22 citations · 22 across the 16 of their papers we have counts for
17 papers
Wishart Matrices and Quantum Geometry: Foundations and Applications in Quantum Information
Noémie C. Combe
We present a unified framework for the study of Wishart matrices (W_p(n,Σ)), which generalize the chi-squared distribution to matrix-variate settings and model the covariance struc…
On Duality, Legendre Bundles and Deformations
N. C. Combe, P. G. Combe, H. K. Nencka
We introduce the Legendre bundle, a geometric structure encoding the essential duality of dually flat (Hessian) manifolds, and demonstrate that both exponential families in informa…
The Infinite Sphere and Galois Belyi maps
Noémie C. Combe
We show that the space of Belyi maps admits a natural parametrization by an infinite-dimensional sphere arising from Voiculescu's theory of noncommutative probability spaces. We sh…
Shape Theory via the Atiyah--Molino Reconstruction and Deformations
Noémie C. Combe, Hanna N. Nencka
Reconstruction problems lie at the very heart of both mathematics and science, posing the enigmatic challenge: \emph{How does one resurrect a hidden structure from the shards of in…
Proving the Grothendieck--Teichmüller Conjecture for Profinite Spaces & The Galois Grothendieck Path Integral
Noémie C. Combe
We establish that the Grothendieck-Teichmuller conjecture, which predicts an isomorphism between the Grothendieck-Teichmuller group GT and the absolute Galois group of rational num…
Quantum Geometry insights in Deep Learning
Noémie C. Combe
In this paper, we explore the fundamental role of the Monge-Ampère equation in deep learning, particularly in the context of Boltzmann machines and energy-based models. We first re…
Exploring information geometry: Recent Advances and Connections to Topological Field Theory
Noémie C. Combe, Philippe G. Combe, Hanna K. Nencka
This introductory text arises from a lecture given in Göteborg, Sweden, given by the first author and is intended for undergraduate students, as well as for any mathematically incl…
On Multiquantum Bits, Segre Embeddings and Coxeter Chambers
Noémie C. Combe
This work explores the interplay between quantum information theory, algebraic geometry, and number theory, with a particular focus on multiqubit systems, their entanglement struct…
Maximum Likelihood, permutohedra and Associativity Equations
Noémie C. Combe
We consider the cone of concentration matrices related to linear concentration models and Wishart laws. We prove that this cone is a Monge--Ampère domain and that the log-likelihoo…
On the geometry of Kähler--Frobenius manifolds and their classification
Noémie. C. Combe
The purpose of this article is to show that flat compact Kähler manifolds exhibit the structure of a Frobenius manifold, a structure originating in 2D Topological Quantum Field The…
Wishart cones and quantum geometry
Noemie C. Combe
An important object appearing in the framework of the Tomita--Takesaki theory is an invariant cone under the modular automorphism group of von Neumann algebras. As a result of the…
Learning on hexagonal structures and Monge-Ampère operators
Noémie C. Combe
Dually flat statistical manifolds provide a rich toolbox for investigations around the learning process. We prove that such manifolds are Monge-Ampère manifolds. Examples of such m…
Landau-Ginzburg models, Monge-Ampère domains and (pre-)Frobenius manifolds
Noémie C. Combe
Kontsevich suggested that the Landau-Ginzburg model presents a good formalism for homological mirror symmetry. In this paper we propose to investigate the LG theory from the viewpo…
On Frobenius structures in symmetric cones
Noemie C. Combe
We prove that in any strictly convex symmetric cone there exists a non empty locus where the WDVV equation is satisfied (i.e. there exists a hyperplane being a Frobenius manifo…
Manin conjecture for statistical pre-Frobenius manifolds, hypercube relations and motivic Galois group in coding
N. C. Combe
This article develops, via the perspective of (arithmetic) algebraic geometry and category theory, different aspects of geometry of information. First, we describe in the terms of…
Quantum Operads
Noemie Combe, Yuri Manin, Matilde Marcolli
The most standard description of symmetries of a mathematical structure produces a group. However, when the definition of this structure is motivated by physics, or information the…
Mind2Mind : transfer learning for GANs
Yaël Frégier, Jean-Baptiste Gouray
Training generative adversarial networks (GANs) on high quality (HQ) images involves important computing resources. This requirement represents a bottleneck for the development of…