arXiv research

Dimensionality reduction on Riemannian manifolds is challenging due to the complex nonlinear data structures. While probabilistic principal geodesic analysis~(PPGA) has been proposed to generalize conventional principal component analysis (PCA) onto manifolds, its effectiveness is limited to data with a single modality…

This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.

R-PCA extends PCA to Riemannian manifolds for structured data.

Polynomial chaos expansions on Grassmannian submanifolds for high-dimensional stochastic systems.

We develop a novel analogue of Euclidean PCA (principal component analysis) for data taking values on a Riemannian symmetric space, using totally geodesic submanifolds as approximating lower dimnsional submanifolds. We illustrate the technique on n-spheres, Grassmannians, n-tori and polyspheres.

A new dynamical formulation of log-PCA captures local principal modes of geodesic variations.

We propose a fair principal component analysis method that balances reconstruction error and subgroup fairness.

Unified PCA framework on flag manifolds for robust data analysis.

Study geodesics on nested non-holonomic systems.

Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family. We propose to study this problem under the optimal transport (Wasserstein) geometry, using curves that are re…

Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.

Study normal operators of double fibration transforms with conjugate points.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of $k+1$ reference points.…

In this paper we study the shape space of curves with values in a homogeneous space $M = G/K$, where $G$ is a Lie group and $K$ is a compact Lie subgroup. We generalize the square root velocity framework to obtain a reparametrization invariant metric on the space of curves in $M$. By identifying curves in $M$ with thei…

New equations reveal how cylinder power in progressive lenses depends on geodesic curvature.

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmuller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmuller ge…

We consider random walks on the mapping class group whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anoso…

Study of logarithms in SVD-closed subgroups of unitary group.

The paper analyzes the excess risk of PCA and provides a precise characterization.

The study examines surfaces in hyperbolic 3-manifolds that become nearly flat.

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we p…

Geometric approach to quantum thermodynamics models state spaces and processes.

We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.

We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds…

In this note we construct an explicit example of a (compact) conformally flat Riemannian manifold which admits a totally geodesic foliation of codimension one with no isoparametric leaves. This answers negatively the question: is every hypersurface with constant principal curvatures isoparametric?

Geodesic flows with specific integrals are linked to special 4-webs.

Mannheim curves are defined for immersed curves in 3-dimensional sphere S^3 . The definition is given by considering the geodesics of S^3. First, two special geodesics, called principal normal geodesic and binormal geodesic, of S^3 are defined by using Frenet vectors of a curve immersed in S^3. Later, the curve alpha i…

The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.

Efficient private matrix analysis algorithms for recent variants.

While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods are not amenable to the analysis of such manifolds. This is mainly due to the necessary spectral decomposition step, which limits control over the…

Study non-existence of complex ball quotients in Torelli locus.

The Bratteli diagram is an infinite graph which reflects the structure of projections in a C*-algebra. We prove that every strictly ergodic unimodular Bratteli diagram of rank 2g+m-1 gives rise to a minimal geodesic lamination with the m-component principal region on a surface of genus g greater or equal to 1. The proo…

In the recent years, Riemannian shape analysis of curves and surfaces has found several applications in medical image analysis. In this paper we present a numerical discretization of second order Sobolev metrics on the space of regular curves in Euclidean space. This class of metrics has several desirable mathematical …

The paper introduces a method for interpretable principal component analysis of high-dimensional time series.

We describe min-max formulas for the principal eigenvalue of a $V$-drift Laplacian defined by a vector field $V$ on a geodesic ball of a Riemannian manifold $N$. Then we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field, under p…

Proofs and descriptions of totally geodesic submanifolds in symmetric spaces.

The statistical analysis of data lying on a differentiable, locally Euclidean, manifold introduces a variety of challenges because the analogous measures to standard Euclidean statistics are local, that is only defined within a neighbourhood of each datapoint. This is because the curvature of the space means that the c…

Geodesic completeness and optimal Sobolev index proven for Minkowski spacetimes.

Paper finds principal eigenvalue for infinity Laplacian in metric spaces.

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic…

It's well-known in \kahler geometry that the infinite dimensional symmetric space $\hcal$ of smooth \kahler metrics in a fixed \kahler class on a polarized \kahler manifold is well approximated by finite dimensional submanifolds $\bcal_k \subset \hcal$ of Bergman metrics of height $k$. Then it's natural to ask whether …

Given a vector field $X$ in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to $X$ if the projection of $X$ onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful charac…

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given i…

A real hypersurface in the complex quadric $Q^m=SO_{m+2}/SO_mSO_2$ is said to be $\mathfrak A$-principal if its unit normal vector field is singular of type $\mathfrak A$-principal everywhere. In this paper, we show that a $\mathfrak A$-principal Hopf hypersurface in $Q^m$, $m\geq3$ is an open part of a tube around a t…

A neural network learns efficient parametrizations of product shape spaces.

Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two tangents coincide with a given tangent, then we obtain the normal curvature of t…

If we consider the moduli space of flat connections of a non trivial principal SO(3)-bundle over a surface, then we can define a map from the set of perturbed closed geodesics, below a given energy level, into families of perturbed Yang-Mills connections depending on a small parameter. In this paper we show that this m…

We study the Jacobi osculating rank of geodesics on naturally reductive homogeneous manifolds and we apply this theory to the 3-dimensional case. Here, each non-symmetric, simply connected naturally reductive 3-manifold can be given as a principal bundle over a surface of constant curvature, such that the curvature of …