In high dimensions, the mean and geometric median are nearly identical.
problem Understanding the relationship between mean and geometric median in high-dimensional spaces.
method Analytical derivation and simulation of the distance between mean and geometric median.
result The distance between mean and geometric median vanishes with dimensionality in high dimensions.
New method beats volumetric barrier for manifold recovery.
problem Reconstructing latent geometry from noisy distances.
method Orthogonal Ring Distance Estimation Routine (ORDER).
result Achieves pointwise distance estimation of order n−2/(d+5). A new metric mav offers a practical alternative to costly Riemannian distance.
problem Efficiently compute Riemannian distance on SE(3) invariant metrics.
method Propose mav distance, defined as Riemannian length of a curve.
result Mav distance offers a trainable invariant for geometric deep learning.
A new invariant captures geometric features of circle embeddings.
problem Capturing geometric features of circle embeddings invariantly.
method Chordal distance transform and persistent homology.
result Persistent homology of chordal distance transform is invariant.
Neural networks can learn distance metrics affecting model performance.
problem Understanding how neural networks learn and represent data.
method Experiments with six MNIST architectures, constrained to learn either distance or intensity representations.
result Distance-based learning affects model performance, validating the geometric framework.
Geometric insights improve convergence of implicit generative models.
problem Improving convergence of implicit generative models.
method Analyzing geometries induced by Wasserstein distance and other criteria.
result Established surprising approximate global convergence guarantees for the 1-Wasserstein distance.
The paper explores selecting the parameter α for Fermat distance to balance geometry and noise.
problem Choosing the optimal parameter α for Fermat distance to navigate geometry and noise.
method Theoretical and simulation studies to determine the best α value.
result An optimal α value is identified to balance geometry and noise.
Estimates distances between latent points in random geometric graphs.
problem Estimating distances between latent points in random geometric graphs.
method Spectral estimator of pairwise distances.
result Rate of convergence is the same as nonparametric estimation on the sphere, up to a logarithmic factor.
New curves in differentiable distance spaces mimic Finsler geodesics.
problem Distance functions are not differentiable in general.
method Using distance spheres and direct geometric considerations.
result Curves in differentiable distance spaces behave like Finsler geodesics.
Study shows vanishing distance in fluid dynamics equations.
problem Understanding the geometric origins of fluid dynamics equations.
method Analyzing geodesic distances on diffeomorphism and symplectomorphism groups.
result Modified Constantin-Lax-Majda and surface quasi-geostrophic equations arise from metrics with vanishing geodesic distance.
Sharp bounds found for distances between specific geometric shapes in hyperbolic space.
problem Finding effective distances between specific geometric shapes (tori) in hyperbolic 3-manifolds.
method Sharp, effective bounds on distances between tori of fixed injectivity radius.
result Effective bounds on distances between specific geometric shapes in hyperbolic space.
Theory of space-time currents for geometric evolutions.
problem Analysis of geometric evolutions driven by dislocations.
method Development of space-time integral currents with bounded variation, introduction of Lipschitz deformation distance.
result Agreement of Lipschitz deformation distance with integral Whitney flat metric for boundaryless currents.
Adapts Stein's method for geometric inequalities, addressing boundary terms.
problem Geometric inequalities and their stability under constraints.
method Uses elliptic PDE with oblique boundary condition to handle boundary terms.
result Stability results for various geometric inequalities with respect to a new distance.
The paper explains geometrically why certain mappings have singular points.
problem Understanding singular points in mappings from R^2 to R^3 and higher.
method Analyzing full rank matrices constructed from coefficients of mappings.
result Mappings have only one singular point when ℓ=3 and no singular points when ℓ>3.
The study compares Euclidean and cosine distances in medical drug prescription prediction.
problem Comparing Euclidean and cosine distances in medical drug prescription prediction.
method Established geometric properties and compared distances in real-world medical data.
result Different distances lead to different optimizing nonlinear kernel embedding frameworks.
Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefini…
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
CDOT optimizes transport between domains preserving both feature and geometric structure.
problem Optimizing transport between heterogeneous domains with preserved feature and geometric structure.
method CDOT uses operator-based regularization to align distance structures, proving pseudometric properties.
result CDOT improves robustness to local geometric variations and is provably convex.
New method calculates Ricci curvature from distances between weighted volumes.
problem Calculating Ricci curvature for weighted Riemannian manifolds.
method Asymptotic retrieval of generalized Ricci tensor from scaled metric derivatives of Wasserstein 1-distances.
result Limiting coarse curvature of random graphs converges to generalized Ricci tensor.
The paper studies a flow related to Ricci flow on manifolds with geometric singularities.
problem Analyzing geometric flows on manifolds with singularities.
method Introduced geometric flow on smooth compact manifolds with rough metrics, providing a regularity theory and demonstrating distance consistency.
result The flow on spaces with singularities preserves the distance metric as in smooth cases, maintaining smoothness away from singular points.
The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometric…
Research explores Lorentzian distances on a specific geometric plane.
problem Investigating Lorentzian structures on a 2D geometric plane.
method Analyzes sectional curvature, attainable sets, and Lorentzian length maximizers.
result Describes distance properties and spheres in the context of Lorentzian geometry.
New method recovers manifold distances from noisy data.
problem Reconstructing manifold geometry from noisy distance measurements.
method Develops new framework to estimate L2-norms of expectation-functions, uses geometric clusters to recover distances.
result Recovery of true distances up to an additive error of O(ε log ε⁻¹) under mild geometric assumptions.
New geometrical method optimizes portfolio with minimal risk.
problem Optimizing portfolios with risk aversion and constraints.
method Computing generalized Euclidean distance to a simplex.
result Determines portfolios with minimal risk and high returns.
Establishes exponential contraction in Wasserstein distance on manifolds and flows.
problem Analyzing contraction rates in Wasserstein distance on manifolds and their evolution.
method Explicit estimates and extension to evolving manifolds under geometric flow.
result Gradient estimates with exponential contraction rate under weak curvature conditions.
New geometric structures defined on SPD matrices for better understanding.
problem Understanding SPD matrices and their geometric properties.
method Introducing Finslerian and dual information-geometric structures on James' bicone domain.
result Geodesics correspond to straight lines in coordinate systems, and new dissimilarities generalize existing ones.
Study geometric properties of symmetric matrices with repeated eigenvalues.
problem Investigate geometric properties of symmetric matrices with repeated eigenvalues.
method Explicitly compute the volume of the intersection with the sphere and prove an Eckart-Young-Mirsky-type theorem.
result Prove connections to Real Algebraic Geometry and Random Matrix Theory.
The paper studies parallel surfaces of cuspidal cross caps and their degeneracy.
problem Investigating the geometry and singularities of parallel surfaces of cuspidal cross caps.
method Established a criterion for the degeneracy of the distance squared function using geometric invariants.
result Parallel surfaces degenerate into a degenerated cuspidal S1 singularity at specific distances.
Study of metrics on positive-definite matrices from power potential, linking to power means.
problem Understanding metrics on positive-definite matrices derived from power potential.
method Explicit expressions for geodesics and distance function derived from Hessian of power potential.
result Geodesics and distance function converge to weighted matrix geometric mean as β tends to zero.
A new robust metric compares distributions more accurately than existing methods.
problem Sensitivity to outliers and sampling discrepancy in Wasserstein distances.
method Introducing k-RPW, a partial p-Wasserstein distance.
result k-RPW converges faster to true distance and is more robust to outliers.
New geometric invariant from min-max width of spheres on Riemannian 2-spheres.
problem Understanding the min-max width of spheres associated to distance functions.
method Application of min-max methods to pairs of points on Riemannian 2-spheres.
result The min-max width does not always equal half the length of a simple closed geodesic.
