Related papers: Learning with symmetric positive definite matrices…
Classical W-symmetry is globally parametrized by the Grassmannian Manifold which is associated with the non-relativistic fermions. We give the bosonization rule which defines the natural higher coordinates system to describe the W-geometry.…
A growing number of generative statistical models do not permit the numerical evaluation of their likelihood functions. Approximate Bayesian computation (ABC) has become a popular approach to overcome this issue, in which one simulates…
This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures $\mathcal{P}_{\mathrm{a.c.}}(\Omega)$ with $\Omega$ a compact and convex subset of…
This paper proposes and analyzes Riemannian optimization algorithms on the manifold of unitary and symmetric matrices, denoted ${\cal {U}}_s$, which naturally models the scattering matrices of passive and reciprocal devices such as…
We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the…
Gaussian distributions can be generalized from Euclidean space to a wide class of Riemannian manifolds. Gaussian distributions on manifolds are harder to make use of in applications since the normalisation factors, which we will refer to as…
In this article, we derive Stein's method for approximating a spatial random graph by a generalised random geometric graph, which has vertices given by a finite Gibbs point process and edges based on a general connection function. Our main…
In this work a thorough study of a number of specific supersymmetric sigma-models with extended supersymmetry is performed within the context of generalised complex geometry. More specifically the supersymmetric Wess-Zumino-Witten model on…
Post-data statistical inference concerns making probability statements about model parameters conditional on observed data. When a priori knowledge about parameters is available, post-data inference can be conveniently made from Bayesian…
We study the natural gradient method for learning in deep Bayesian networks, including neural networks. There are two natural geometries associated with such learning systems consisting of visible and hidden units. One geometry is related…
We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability…
Flow matching has recently emerged as a flexible and efficient framework for generative modelling by learning deterministic transport dynamics between probability measures. In this work, we extend flow matching to the space of probability…
Gromov-Wasserstein (GW) distances are combinations of Gromov-Hausdorff and Wasserstein distances that allow the comparison of two different metric measure spaces (mm-spaces). Due to their invariance under measure- and distance-preserving…
We propose a projected Wasserstein gradient descent method (pWGD) for high-dimensional Bayesian inference problems. The underlying density function of a particle system of WGD is approximated by kernel density estimation (KDE), which faces…
We propose min Generalized Sliced Gromov--Wasserstein (min-GSGW), a sliced formulation for the Gromov--Wasserstein (GW) problem using expressive generalized slicers. The key idea is to learn coupled nonlinear slicers that assign compatible…
Symmetric positive-definite (SPD) matrix datasets play a central role across numerous scientific disciplines, including signal processing, statistics, finance, computer vision, information theory, and machine learning among others. The set…
This work considers the problem of computing distances between structured objects such as undirected graphs, seen as probability distributions in a specific metric space. We consider a new transportation distance (i.e. that minimizes a…
A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…
Image reconstruction based on indirect, noisy, or incomplete data remains an important yet challenging task. While methods such as compressive sensing have demonstrated high-resolution image recovery in various settings, there remain issues…
Stochastic optimization powers the scalability of modern artificial intelligence, spanning machine learning, deep learning, reinforcement learning, and large language model training. Yet, existing theory remains largely confined to Hilbert…