arXiv research

Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.

Extended Rank-One Theorem to special metric spaces.

Affine maps reveal higher rank structures in certain spaces.

In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher filling functions. For a proper, cocompact, simply-connected geodesic metric space of non-curvature in the sen…

Improved deep learning performance in financial markets by using rank space.

Right inverse found for Cartan differential in rank-1 symmetric spaces.

In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.

Classifies foliations on specific symmetric spaces.

New estimator GMIPS reduces variance in ranking policy evaluation.

Geometrically, tensors of fixed rank form a minimal submanifold.

No Einstein hypersurfaces found in Damek-Ricci spaces.

Characterizes higher rank model geometries using antipodal sets.

Constructs explicit p-harmonic functions on specific Lie groups.

We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded nonpositive flag curvature with rank at least $2$ whose universal cover is irreducible, is a locally symmetric space or a locall…

Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.

Proves conditions for Fourier transforms in rank 1 symmetric spaces.

Researchers classify geodesic orbit spaces for compact Lie groups of rank two.

The study proves conditions for CAT(0) spaces with higher rank rigidity.

Characterizes Stein surfaces with finite homotopy rank-sum.

The aim of this paper is to study the spectrum of the $L^p$ Laplacian and the dynamics of the $L^p$ heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in t…

We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.

New concept of coarse medians for higher rank symmetric spaces.

Convex cores found for group actions on median spaces.

We prove a version of the Tits alternative for groups acting on complete, finite rank median spaces. This shows that group actions on finite rank median spaces are much more restricted than actions on general median spaces. Along the way, we extend to median spaces the Caprace-Sageev machinery and part of Hagen's theor…

We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank one isometries for Hilbert geometries and characterize them as being equivalent to contracting elem…

Compact RCD(K,N) spaces with maximal rank are homeomorphic to infranilmanifolds.

New rigidity result for CAT(0) spaces of higher rank.

Learning to rank is a supervised learning problem where the output space is the space of rankings but the supervision space is the space of relevance scores. We make theoretical contributions to the learning to rank problem both in the online and batch settings. First, we propose a perceptron-like algorithm for learnin…

We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of ${\rm CAT}(0)$ cube complexes. Examples of median spaces with compact intervals include all finite rank median spaces and all proper median spaces of infinite rank. Our methods also work for general median algebra…

Compact rank one symmetric spaces are rigid under certain curvature conditions.

Differentiable sorting and rank normalization are incompatible, with specific conditions for admissibility.

IRMAE learns compact latent spaces by minimizing rank.

Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and po…

The problem of recovering a low $n$-rank tensor is an extension of sparse recovery problem from the low dimensional space (matrix space) to the high dimensional space (tensor space) and has many applications in computer vision and graphics such as image inpainting and video inpainting. In this paper, we consider a new …

Study shows how certain spaces can be mapped to R^n with specific properties.

We show that polar actions of cohomogeneity two on simple compact Lie groups of higher rank, endowed with a biinvariant Riemannian metric, are hyperpolar. Combining this with a recent result of the second-named author, we are able to prove that polar actions induced by reductive algebraic subgroups in the isometry grou…

Proves homological inequality for cycles in Hadamard spaces of asymptotic rank 2.

New random walk results on rank one symmetric spaces.

We show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most of the higher rank symmetric spaces. As a consequence, we prove that the locally finite simplicial volume of most $\mathbb Q$-rank $1$ locally symmetric spaces is positive, which has been open for many y…

We prove an obstruction at the level of rational cohomology in small degrees to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As an application, we provide evidence for a generalized conjecture of Hopf that says that no symm…

In this paper, we show that there exists no equifocal submanifold with non-flat section in four irreducible simply connected symmetric spaces of compact type and rank two. Also, we show a fact for the sections of equifocal submanifolds with non-flat section in other irreducible simply connected symmetric spaces of comp…

Develops sublinear Morse theory in symmetric spaces.

Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is…

Volume comparison theorem for rank 1 symmetric spaces proved.

Study on diagonal and separating coordinates for symmetric spaces of rank 1.

New tools compute index of symmetry in homogeneous fibrations.

Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.