Coarse density of subspaces in moduli space of Riemann surfaces.
problem Characterizing subspaces of moduli space that are coarsely dense.
method Using Teichmüller metric and projections of orbit closures in abelian differentials.
result Projections of certain strata in abelian differentials are coarsely dense in moduli space.
Combines coarse learners for nonparametric probabilistic regression.
problem Predicting full probability density functions without strong assumptions.
method Combining gradient boosted forests trained on coarsened target values.
result Prediction intervals have high fidelity and provide valuable insights.
Temporal coarse-graining of latent default paths explains effective correlation in corporate defaults.
problem Understanding effective default correlation in corporate defaults.
method Temporal coarse-graining of latent default-probability paths, applied to corporate default-count data.
result Temporal coarse-graining provides a scale-consistent baseline that improves identifiability and reduces over-allocation of long-horizon fluctuations.
Temporal aggregation reveals latent default correlation from monthly data.
problem Understanding effective default correlation from monthly default data.
method Temporal coarse-graining of latent default-probability paths.
result Temporal coarse-graining improves identifiability and reduces over-allocation of long-horizon fluctuations.
Diffusion models can generalize well even with coarse scores, thanks to the manifold hypothesis.
problem Understanding why diffusion models generate novel samples with coarse scores.
method Exploring the manifold hypothesis to explain diffusion model behavior.
result Diffusion models trained with coarse scores can achieve near-parametric rates of generalization, faster than estimating the full data distribution.
New method approximates controllability of large networks from coarse summaries.
problem Controlling large-scale linear dynamical systems with incomplete network information.
method Algorithm using stochastic block model to estimate controllability from coarse summaries.
result Average controllability of fine-scale system can be well approximated by coarse-scale system.
The paper tackles physical constraints in probabilistic machine learning for CG models of high-dimensional systems.
problem Introducing physical constraints in probabilistic machine learning objectives for coarse-graining dynamical systems.
method Formulating coarse-graining process using probabilistic state-space model and accounting for constraints as virtual observables.
result Probabilistic inference tools can identify coarse-grained variables without needing a fine-to-coarse projection or time-derivatives.
Tensor network architecture for classification and regression using wavelet transformations.
problem Efficiently performing classification and regression tasks on complex data.
method Tensor network layers based on MERA and MPS, with adaptive fine-graining.
result Adaptive fine-graining improves model performance without loss in accuracy.
Study semi-coarse spaces' homotopy and homology, extending coarse geometry.
problem Extend homotopy and homology concepts to semi-coarse spaces.
method Analyze homotopy and construct homology groups invariant under semi-coarse homotopy equivalence.
result Show semi-coarse homology is isomorphic to Vietoris-Rips homology for graphs.
We propose a method based on finite mixture models for classifying a set of observations into number of different categories. In order to demonstrate the method, we show how the component densities for the mixture model can be derived by using the maximum entropy method in conjunction with conservation of Pythagorean m…
Machine learning discovers equations from simulated data.
problem Discovering equations from computer-generated data.
method Sparse regression for equation learning.
result Machine learning can discover equations from complex data.
We define a notion of free product for coarse spaces that generalizes the corresponding notion of a free product for groups. We show that free products preserve coarse properties such as coarse property C, finite coarse decomposition complexity, and coarse property A. We also give an upper bound estimate on the dimensi…
Defines coarse cohomology of space complements, proving new duality results.
problem Defining and studying coarse cohomology of space complements.
method Introducing a model space, new approach to PD spaces, homological criterion.
result Proves new versions of coarse Poincaré duality and Alexander duality.
Geodesic currents in strongly hyperbolic spaces are dense.
problem Characterizing geodesic currents with strongly hyperbolic dual pseudometrics.
method Combining finite-cover argument and boundary data characterization.
result Dense subset of geodesic currents with strongly hyperbolic dual pseudometrics.
For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results generalize known results for the analytic coarse assembly map for K-homology to general …
New method optimizes complex models with minimal data, proving global optimality.
problem Optimizing complex models with unknown cost functions and prior distributions.
method Introduces 'coarse learnability' and an iterative MBO algorithm with sample correction.
result Achieves global optimality with polynomial sample complexity.
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an ass…
We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar w…
Interprets coarse symbol and index classes for Callias type operators.
problem Understanding coarse geometry and index classes for Callias type operators.
method Interprets coarse symbol and index classes in terms of K-theory classes of coarse corona.
result Local positivity and invertibility conditions are incorporated into support conditions in K-theory.
The coarse category was established by Roe to distill the salient features of the large-scale approach to metric spaces and groups that was started by Gromov. In this paper, we use the language of coarse spaces to define coarse versions of asymptotic property C and decomposition complexity. We prove that coarse propert…
We show that coarse property C is preserved by finite coarse direct products. We also show that the coarse analog of Dydak's countable asymptotic dimension is equivalent to the coarse version of straight finite decomposition complexity and is therefore preserved by direct products.
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…
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…
New concept of coarse medians for higher rank symmetric spaces.
problem Understanding medians in higher rank symmetric spaces.
method Introducing coarse r-median spaces and proving their existence. result Existence of coarse higher medians on divisible and quasi-homogeneous convex domains.
