arXiv research

Introduces resemblance structure for large scale geometry.

Introduces bounded scale measure and generalizes property A.

Let M be a compact manifold. We show the identity component $\mathrm{Homeo}_0(M)$ of the group of self-homeomorphisms of M has a well-defined quasi-isometry type, and study its large scale geometry. Through examples, we relate this large scale geometry to both the topology of M and the dynamics of group actions on M. T…

Study large-scale geometry of infinite type surface mapping class groups.

We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type FP_\infty by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups.

Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.

Study of graphs interpolating curve and pants graphs, providing formulae and geometry classifications.

A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an or…

We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (all scale uniform equivalences). We prove that these equivalences can be characterized with parallel constructions using a combinatoric tool called common zi…

Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.

Study large-scale geometry of graph braid groups via cubical structures.

We study the large scale geometry of the upper triangular subgroup of PSL(2,Z[1/n]), which arises naturally in a geometric context. We prove a quasi-isometry classification theorem and show that these groups are quasi-isometrically rigid with infinite dimensional quasi-isometry group. We generalize our results to a lar…

In the framework of nonassociative geometry (hep-th/0003238) a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called "diodular discrete structure" which at large spacetime scales `looks like' a differentiable manifold. After a…

Introduces halo products and studies their geometric properties.

FibeRed reduces complex data dimensions while preserving topology.

We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many 3-manifolds, including Euclidean 3-space, hyperbolic 3-space and the product of the h…

This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.

We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gr…

In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT(0) metric space or an…

Let G be a finitely presented group, and G' its commutator subgroup. Let C be the Cayley graph of G' with all commutators in G as generators. Then C is large scale simply connected. Furthermore, if G is a torsion-free nonelementary word-hyperbolic group, C is one-ended. Hence (in this case), the asymptotic dimension of…

Uniformity and proximity are two different ways for defining small scale structures on a set. Coarse structures are large scale counterparts of uniform structures. In this paper, motivated by the definition of proximity, we develop the concept of asymptotic resemblance as a relation between subsets of a set to define a…

SEMASIA provides a large dataset of latent representations for model comparison.

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using parti…

We study the large-scale geometry of 3-manifolds with nontrivial 2-dimensional bounded cohomology, with a view to proving a weak version of the geometrization conjecture for such manifolds.

We present a large scale hyperbolic recommender system. We discuss why hyperbolic geometry is a more suitable underlying geometry for many recommendation systems and cover the fundamental milestones and insights that we have gained from its development. In doing so, we demonstrate the viability of hyperbolic geometry f…

VB uses natural gradients in information geometry.

New method computes affine normal directions efficiently for sparse polynomials.

We describe tests validating progress made toward acceleration and automation of hydrodynamic codes in the regime of developed turbulence by three Deep Learning (DL) Neural Network (NN) schemes trained on Direct Numerical Simulations of turbulence. Even the bare DL solutions, which do not take into account any physics …

Characterizes fundamental groups of disjointly tree-graded spaces.

New algorithm speeds up large-scale statistical inference.

The note proves a metric equivalence for stable bundles on surfaces.

Study shows saddle connection graph's geometry and quasi-isometry properties.

Consider a connected orientable surface $S$ of infinite topological type, i.e. with infinitely-generated fundamental group. We describe the large-scale geometry of arbitrary connected subgraphs of the arc complex $A(S)$ and curve complex $C(S)$ of $S$, provided they are invariant under a sufficiently big subgroup of th…

We provide a numerically robust and fast method capable of exploiting the local geometry when solving large-scale stochastic optimisation problems. Our key innovation is an auxiliary variable construction coupled with an inverse Hessian approximation computed using a receding history of iterates and gradients. It is th…

The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.

In the study manifolds of Ricci curvature bounded below, a stumbling obstruction is the lack of links between large-scale geometry and small-scale geometry at a fixed reference point. There have been few links (volume, dimension) when the unit ball at the point is not collapsed, that is, $\mathrm{vol}(B_1(p))\ge v>0$. …

We propose to jointly learn multi-view geometry and warping between views of the same object instances for robust cross-view object detection. What makes multi-view object instance detection difficult are strong changes in viewpoint, lighting conditions, high similarity of neighbouring objects, and strong variability i…

Extracts geometric information from point-clouds for multiclass classification.

Study the geometry of graph product extension graphs.

We study the asymptotic cones of the universal covering spaces of closed 4-dimensional nonpositively curved real analytic manifolds. We show that the existence of nonstandard components in the Tits boundary, discovered by Christoph Hummel and Victor Schroeder, depends only on the quasi-isometry type of the fundamental …

We prove the Farrell-Jones Conjecture for mapping class groups. The proof uses the Masur-Minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of Teichmueller space. The proof is presented in an axiomatic setup, extending the projection axioms of Bestvina-Bromberg-Fujiwara…

eDCF estimates intrinsic dimension using local connectivity.

Develops a new framework for large-scale geometry.

Study shows one-dimensional location-scale-shape models are flat in Wasserstein geometry.

In this paper we explore coarse properties of cusp-decomposable manifolds first defined by Nguyên Phan. We describe the large scale geometry of the universal cover of a cusp-decomposable manifold and of quasi-isometries between two such universal covers. This description will provide us the tools to prove quasi-isometr…

While matrix factorisation models are ubiquitous in large scale recommendation and search, real time application of such models requires inner product computations over an intractably large set of item factors. In this manuscript we present a novel framework that uses the inverted index representation to exploit struct…

Algorithm tackles large-scale portfolio optimization with higher moments, improving computational efficiency.

Generative model for joint discrete distributions using randomized assignment flows.