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 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.
The purpose of these notes is to explain parts of Gromov's survey of Carnot-Carathedory spaces, in the light of subsequent results of M. Rumin. Among the rich material provided by Gromov, most of which pertains to analysis on metric spaces, we choose to concentrate on the H{ö}lder equivalence problem for Carnot manifol…
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.
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.
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.
A novel Gromov-Wasserstein learning framework is proposed to jointly match (align) graphs and learn embedding vectors for the associated graph nodes. Using Gromov-Wasserstein discrepancy, we measure the dissimilarity between two graphs and find their correspondence, according to the learned optimal transport. The node …
In this paper we prove: if the complete Kähler-Einstein metric on a bounded convex domain (with no boundary regularity assumptions) is Gromov hyperbolic, then the ∂ˉ-Neumann problem satisfies a subelliptic estimate. This is accomplished by constructing bounded plurisubharmonic function whose Hessian grows…
We prove Cheeger-Gromov convergence for a subsequence of a given sequence of manifolds-with-boundary of bounded geometry. The method of the proof is to reduce, via height functions, the problem to the setting of Hamilton's compactnes theorem for manifolds without boundary.
Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we pro…
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.
Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdo…
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.
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we…
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.
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…
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.
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.
In the present paper we calculate the Gromov-Hausdorff distance between an arbitrary simplex (a metric space all whose non-zero distances are the same) and a finite metric space whose non-zero distances take two distinct values (so-called 2-distance spaces). As a corollary, a complete solution to generalized Borsuk p…
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.
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.
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.
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…
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. 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.
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.
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 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.
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.
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…
The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.
problem Stability in compact finite dimensional Alexandrov spaces.
method Equivariant Gromov--Hausdorff convergence and almost commutative diagrams.
result Stability result in compact finite dimensional Alexandrov spaces.
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.
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.
In 1969 M. Gromov in his PhD thesis greatly generalized Smale-Hirsch-Phillips immersion-submersion theory by proving what is now called the h-principle for invariant open differential relations over open manifolds. Gromov extracted the original geometric idea of Smale and put it to work in the maximal possible generali…
Develop criteria to distinguish Gromov-Thurston manifolds using algebraic Dehn fillings.
problem Distinguishing homotopy types of Gromov-Thurston manifolds
method Using virtual Dehn fillings of relatively hyperbolic groups
result Criteria to distinguish homotopy types
Adapts DR objectives for both sample and feature size reduction.
problem Simultaneously reduce sample and feature sizes.
method Semi-relaxed Gromov-Wasserstein optimal transport.
result OT plan delivers competitive hard clustering.
Extends Gromov's theorem with amenable covers.
problem Gromov's Vanishing Theorem limitations.
method Gromov's multicomplex theory, relative bounded cohomology.
result Relative version of Gromov's Vanishing Theorem.