arXiv research

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from the Satake diagram, in a way that is suited for the use with comput…

This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, F…

Paper constructs subcomplexes from filtered Riemannian manifolds.

We provide the first experimental results on non-synthetic datasets for the quasi-diagonal Riemannian gradient descents for neural networks introduced in [Ollivier, 2015]. These include the MNIST, SVHN, and FACE datasets as well as a previously unpublished electroencephalogram dataset. The quasi-diagonal Riemannian alg…

We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.

Method approximates Riemannian barycenter on manifolds.

RieCUR improves Robust PCA by combining Riemannian optimization and CUR decompositions.

Computes minimal polynomials for generalized Heisenberg groups.

Bayesian quadrature improves integration on Riemannian manifolds.

We introduce a new approach for computing curvature of sub-Riemannian manifolds. Curvature is here meant as symplectic invariants of Jacobi curves of geodesics, as introduced by Zelenko and Li. We describe how they can be expressed using a compatible affine connection and induced tensors, without any restriction on our…

Paper proposes a method to classify EEG signals with missing data.

We introduce geomstats, a python package that performs computations on manifolds such as hyperspheres, hyperbolic spaces, spaces of symmetric positive definite matrices and Lie groups of transformations. We provide efficient and extensively unit-tested implementations of these manifolds, together with useful Riemannian…

Paper presents efficient computation of robust Wasserstein distance using Riemannian optimization.

We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure

Let $(M,g)$ be a Riemannian manifold, $L(M)$ its frame bundle. We construct new examples of Riemannian metrics on $L(M)$, which are obtained from Riemannian metrics on the tangent bundle $TM$. We compute the Levi--Civita connection and curvatures of these metrics.

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations exist, it is sometimes difficult to gain an overview how differential geometry and…

Neural networks learn discrete tasks on continuous data via emergent geometry.

Study uniquely determines Riemannian metric derivatives from boundary data.

Study on properties of tangential hypersurfaces in product-like manifolds.

The Riemannian barycentre is one of the most widely used statistical descriptors for probability distributions on Riemannian manifolds. At present, existing algorithms are able to compute the Riemannian barycentre of a probability distribution, only if i.i.d. samples of this distribution are readily available. However,…

Study geodesic complexity in homogeneous Riemannian manifolds.

The paper calculates how random changes affect paths on a complex geometric space.

The main result of the paper is a computation of the Ricci curvature of $\DS/S^1$. Unlike earlier results on the subject, we do not use the Kähler structure symmetries to compute the Ricci curvature, but rather rely on classical finite-dimensional results of Nomizu et al on Riemannian geometry of homogeneous spaces.

Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.

The curvature tensor and the scalar curvature are computed in the space of positive definite real matrices endowed by the Kubo-Mori inner product as a Riemannian metric.

GEORCE computes geodesics quickly and accurately.

Riemannian geometry improves protein dynamics analysis.

We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Rieman…

In the previous paper [GLM2018], we showed that the theory of harmonic maps between Riemannian manifolds may be discretized by introducing triangulations with vertex and edge weights on the domain manifold. In the present paper, we study convergence of the discrete theory to the smooth theory when taking finer and fine…

The analysis of manifold-valued data requires efficient tools from Riemannian geometry to cope with the computational complexity at stake. This complexity arises from the always-increasing dimension of the data, and the absence of closed-form expressions to basic operations such as the Riemannian logarithm. In this pap…

New formulas for geodesics on Stiefel and flag manifolds using trust-region method.

A new retraction on Stiefel manifold with a closed-form inverse.

We establish a few formulas that compute the volume of the zero-set (or nodal set) of a function on a compact Riemannian manifold as integrals of functionals of the function and its derivatives.

For a Riemannian $G$-structure, we compute the divergence of the vector field induced by the intrinsic torsion. Applying the Stokes theorem, we obtain the integral formula on a closed oriented Riemannian manifold, which we interpret in certain cases. We focus on almost harmitian and almost contact metric structures.

GEORCE-FM algorithm optimizes Fréchet means and distances efficiently.

Study finds conditions for existence of specific pseudo-Riemannian cobordisms.

New quasi-geodesics for Stiefel manifold simplify complex computations.

Introduces canonical connections for sub-Riemannian manifolds with constant symbol.

Control Contraction Metrics (CCMs) provide a nonlinear controller design involving an offline search for a Riemannian metric and an online search for a shortest path between the current and desired trajectories. In this paper, we generalize CCMs to Finsler geometry, allowing the use of non-Riemannian metrics. We provid…

It is explained how to find the de~Rham decomposition of a Riemannian manifold and the Wu decomposition of a Lorentzian manifold. For that it is enough to find parallel symmetric bilinear forms on the manifold, and do some linear algebra. This result will allow to compute the connected holonomy group of an arbitrary Ri…

Develops Riemannian geometry for optimization on manifolds with detailed derivations.

Paper derives sub-Riemannian versions of Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for twisted BCV spaces.

Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree $N$ to be thos…

Computes tube formulas for valuations in complex space forms.

Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.

New metrics defined for full-rank correlation matrices, ensuring unique operations.

Researchers calculate entropy of heat kernel on manifolds for very small times.

The Navier-Stokes equations on certain manifolds can perform universal computation.