This is an exposition of the theory of differentiable structures on metric measures spaces, in the sense of Cheeger and Keith.
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We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a measurable differentiable structure assuming that one can control the pointwise up…
In General Relativity the metric can be recovered from the structure of the lightcones and a measure giving the volume element. Since the causal structure seems to be simpler than the Lorentzian manifold structure, this suggests that it is more fundamental. But there are cases when seemingly healthy causal structure an…
Study stability of curvature-dimension condition for negative dimensions.
problem Stability of curvature-dimension condition with negative dimension parameters.
method Introduced CD(K, N)-condition for N < 0, defined distance d_{\mathsf{iKRW}}, proved convergence stability.
result Limit structure of converging metric measure spaces remains CD(K, N) for N < 0.
Paper discusses the Fisher metric and differentiability in statistical models.
problem Understanding the relationship between Fisher metric and differentiability in statistical models.
method Comparison of different concepts and models in Information Geometry, mathematical statistics, and measure theory.
result Discussion of various models and their differentiability properties.
Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy…
Implementing -NN classification using Gromov--Wasserstein distances
problem Comparing metric measure spaces
method Gromov--Wasserstein and fused Gromov--Wasserstein distances
result Universal consistency of -NN classifiers
The study examines non-continuous Riemannian metrics on manifolds and their infinitesimal properties.
problem Investigating non-continuous Riemannian metrics and their infinitesimal structure.
method Constructing examples of metric measure spaces with discontinuous metrics.
result Examples show failure of infinitesimal Hilbertian or quasi-Riemannian properties.
Optimal transport for measures on noisy tree metrics is solved with robust approach.
problem Optimal transport problem for measures on noisy tree metrics.
method Max-min robust optimal transport approach considering uncertainty sets of tree metrics.
result Robust optimal transport admits a closed-form expression for fast computation.
Study classifies Einstein spaces and warped products in weighted geometry.
problem Characterizing geometric structures of weighted Einstein spaces.
method Complete local classification of weighted Einstein spaces with harmonic Weyl tensor.
result Spaces decompose into Einstein or specific warped products.
The paper explores geometry of probability measures and barycenter maps.
problem Understanding the space of probability measures and their barycenter.
method Information geometry, Fisher metric, dualistic structures, divergences, geodesics.
result Recent developments in the geometry of probability measures and barycenter.
A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its…
Gromov-Wasserstein (GW) is a powerful tool to compare probability measures whose supports are in different metric spaces. GW suffers however from a computational drawback since it requires to solve a complex non-convex quadratic program. We consider in this work a specific family of cost metrics, namely \textit{tree me…
Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.
problem Solving the Calabi-Yau equation on symplectic manifolds.
method Global deformation of almost complex structures compatible with symplectic form, constructing measurable Lipschitz Kahler metric.
result Existence theorem for solutions to the one-form type Calabi-Yau equation on closed symplectic manifolds.
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …
We study several problems concerning conformal transformation on metric measure spaces, including the Sobolev space, the differential structure and the curvature-dimension condition under conformal transformations. This is the first result about preservation of lower curvature bounds under perturbation, which is new ev…
Performance metrics (error measures) are vital components of the evaluation frameworks in various fields. The intention of this study was to overview of a variety of performance metrics and approaches to their classification. The main goal of the study was to develop a typology that will help to improve our knowledge a…
Paper investigates conditions for independence of weak gradients on metric spaces.
problem Dependence of weak gradients on in arbitrary metric measure spaces.
method Investigates the Bounded Interpolation Property to ensure independence of weak gradients.
result Bounded Interpolation Property guarantees independence of weak gradients.
Study 2D spaces with curvature, focusing on structure and approximations.
problem Characterize and understand 2D metric spaces with curvature constraints.
method Lipschitz homotopy approximations, curvature measures, convergence analysis.
result Established Gauss-Bonnet Theorem and characterized spaces.
New theory approximates functions between metric spaces using random probability measures.
problem Building universal functions approximators between arbitrary metric spaces.
method Using elementary functions between Euclidean spaces, randomization to output discrete probability measures over target space.
result Very general qualitative guarantees and quantitative guarantees for Hölder-like maps.
This paper introduces a new formulation of the Conic Gromov-Wasserstein distance for comparing complex network structures.
problem Comparing measures of unequal mass and complex network structures.
method Novel semi-coupling formulation and extension to hypernetworks.
result Establishes fundamental properties and robustness of CGW metric.
Extends Lipschitz functions while preserving local constants.
problem Extending Lipschitz functions on metric spaces while maintaining local constants.
method Extends Lipschitz functions on metric spaces while locally preserving the asymptotic Lipschitz constant.
result Sobolev spaces on metric measure spaces are invariant under isomorphism of mm-structures.
This paper shows moduli spaces of RCD(0,2) structures are contractible.
problem Understanding moduli spaces of RCD(0,2) structures.
method Established a list of compact topological spaces admitting RCD(0,2) structures and described their associated moduli spaces.
result All moduli spaces of RCD(0,2) structures are contractible.
New calculus on spacetimes for nonlinear differential equations.
problem Nonlinear differential equations on metric measure spacetimes.
method Introduces maximal weak subslope and variational calculus.
result Establishes a comparison theorem for nonlinear -d'Alembertian.
Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.
problem Unified framework for Riemannian and sub-Riemannian geometries.
method Study of gauge metric measure spaces.
result Unified synthetic Ricci curvature lower bounds for both Riemannian and sub-Riemannian structures.
