arXiv research

Unique median structures found in hyperbolic spaces.

Groups with specific properties have similar cubulations and coarse median structures.

New concept of coarse medians for higher rank symmetric spaces.

Study of automorphisms and splittings of special groups, showing infinite groups under certain conditions.

The paper studies automorphisms of RAAGs and RACGs, proving properties of their fixed subgroups.

Study on connectivity and geometry of random Coxeter groups.

Generators found for automorphisms of special groups.

We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have the coarse median property defined by Bowditch. Consequences for such graphs include a distance formula analogous to Mas…

Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3--manifold groups. We investigate strongly quasiconvex subsets in this class and characterize them in terms of their contracting pro…

Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.

The paper studies properties of group relations induced by compatible coarse structures.

Study semi-coarse spaces' homotopy and homology, extending coarse geometry.

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the group-compact coarse structure generalizing several familiar results for discrete group…

We study the concept of coarse disjointness and large scale $n$-to-$1$ functions. As a byproduct, we obtain an Ostrand-type characterization of asymptotic dimension for coarse structures. It is shown that properties like finite asymptotic dimension, coarse finitism, large scale weak paracompactness, ect. are all invari…

Upper bound for Hausdorff distance between hyperbolic space and its medianization.

New graph properties inherited by Frechet mean and median.

Defines coarse cohomology of space complements, proving new duality results.

In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of cone of a compa…

New method approximates controllability of large networks from coarse summaries.

Proper actions on bornological spaces are characterized with compatible coarse structures.

Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.

Atomistic or ab-initio molecular dynamics simulations are widely used to predict thermodynamics and kinetics and relate them to molecular structure. A common approach to go beyond the time- and length-scales accessible with such computationally expensive simulations is the definition of coarse-grained molecular models.…

New method for summarizing ranking distributions using consensus ranking distributions.

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…

Study recovers community structure from coarse graph measurements.

The main results of the paper are: \begin{Prop}\label{GenSvarc-Milnor} A group $G$ acting coarsely on a coarse space $(X,\CC)$ induces a coarse equivalence $g\to g\cdot x_0$ from $G$ to $X$ for any $x_0\in X$. \end{Prop} Theorem: \label{GenGromovThm} Two coarse structures $\CC_1$ and $\CC_2$ on the same set $X$ are equ…

Proposes a method for coarse graph alignment using sparse partial least squares.

This paper is devoted to introducing coarse structures in a very simple way, namely as an equivalence relation on the set of simple ends. As an application we show that Gromov boundary of every hyperbolic space is an example of a Higson corona and each Freundenthal compactification is an example of a Higson compactific…

Classifies Legendrian Hopf links in lens spaces.

Bornological metrics on groups are studied, showing equivalence classes and constructing non-equivalent improper metrics.

We prove a geometric model for HHS hierarchies as CAT(0) cube complexes.

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 show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of compatibility possible with the median geometry. Our theorem is also relevant for potential …

Simplified calculus for manifold operators, proving index theorems.

Convex cores found for group actions on median spaces.

Many practical techniques for probabilistic inference require a sequence of distributions that interpolate between a tractable distribution and an intractable distribution of interest. Usually, the sequences used are simple, e.g., based on geometric averages between distributions. When models are expressed as probabili…

We introduce and begin to explore the mean and median of finite sets of shapes represented as integral currents. The median can be computed efficiently in practice, and we focus most of our theoretical and computational attention on medians. We consider questions on the existence and regularity of medians. While the me…

The paper develops glueing theory for topological spaces and applies it to compactifications.

Graph neural networks (GNNs) have been shown to replicate convolutional neural networks' (CNNs) superior performance in many problems involving graphs. By replacing regular convolutions with linear shift-invariant graph filters (LSI-GFs), GNNs take into account the (irregular) structure of the graph and provide meaning…

This work achieves exponential concentration in heavy-tailed data over CAT(κ) spaces using the Fréchet median.

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…

Financial markets analyzed by reducing correlation matrix complexity.

A method for estimating the median of gradients in stochastic optimization.

This paper introduces online algorithms to estimate robust geometric median in large data streams.

The study finds that maximizing median returns is the only viable strategy in portfolio selection.

The study confirms Gromov's speculation and provides bounds for taming symplectic structures.

Sparsity helps reduce the computational complexity of deep neural networks by skipping zeros. Taking advantage of sparsity is listed as a high priority in next generation DNN accelerators such as TPU. The structure of sparsity, i.e., the granularity of pruning, affects the efficiency of hardware accelerator design as w…

The consistency of Fréchet medians is proved for probability measures in proper metric spaces. In the context of Riemannian manifolds, assuming that the probability measure has more than a half mass lying in a convex ball and verifies some concentration conditions, the positions of its Fréchet medians are estimated. It…