in logistics that is addressed here with a continuous model relying on optimal transport theory. A physics informed neuralnetwork method is advocated…

Many techniques in machine learning attempt explicitly or implicitly to infer a low-dimensional manifold structure of an underlying physical phenomenon from measurements without an explicit model of the phenomenon or the measurement…

Memory-based meta-learning is a technique for approximating Bayes-optimal predictors. Under fairly general conditions, minimizing sequential prediction error, measured by the log loss, leads to implicit meta-learning. The goal of this work…