Identity testing for reversible Markov chains without symmetry assumption.
problem Identity testing of reversible Markov chains.
method Using distance notion from Daskalakis et al. [2018a], testing without symmetry assumption.
result It is possible to perform identity testing under weaker assumption of reversibility.
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…
Different distances on symmetrical domains in complex space.
problem Comparing distances on specific complex domains.
method Examined Carathéodory pseudo-distance and Kähler-Einstein metric distances.
result Found the distances differ on certain complex domains.
Generates valid Euclidean distance matrices for molecular structures.
problem Generating point clouds in arbitrary rotations and translations is challenging.
method Developed a neural network architecture that produces valid Euclidean distance matrices invariant to rotations and translations.
result The architecture can generate molecular structures in a one-shot fashion by producing Euclidean distance matrices with a three-dimensional embedding.
Study shows spherical hyperbolic manifolds almost rigidly converge to hyperbolic space.
problem Almost rigidity of positive mass theorem for spherical hyperbolic manifolds.
method Intrinsic flat distance to prove convergence.
result Spherically symmetric asymptotically hyperbolic manifolds converge to hyperbolic space if mass limit is zero.
BERET improves binary expansion test for multivariate independence.
problem Testing independence of random vectors in arbitrary dimensions.
method Ensemble approach using sum of squared symmetry statistics and distance correlation.
result Improves power while preserving interpretability.
We prove that, if Ω⊂Rn is an open bounded starshaped domain of class C2, the constancy over ∂Ω of the function φ(y)=∫0λ(y)∏j=1n−1[1−tκj(y)]dt implies that Ω is a ball. Here kj(y) and λ(y) denote respectively the principal curvatures and the cut v…
In this paper we prove a symmetry result on submanifolds of codimension one in a n + 1-dimensional space form, related to the geodesic distance function and to the normal curvature of some fixed vector field. As applications we will prove sphere characterization type theorems for Kahler manifolds endowed with a toric g…
New measures found in 3-uniform geometry.
problem Understanding non-flat uniform measures in geometric measure theory.
method Combining combinatorial methods and distance symmetry properties.
result Infinite family of 3-uniform measures constructed.
Deep neural networks favor symmetric structures, enabling multilevel symmetries.
problem Understanding and optimizing deep neural networks.
method Formulating DNN training as convex Lasso problems with geometric algebra.
result Deep networks inherently favor symmetric structures, enabling multilevel symmetries.
We introduce SARR for symmetric object pose estimation, improving CNN performance.
problem Ambiguities in symmetric object orientations hinder deep learning pose estimation.
method Numeric rotation representation using symmetry-derived trigonometric identities.
result SARR enables standard CNNs to achieve state-of-the-art performance.
Bayesian framework detects symmetries in chaotic dynamical systems.
problem Detecting symmetries in chaotic attractors for insights into dynamical system structure.
method Bayesian framework using Gibbs posterior constructed from Wasserstein distances.
result Bayesian framework accurately recovers symmetries under high noise and small sample sizes.
We consider least energy solutions to the nonlinear equation −Δgu=f(r,u) posed on a class of Riemannian models (M,g) of dimension n≥2 which include the classical hyperbolic space Hn as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is …
Graph neural network learns graph distances effectively.
problem Maintaining graph distance metric properties.
method GRAPH-BERT based semi-supervised distance metric learning.
result GB-DISTANCE outperforms existing methods.
Bispectral OT improves dataset comparison by preserving intrinsic coherence.
problem Ignoring intrinsic coherence in dataset comparisons using pairwise geometric distances.
method Introduces Bispectral Optimal Transport, a symmetry-aware extension of discrete OT.
result Transport plans computed with Bispectral OT achieve greater class preservation accuracy.
Paper relaxes symmetry conditions for universal feature selection in noisy data.
problem Feature selection in noisy data with weak symmetry.
method Developed a universal feature selection framework using singular value decomposition of canonical dependence matrix.
result Selected features achieve asymptotically optimal error exponents up to a residual term.
Proves transversality for local Morse homology with symmetries.
problem Defining local Morse chain complexes with finite cyclic group symmetry.
method Special regularized distance functions and inductive perturbation process.
result Global existence theorem for symmetric Morse-Smale pairs.
Symmetric hypersurfaces found between parallel hyperplanes with specific curvature conditions.
problem Finding symmetric hypersurfaces between parallel hyperplanes with curvature constraints.
method Analyzing hypersurfaces with mean curvature dependent on distance to parallel planes and proving symmetry under various conditions.
result Symmetric hypersurfaces found under different curvature and boundary conditions.
A consistent theory of quantum gravity (QG) at Planck scale almost sure contains manifestations of Lorentz local symmetry violations (LV) which may be detected at observable scales. This can be effectively described and classified by models with nonlinear dispersions and related Finsler metrics and fundamental geometri…
New fault-tolerant quantum gates for homological LDPC codes with constant or almost-constant rate.
problem Fault-tolerant quantum computing for homological LDPC codes with constant or almost-constant encoding rate.
method Derive generic formula for transversal and logical gates acting on 3-manifolds, using higher symmetries and cup product cohomology.
result Parallelizable logical gates for homological LDPC codes with constant or almost-constant rate.
