Note on the computational complexity of Gromov-Wasserstein distance.
problem Computational difficulty of Gromov-Wasserstein distance.
method Analysis of the optimization problem structure and providing explicit examples.
result Gromov-Wasserstein distance optimization problem is non-convex quadratic.
A new sliced IGW distance for Gromov-Wasserstein alignment.
problem Scalability issues in Gromov-Wasserstein alignment for high-dimensional problems.
method Proposed a sliced IGW distance with rotational invariance.
result Natural rotational invariance of the sliced IGW distance.
Survey solves curvature problems with hyperbolic spaces.
problem Singularities in hypersurface geometry.
method Hyperbolic unfolding correspondence linking hypersurfaces to Gromov hyperbolic spaces.
result Eliminates hypersurface singularities in scalar curvature geometry.
Paper calculates Gromov-Hausdorff distance between simplexes and 2-distance spaces.
problem Calculating Gromov-Hausdorff distance between simplexes and 2-distance spaces.
method Formulas derived for clique covering number and chromatic number of graphs.
result Complete solution to generalized Borsuk problem for 2-distance spaces.
Paper solves Gromov-Wasserstein for point clouds efficiently.
problem Quantifying similarity between two formations or shapes.
method Reformulates QAP as low-rank concave quadratic optimization problem.
result Global solution for large-scale problems with thousands of points.
Explains Gromov's and Rumin's work on Carnot manifolds.
problem Hölder equivalence problem for Carnot manifolds
method PDE techniques and Rumin's complex
result Both methods yield similar conclusions for the H{ö}lder equivalence problem
Paper introduces new Gromov-type distances for comparing Gaussian mixture models.
problem Comparing distributions across different metric spaces using Gromov-Wasserstein distances.
method Incorporates invariance properties into MW2, introducing MGW2 and EW2.
result MGW2 and EW2 are efficient for estimating distances between GMMs in practical applications.
Extends subspace detour method to Gromov-Wasserstein problem.
problem Matching shapes using Gromov-Wasserstein distance.
method Project measures onto a subspace, then compute optimal transport plan.
result Connections with Knothe-Rosenblatt rearrangement.
The paper solves a problem posed by Gromov for the Heisenberg group.
problem Solving Gromov's Hölder equivalence problem for the Heisenberg group.
method Using maps with Hölder exponent greater than 1/2 and properties of quasi-convex metric spaces.
result Injectivity of maps with Hölder exponent > 1/2 from certain metric spaces into the Heisenberg group is impossible.
Solves linearity problem for acyclic groups, bounds Cheeger-Gromov ρ-invariants.
problem Linearity problem for acyclic groups and Cheeger-Gromov ρ-invariants.
method Quantitative algebraic and geometric techniques over simplicial classifying spaces.
result Universal linear bound for Cheeger-Gromov ρ-invariants of PL (4k-1)-manifolds.
The paper proves isoperimetric regions on Riemannian manifolds with Ricci bounded below.
problem Proving the existence of isoperimetric regions in Riemannian manifolds.
method Gromov-Hausdorff asymptotic analysis to study perimeter-minimizing sequences.
result Existence of isoperimetric regions in noncollapsed Riemannian manifolds with Ricci curvature bound.
This paper tackles Gromov's filling area conjecture using discrete graph theory.
problem Finding the smallest surface area for isometrically filling a circle.
method Using graph-theoretic tools like Menger's theorem to derive bounds.
result Discrete bounds translate to a lower bound of 1.36π for the hemisphere's surface area.
The paper reconstructs Lorentzian spacetimes from causal sets.
problem Reconstructing Lorentzian spacetimes from causal sets.
method Introduced a concept of isomorphy and three types of convergence.
result Established Gromov's reconstruction theorem in Lorentzian geometry.
Compactness theorem for manifolds with boundary proved.
problem Proving convergence of manifolds with boundary.
method Reduction to Hamilton's compactness theorem for manifolds without boundary.
result Cheeger-Gromov convergence for a subsequence of manifolds with bounded geometry.
Proposes a novel framework for graph matching and node embedding.
problem Graph matching and node embedding in real-world networks.
method Gromov-Wasserstein discrepancy for graph dissimilarity, optimal transport for correspondence, structural regularizers for learning.
result Superior performance in graph matching compared to alternative approaches.
