Related papers: Categorical and geometric methods in statistical, …

We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d…

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques…

We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…

Latent representations are an important theme in modern machine learning. Any network training with the notion of locality introduces a latent geometry which we can analyze with the help of differential geometry, specifically information…

Probabilistic graphical modeling is a branch of machine learning that uses probability distributions to describe the world, make predictions, and support decision-making under uncertainty. Underlying this modeling framework is an elegant…

Probabilistic graphical models (PGMs) are widely used to discover latent structure in data, but their success hinges on selecting an appropriate model design. In practice, model specification is difficult and often requires iterative…

Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic…

This chapter collects several probabilistic tools that proved to be useful in the analysis of randomized search heuristics. This includes classic material like Markov, Chebyshev and Chernoff inequalities, but also lesser known topics like…

We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We…

Machine learning (ML) has emerged as a powerful tool for tackling complex regression and classification tasks, yet its success often hinges on the quality of training data. This study introduces an ML paradigm inspired by domain knowledge…

We describe algorithms based on invariant theory to solve problems on the geometry of curves, mainly those of genus 2, 3 and 4. New theoretical results building on the first author's PhD thesis are also included.

The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years in biology as the cornerstone of attempts to infer and reconstruct the ancestral…

In this paper we formulate a probabilistic model for class-specific discriminant subspace learning. The proposed model can naturally incorporate the multi-modal structure of the negative class, which is neglected by existing class-specific…

This text introduces geometric quantization on orbifolds. After reviewing the necessary background, it develops new treatments of prequantization, polarizations, and metaplectic correction for symplectic orbifolds.

We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly…

We introduce the Locally Linear Latent Variable Model (LL-LVM), a probabilistic model for non-linear manifold discovery that describes a joint distribution over observations, their manifold coordinates and locally linear maps conditioned on…

Predictions and forecasts of machine learning models should take the form of probability distributions, aiming to increase the quantity of information communicated to end users. Although applications of probabilistic prediction and…

Symplectic and Poisson geometry emerged as a tool to understand the mathematical structure behind classical mechanics. However, due to its huge development over the past century, it has become an independent field of research in…

The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…

In the last quarter of a century, algebraic statistics has established itself as an expanding field which uses multilinear algebra, commutative algebra, computational algebra, geometry, and combinatorics to tackle problems in mathematical…