Related papers: Restricted Riemannian geometry for positive semide…

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive…

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…

The space of all non degenerate bilinear structures on a manifold $M$ carries a one parameter family of pseudo Riemannian metrics. We determine the geodesic equation, covariant derivative, curvature, and we solve the geodesic equation…

Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties…

We prove that the recently shown cohomological obstruction for quasiregular ellipticity has a generalization in the theory of quasiregular values. More specifically, if $M$ is a closed, connected, and oriented Riemannian $n$-manifold, and…

This short survey reviews some aspects of spaces of positive-definite self-adjoint linear transformations on R^n and on C^n, including the standard Riemannian metric and the relation with the exponential mapping acting on self-adjoint…

In the domain of image-set based classification, a considerable advance has been made by representing original image sets as covariance matrices which typical lie in a Riemannian manifold. Specifically, it is a Symmetric Positive Definite…

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 paper explores the Riemannian geometry of the Wasserstein space of the circle, namely $P(S^{1})$, the set of probability measures on the unit circle endowed with the 2-Wasserstein metric. Building on the foundational work of Otto,…

We show that unconstrained quadratic optimization over a Grassmannian $\operatorname{Gr}(k,n)$ is NP-hard. Our results cover all scenarios: (i) when $k$ and $n$ are both allowed to grow; (ii) when $k$ is arbitrary but fixed; (iii) when $k$…

In this work we study the set of strictly accretive matrices, that is, the set of matrices with positive definite Hermitian part, and show that the set can be interpreted as a smooth manifold. Using the recently proposed symmetric polar…

Although many machine learning algorithms involve learning subspaces with particular characteristics, optimizing a parameter matrix that is constrained to represent a subspace can be challenging. One solution is to use Riemannian…

In this paper, we propose a novel space-time geometric representation of human landmark configurations and derive tools for comparison and classification. We model the temporal evolution of landmarks as parametrized trajectories on the…

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the…

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…

Jung et al. (2015) introduced a geometric structure on ${\rm Sym}^+(p)$, the set of $p \times p$ symmetric positive-definite matrices, based on eigen-decomposition. Eigenstructure determines both a stratification of ${\rm Sym}^+(p)$,…

Given a smooth, complete Riemannian manifold $M$ with bounded Ricci curvature and positive injectivity radius, we derive a sharp Sobolev inequality for the embedding of $W^{1,p}(M)$ into $L^{\frac{np}{n-p}}(M)$, when $1\le p< n$. We will…

In this work, we aim to prove algebra properties for generalized Sobolev spaces $W^{s,p} \cap L^\infty$ on a Riemannian manifold, where $W^{s,p}$ is of Bessel-type $W^{s,p}:=(1+L)^{-s/m}(L^p)$ with an operator $L$ generating a heat…

Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded, concave, connected open set $U$ with $C^0$ boundary and $M\setminus U$ is connected. We assume that either the relative homotopy set $\pi_1(M,M\setminus U)=0$ or the…

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best…