Unified framework for DR and clustering using Gromov-Wasserstein.
problem Capturing structure in high-dimensional datasets.
method Distributional reduction framework using Gromov-Wasserstein.
result Unified approach recovers DR and clustering as special cases.
A new conformal prediction framework for graph-valued outputs using Z-Gromov-Wasserstein distances.
problem Lack of principled uncertainty quantification for graph-valued supervised prediction.
method Proposes a conformal prediction framework using Z-Gromov-Wasserstein distances for graph-valued outputs.
result Provides distribution-free coverage guarantees for graph-valued outputs.
Paper proves subelliptic estimates for Kähler-Einstein metrics on convex domains.
problem Proving subelliptic estimates for Kähler-Einstein metrics on convex domains.
method Proved subelliptic estimates by constructing bounded plurisubharmonic functions.
result Subelliptic estimates for Kähler-Einstein metrics on convex domains.
Polynomial algorithm estimates mGH distance between unweighted graphs.
problem Efficiently measuring shape difference between unweighted graphs.
method Polynomial algorithm for mGH distance estimation.
result Algorithm finds mGH distances exactly on most scale-free graphs.
The study connects hyperbolicity and isoperimetric inequality in manifolds and graphs.
problem Understanding the relationship between hyperbolicity and isoperimetric inequality.
method Characterization of hyperbolic manifolds and graphs with isoperimetric inequality.
result Characterization of trees with isoperimetric inequality and solvability of Dirichlet problem at infinity.
New method learns disentangled representations using Gromov-Monge maps.
problem Learning disentangled representations from unlabelled data.
method Introduces a novel approach based on Gromov-Monge maps to preserve geometric features while aligning data distributions.
result Demonstrates effectiveness on four benchmarks, outperforming other methods.
New DR method uses Gromov-Wasserstein distance for high-dimensional data.
problem Analyzing relationships between high-dimensional objects.
method Optimal transportation theory and Gromov-Wasserstein distance.
result Robust and efficient solution for complex high-dimensional datasets.
Study on simplicial volume and Euler characteristic of aspherical manifolds.
problem Whether vanishing simplicial volume implies vanishing Euler characteristic.
method Various strategies for both affirmative and negative answers, context with other problems, and comparative analysis of additivity properties.
result Found counterexamples among aspherical spaces that are homology equivalent to manifolds but not manifolds themselves.
New model optimizes feature alignment, improving statistical and computational efficiency.
problem Statistical and computational challenges in feature alignment.
method Covariance alignment model, Gromov-Wasserstein algorithm.
result Gromov-Wasserstein algorithm is minimax optimal and practical for covariance alignment.
Study of flat bundles from Calabi-Yau theory converging to Riemann-Hilbert solutions.
problem Understanding flat bundles and their limits in Calabi-Yau theory.
method Analysis of variations of BPS structures and convergence to Riemann-Hilbert problems.
result Expression for Gromov-Witten partition function in terms of confluent hypergeometric equations.
Proposes an efficient lower bound for Gromov-Wasserstein discrepancy.
problem Comparing structured data from different metric-measure spaces.
method Orthogonal Gromov-Wasserstein (OGW) discrepancy with efficient closed-form lower bound.
result Efficient and tight lower bounds for Gromov-Wasserstein discrepancy.
Differentiable optimization bridges arbitrary metrics to tree metrics.
problem Designing algorithms to convert arbitrary metrics to tree metrics with guarantees.
method DeltaZero framework, leveraging differentiable Gromov hyperbolicity.
result DeltaZero consistently achieves state-of-the-art distortion on synthetic and real-world datasets.
Lorentzian versions of classical Riemannian volume comparison theorems by Gunther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case, which is extensively discussed. For the general semi-Riemannian case, a local v…
Survey on scalar curvature and convergence of manifolds.
problem Understanding convergence of manifolds with scalar curvature bounds.
method Intrinsic Flat convergence, developed with Wenger.
result Successfully applied to study sequences in General Relativity.
The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.
problem Investigating Gromov hyperbolicity of specific metrics.
method Using isoperimetric inequalities to characterize Gromov hyperbolicity.
result Characterization of domains where these metrics are Gromov hyperbolic.
Proposes a new metric for comparing shapes in different spaces.
problem Comparing shapes in different metric spaces with unequal mass.
method Developed a Partial Gromov-Wasserstein (PGW) metric and algorithms to solve it.
result PGW is a well-defined metric between metric measure spaces.