Study uses CNNs to upscale wind speed data from 100 km to 3 km, improving subgrid-scale variability.
problem Recovering fine-scale wind speed information from coarse data.
method Convolutional neural networks (CNNs) with different input configurations (coarse wind speed, fine-scale topography, diurnal cycle) were tested.
result CNN models with coarse wind and fine topography inputs perform best in generalizing to unseen regions.
We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric …
The study explores ends in coarse homotopy of proper geodesic spaces.
problem Understanding ends in coarse homotopy of proper geodesic spaces.
method Recontextualizing ends as a functor and proving properties of coarse path components.
result Existence of a natural surjection from coarse path components to ends, not always an injection.
Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the_Roe (C*-)algebras_ C*(X) and their K…
Classifies homeomorphism groups of countable Stone spaces up to coarse equivalence.
problem Classifying non-locally compact topological groups using geometric group theory.
method Classification based on coarsely bounded sets and quasi-isometry.
result Groups in the second class are quasi-isometric to the Hamming cube.
The paper studies properties of group relations induced by compatible coarse structures.
problem Properties of asymptotic resemblance relations on groups.
method Generalization of asymptotic dimension and introduction of set theoretic coupling.
result Groups with compatible coarse structures that admit a set theoretic coupling are asymptotic equivalent.
We investigate the coarse homology of leaves in foliations of compact manifolds. This is motivated by the observation that the non-leaves constructed by Schweitzer and by Zeghib all have non-finitely generated coarse homology. This led us to ask whether the coarse homology of leaves in a compact manifold always has to …
Paper relates asymptotic dimension to cofinal dimension using coarse proximities.
problem Relating asymptotic dimension to cofinal dimension in metric spaces.
method Introducing coarse proximities and inverse limit constructions.
result Asymptotic dimension is bounded by coarse cofinal dimension and cofinal dimension of Higson corona.
Unique median structures found in hyperbolic spaces.
problem Uniqueness of median structures in hyperbolic spaces.
method Analyzing product of hyperbolic spaces and properties of relative hyperbolicity.
result Non-hyperbolic pants graphs can have unique median structures.
Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
This paper extends boundary embedding results to coarsely convex spaces.
problem Generalizing boundary embedding results to coarsely convex spaces.
method Generalizing Dydak and Virk's work on Gromov hyperbolic spaces to coarsely convex spaces.
result Maps between coarsely convex spaces induce continuous maps between their boundaries.
The paper shows how coarse embeddings affect homological Dehn functions.
problem Characterizing groups with coarse embeddings into hyperbolic groups.
method Demonstrates a coarse embedding theorem for homological filling functions.
result Characterizes groups with coarse embeddings into hyperbolic groups of geometric dimension 2.
Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective ma…
We construct the coarse index class with support condition (as an element of coarse K-homology) of an equivariant Dirac operator on a complete Riemannian manifold endowed with a proper, isometric action of a group. We further show a coarse relative index theorem and discuss the compatibility of the index with the sus…
Generalizes Bestvina's Z-boundaries to coarse Z-boundaries.
problem Establishing properties of Z-boundaries for groups. method Introducing a new concept of a 'coarse Z-boundary' and proving theorems about it. result Admitting a coarse Z-boundary is a pure quasi-isometry invariant. 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 investigate the fixed point property of the group actions on a coarse space and its Higson corona. We deduce the coarse version of Brouwer's fixed point theorem.
Groups with specific properties have similar cubulations and coarse median structures.
problem Understanding the structure of certain groups through cubical coarsening.
method Analyzing right-angled Artin and Coxeter groups, focusing on automorphisms and cubulations.
result Automorphisms of specific groups preserve coarse median structures and have nice fixed subgroups.
The paper characterizes arithmetic metrics in coarsely geometric settings.
problem Characterizing arithmetic metrics in coarsely geometric settings.
method Using coarse-geometric commensurators and under the Hilbert-Smith conjecture.
result Positive answer in general and unconditional for specific cases.
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…
Paper proves a relative version of coarse Alexander duality and applies it to Jordan cycles.
problem Proving a relative version of coarse Alexander duality.
method Introduced a relative Čech homology satisfying Eilenberg-Steenrod Exactness Axiom.
result Jordan cycle invariant in proving existence of certain 3-manifold groups.
Develops Patterson-Sullivan theory for coarse cocycles.
problem None explicitly stated in the abstract.
method Theory of Patterson--Sullivan measures for coarse cocycles of convergence groups.
result Existence, uniqueness, and ergodicity results for Patterson-Sullivan measures under geometric assumptions.
Machine learning generates coarse-grained force fields for molecular dynamics.
problem Creating thermodynamically consistent coarse-grained models for larger systems.
method Hybrid architecture using graph neural networks to learn molecular features.
result Framework reproduces thermodynamics for small biomolecular systems.
Characterizes quasi-isometric embeddings in coarsely Lipschitz category.
problem Understanding quasi-isometric embeddings in geometric terms.
method Formalizes quasi-isometric embeddings as regular monomorphisms in coarsely Lipschitz category.
result Quasi-isometric embeddings are equivalently characterised as effective, strong, or extremal monomorphisms.