Classifies points in quaternionic hyperbolic spaces up to congruence.
problem Classifying points in quaternionic hyperbolic spaces up to congruence.
method Introduces geometric invariants and distance formulas to classify points.
result Congruence classes are described by quaternionic Cartan's angular invariants and distances.
Unified framework for Brownian motion distances on specific geometric manifolds.
problem Understanding Brownian motion distances on radially isoparametric manifolds.
method Developed a geometric framework and derived drift-window inequalities.
result Unified framework for coadapted Brownian couplings on RIM.
A new method compares synthetic power networks to actual ones using multiscale flat norm.
problem Comparing synthetic power networks to actual ones due to lack of correspondence.
method Proposes a multiscale flat norm approach to compute distance between networks.
result The flat norm distance captures variations more accurately than Hausdorff distance.
This work provides guaranteed bounds on the total variation distance for univariate mixtures.
problem Lack of closed-form expressions for total variation distance between mixtures.
method Two methods: information monotonicity for lower bounds and geometric envelopes for upper bounds.
result Demonstrated tightness of bounds on Gaussian, Gamma, and Rayleigh mixtures.
The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both theories. The concepts of the theory of affine spheres lead to a new definition of an …
Study curves' intersections and distances, with applications in graph and group studies.
problem Understanding intersections and distances of curves.
method Using a relationship between intersection numbers and subsurface projection distances, applications in curve graphs and mapping class groups.
result Explicit quasi-constants for the relationship between intersection numbers and subsurface projection distances.
A new geometric metric identifies true data changes from parametrization artifacts in high-dimensional representations.
problem Quantifying representation drift in high-dimensional data using Euclidean or cosine distances can misattribute changes due to arbitrary parametrizations.
method Introducing the Fubini Study metric to identify representations that differ only by gauge transformations.
result The Fubini Study metric isolates intrinsic evolution by remaining invariant under gauge-induced fluctuations, providing a diagnostic for meaningful structural changes.
Estimates manifold distances using graph Laplacian, proving consistency.
problem Estimating distances in compact Riemannian manifolds.
method Graph Laplacian estimates of the Laplace-Beltrami operator, bounding errors.
result Proof of consistency for manifold distances.
The paper proposes a method to improve data analysis by considering multiple subsets of attributes (views) to enhance geometric information.
problem Distortion of distance metrics in high-dimensional data analysis.
method Partitioning attributes into multiple subsets (views) and using consensus between views to extract geometric information.
result Enhanced geometric information from multiple views improves data analysis.
New Morse theory for shapes at distances.
problem Understanding shapes at distances from a reference point.
method Defining Morse functions and using non-smooth analysis, geometric measure theory.
result Homotopy type changes at critical values, with one cell added per critical point.
Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function whil…
New neural method calculates EMD for particle physics data.
problem Metric for particle collider events based on Wasserstein metric.
method Neural network architecture estimating EMD using Kantorovich-Rubinstein duality.
result Differentiable way to calculate EMD for geometric fitting.
The paper finds inequalities in Grassmannian geometry.
problem Understanding geometric properties of Grassmannians.
method Analyzes inequalities for elements in Grassmannians.
result Law of Cosines and geodesic triangle inequalities.
We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. In doing this we also discuss some geometrical properties of the cutlocus of a point and we compare some different weak notions of Hessian and Laplacian.
Study continuity of limit sets in symmetric spaces.
problem Continuity of limit sets for geometrically finite subgroups in symmetric spaces.
method Extended geometrically finite representations theory.
result Limit sets vary continuously with respect to Hausdorff distance under strong convergence.
In this paper we describe three geometric applications of quandle homology. We show that it gives obstructions to tangle embeddings, provides the lower bound for the 4-move distance between links, and can be used in determining periodicity of links.
A new snake model improves segmentation of SEM images.
problem Efficiently segmenting overlapping electronic structures in SEM images.
method Geodesic tracking on projective line bundle with a geometric criterion for switching between fast spatial snakes and minimizing geodesics.
result Improved robust and automatic segmentation of overlapping electronic structures in SEM images.