The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric sp…
In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we gi…
Study curvature of piecewise metrics using moving frames.
problem Deriving a curvature measure for piecewise-smooth Riemannian metrics.
method Used moving frame techniques to derive curvature, showing it satisfies Cartan structure equations and gauge transformation law.
result Equivalence of the derived curvature to existing densitized distributional curvature.
Paper reinterprets majorizing measure theorem in terms of coding theory.
problem Understanding boundedness of random processes.
method Information-theoretic perspective using variable-length codes.
result Boundedness of random processes linked to efficient coding.
Study shows -NN classifier is not universally consistent on but consistent on discrete and specific measure spaces.
problem Consistency of -NN classifier under Wasserstein distance on measure spaces.
method Analysis of -NN classifier properties under Wasserstein distance, use of -finite metric dimension, geodesic structures of Wasserstein spaces.
result Consistency of -NN classifier on specific measure spaces (discrete, Gaussian, wavelet series) but not on .
A new metric learning scheme for structured data combining graph and feature-space information.
problem Learning a metric from structured data while respecting metric constraints.
method Training metric-constrained linear combinations of dissimilarity matrices, applying graph-based optimization under constraints.
result Our approach can reduce computational complexity by one order of magnitude for some cases.
Study non-Gaussian measures' concentration properties in metric spaces.
problem Concentration properties for non-linear Gaussian functionals with non-Gaussian tails.
method Prove generalised Transportation-Cost Inequalities (TCIs) for specific functionals.
result Extended TCIs for rough volatility and Parabolic Anderson Model.
In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality for some constant , rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-kn…
Measure contraction properties are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension , then is equivalent to Ricci curvature bounded below by . On the other hand, it was ob…
The paper sets limits on the number of ends of certain geometric structures.
problem Limits on the number of ends of smooth metric measure spaces.
method Analyzes the Bakry-Émery Ricci tensor and function degeneration to set limits.
result Establishes gap theorems for ends of smooth metric measure spaces under specific conditions.
We use Bott-Chern cohomology to measure the non-Kählerianity of 6-dimensional nilmanifolds endowed with the invariant complex structures in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa's classification, [Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermit…
CatSIM measures image similarity robustly to small changes.
problem Measuring similarity between images, especially with small perturbations.
method Uses structural similarity image quality paradigm, robust to small location changes.
result Structural similarity between images rated higher when not entirely overlapping.
In the celebrated book entitled Metric Structures for Riemannian and Non-Riemannian Spaces, so-called Green Book, Gromov presented a problem regarding a metric measure space. Gromov posed the question Bound the expansion coefficient from below in terms of the observable diameter. The overall aim of the current study is…
The paper solves the isoperimetric problem on noncompact metric spaces with lower Ricci bounds.
problem Solving the isoperimetric problem on noncompact metric spaces with lower Ricci bounds.
method Establishes a structure theorem for minimizing sequences, proving the limit of such sequences is identified by a finite collection of isoperimetric regions.
result The limit of a minimizing sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov--Hausdorff limits of the ambient space.
Novel algorithm speeds up computation of Sobolev IPM for graph-based probability measures.
problem Efficient computation of Sobolev IPM for graph-based probability measures.
method Established relation between Sobolev norm and weighted -norm, proposed novel regularization, leveraged graph structure.
result Proposed regularized Sobolev IPM provides closed-form expression for fast computation.
We show that the only metric measure space with the structure of an -cone and with two-sided synthetic Ricci bounds is the Euclidean space for integer. This is based on a novel notion of Ricci curvature upper bounds for metric measure spaces given in terms of the short time asymtotic of the h…
Study partial derivatives on non-smooth metric measure structures.
problem Understanding partial derivatives in non-smooth settings.
method Extension of Schwarz's theorem and analysis of Sobolev regularity.
result Complete set of results relating properties of functions in non-smooth spaces.
New findings on geometric flows and equidistribution in Hilbert geometry.
problem Characterizing dynamical and counting results in Hilbert geometry.
method Study of dynamical and counting results in rank-one properly convex projective structures with Hilbert metrics.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
A new metric for comparing measures on tree systems reduces computational burden.
problem Heavy computation in Optimal Transport problems.
method Introducing tree systems and a novel metric (Tree-Sliced Wasserstein distance on Systems of Lines, TSW-SL).
result TSW-SL performs favorably compared to Sliced Wasserstein and its variants.
A new method for transporting unbalanced measures on graphs efficiently.
problem Optimal transport for measures with unequal total masses on graph metric spaces.
method Developed a novel variant of entropy partial transport (Orlicz-EPT) with Orlicz geometric structure, leading to Orlicz-Sobolev transport (OST).
result OST can be efficiently computed by solving a univariate optimization problem, significantly faster than Orlicz-EPT.
Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power of L. In this paper various convergence results are obtained for the correspond…
Synthetic approach to conformal transformations in metric and Lorentzian spaces.
problem Defining consistent conformal transformations in spaces of low regularity.
method Introducing conformal transformations in metric and Lorentzian spaces, focusing on Lorentzian pre-length spaces.
result Established a consistent notion of conformal length and proved its properties.
We propose a family of near-metrics based on local graph diffusion to capture similarity for a wide class of data sets. These quasi-metametrics, as their names suggest, dispense with one or two standard axioms of metric spaces, specifically distinguishability and symmetry, so that similarity between data points of arbi…