Proof of Gromov's theorem on convex polytopes with acute angles.
problem Gromov's conjecture on extremal scalar curvature of convex polytopes.
method Smoothing construction using Dirac operator techniques.
result Detailed proof of Gromov's theorem.
We address two fundamental and well-known problems of Gromov and Lyndon: \demo{Problem A} (Gromov, see [5]). Consider a category Mn of closed manifolds of dimension n with nonzero-degree ways as morphisms. Study a partial order M≥N⇔Mor(M,N)=φ. For which N the degrees of maps $f: M \t…
New theorem using Ricci flow for Gromov almost flat manifolds.
problem Conditions for Gromov almost flat manifolds.
method Employing Ricci flow to derive a new theorem.
result New theorem with weaker condition than Gromov--Ruh Theorem.
Minimal normal curvature immersions in the unit ball studied.
problem Minimal normal curvature immersions in the unit ball.
method Gromov's problem, differentiable sphere theorem, existence result.
result Determined the minimal possible value of the normal curvature of SnimesS1. Improved upper bound for discrete isometric filling of cycles.
problem Finding the minimum number of vertices in a discrete isometric filling of cycle graphs.
method Explicit construction of isometric fillings using concentric annular structures.
result Explicit construction of isometric fillings with \( |V(K_n)| \le \left(\frac{1}{6} + o(1)
ight)n^2 \), improving the upper bound to \( D^* \le \frac{1}{6} \).
Study geometric manifolds in arbitrary dimensions, focusing on maps and diffeomorphisms.
problem Existence and properties of maps and diffeomorphisms in geometric manifolds.
method Analysis of geometric structures and homotopy invariants in arbitrary dimensions.
result Existence of Anosov diffeomorphisms and monotonicity of homotopy invariants.
Proves Gromov's conjecture on metric operators.
problem Non-free isometric immersions and metric operators.
method Proof of a conjecture.
result Infinitesimal invertibility of metric inducing operators proven.
Two tropical gluing formulas help calculate Gromov-Witten invariants.
problem Calculating Gromov-Witten invariants of symplectic manifolds.
method Tropical geometry applied to exploded manifolds.
result Generalizes existing formulas for Gromov-Witten invariants.
Study on 4-manifolds with positive scalar curvature.
problem Characterizing 4-manifolds with positive scalar curvature.
method Geography problem approach focusing on four-dimensional case.
result Survey of mathematical work and strengthening of Carr's result.
Potential theory extended to Gromov hyperbolic spaces.
problem Extending potential theory to a new class of spaces.
method Unified framework for Gromov hyperbolic metric measure spaces.
result Boundary Harnack inequalities and complete classification of positive harmonic functions.
Study bounds expansion coefficient from observable diameter in metric measure spaces.
problem Bound the expansion coefficient from below in terms of the observable diameter.
method Considered concentration of measure phenomenon, connected observable diameter and expansion coefficient, derived upper bound, combined with lower bound to obtain upper bound for observable diameter.
result Obtained upper bound for observable diameter in terms of expansion coefficient.
Gromov's Conjecture states that for a closed n-manifold M with positive scalar curvature the macroscopic dimension of its universal covering M~ satisfies the inequality dimmcM~≤n−2\cite{G2}. We prove this inequality for totally non-spin n-manifolds whose fundamental group is a virtual duali…
Improved spectral clustering via Gromov-Wasserstein Learning.
problem Optimizing graph partitioning performance.
method Bridge spectral clustering and GWL, using heat kernel for stable node correspondences.
result Improved graph partitioning results without compromising theoretical guarantees.
The round sphere is stable among spin manifolds with a specific scalar curvature bound.
problem Stability of the round sphere among spin manifolds with scalar curvature below a certain threshold.
method Showed that if the scalar curvature is bounded from below by n(n−1)−ε, the manifold is C0-close to a finite number of spheres outside a small bad set. result The spherical stability problem is completely solved.
The study proves Gromov hyperbolicity for certain complex domains.
problem Characterizing Gromov hyperbolicity for complex domains.
method Analyzing domains in C2 with finite d'Angelo type and using automorphisms. result Domains in C2 with finite d'Angelo type are Gromov hyperbolic. It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the cl…
Connectivity proven in large rank Gromov boundary of free factor complex.
problem Connectivity of Gromov boundary in large rank free factor complex.
method Analyzing Gromov boundary and free factor complex properties.
result Gromov boundary is path connected and locally path connected in large rank.