Unified understanding of neural representation similarity measures.
problem Fragmented research landscape of neural network similarity measures.
method Observation and exploration of connections between shape distances and normalized Bures similarity.
result Cosine of the Riemannian shape distance equals normalized Bures similarity.
The geometry of oscillatory integrals on manifolds with intermediate symmetry.
problem Classification of curvature conditions in Sogge's program.
method Proposing a classification of curvature conditions.
result No manifolds satisfy the chaotic curvature condition of order 1.
ELD compares graphs by their embedded Laplacian eigenvectors, resolving ambiguities.
problem Comparing graphs of different sizes and structures.
method ELD uses symmetrization and perturbation techniques to compare graph embeddings.
result ELD resolves ambiguities in graph comparisons, making it a natural pseudo-metric.
Data augmented bootstrap unifies various confidence interval construction methods.
problem Constructing confidence intervals from data transformations.
method Data augmented bootstrap (DAB) framework.
result Establishes theoretical coverage results for DAB methods.
Algorithm learns nearest neighbor graph from noisy distance queries.
problem Learning nearest neighbor graph from noisy distance samples.
method Active algorithm to find graph with high probability, analyzing query complexity.
result Empirically and theoretically efficient, needing only O(n log(n)Delta^-2) queries.
The study explores Hesse manifolds and their symmetries in multifield cosmological models.
problem Understanding symmetries in multifield cosmological models.
method Analyzes Hesse functions and their properties on Riemannian manifolds.
result Complete Hesse manifolds are characterized by their index and are hyperbolic.
Study on solutions of conformal equations, proving bounds and profiles.
problem Analyzing solutions of conformally invariant equations on Euclidean domains.
method Established blow-up profiles and heights of solutions around blow-up points.
result Proved bounds on distances and heights of solutions around blow-up points.
Structure-preserving GANs learn distributions with group symmetry efficiently.
problem Learning distributions with group symmetry efficiently.
method Developed structure-preserving GANs by reducing the discriminator space and designing structured generators.
result Significantly improved sample fidelity and diversity in small data regimes.
We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal h…
The paper studies curvature bounds for manifolds with density.
problem Curvature bounds for Riemannian manifolds with density.
method Develops new tools for studying weighted sectional curvature bounds, including a weighted Rauch comparison theorem and a modified convexity notion.
result Improves results for spaces of positive weighted sectional curvature and symmetry.
This work classifies monodromy in vineyards using singularity theory.
problem Understanding and predicting monodromy in vineyards for topological data analysis.
method Using a connection with singularity theory, the study classifies monodromy in vineyards of 1-manifolds in R^2.
result Monodromy in vineyards occurs only if they contain a specific singularity of the distance function.
Paper disproves symmetry of stars at infinity in a specific graph.
problem Symmetry of stars at infinity in a specific graph.
method Defined incidence geometry of stars at infinity; provided an example.
result Relation of one boundary point being included in a star of another is not symmetric.
Proves stability of spacetime Penrose inequality for spherical symmetric initial data.
problem Stability of the Penrose inequality for spherical symmetric spacetimes.
method Formulated and proved stability statement using spherical symmetry and asymptotically flat initial data.
result Initial data must arise from an isometric embedding into a static spacetime close to Schwarzschild spacetime.
3D solitons are shown to be symmetric under specific curvature bounds.
problem Characterizing 3D steady Ricci solitons with curvature decay constraints.
method Proved rotational symmetry for solitons with specified curvature bounds.
result 3D steady Ricci solitons are rotationally symmetric under certain curvature decay conditions.
We prove that a H-surface M in H^2xR, |H| <= 1/2, inherits the symmetries of its boundary when the boundary is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to πFurthermore we prove that the asymptotic boun…
We study the heat kernel of the sub-Laplacian L on the CR sphere S2n+1. An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub- Laplacian -L + n2 that was obtained by Geller [12], and also get an explicit formu…
Optimal stopping times maximize/minimize Brownian motion distance between radially symmetric marginals.
problem Optimal stopping times for Brownian motion between radially symmetric marginals.
method Characterization through Skorohod embeddings and optimal mass transport with subharmonic constraints.
result Optimal stopping times are hitting times of suitable barriers, non-randomized, and unique under radial symmetry.
Proposes new loss functions for GANs to improve estimation accuracy and robustness.
problem Improving the training of GANs to achieve more accurate and robust models.
method Introduces Hellinger-type loss functions and analyzes their statistical properties.
result Demonstrates improved estimation accuracy and robustness of the proposed loss functions.
Study of CR twistor model Q2,2 and its sections.
problem Classify and describe projective lines and hyperplane sections of the CR twistor model.
method Explicit projective methods, classification of lines and sections, use of involution j. result Complete relative classification of smooth quadric sections and explicit non-spherical CR structures.
We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space H2n+1 which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. T…
In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygo…
See http://youtu.be/Mf4IE8gWcJs for a YouTube video showing part of the results in this paper. We consider helicoidal immersions in the Euclidean space whose axis of symmetry is the z-axis that are solutions of the equation 2 H=Λ_0-a 1/2 R^2 where H is the mean curvature of the surface, R is the distance form the point…
New properties for density-based dissimilarity measures in hybrid clustering are proposed and evaluated.
problem Choosing the right dissimilarity measure for hybrid clustering.
method Six data-independent properties for density-based dissimilarity measures are proposed and evaluated.
result A new dissimilarity measure based on Kullback-Leibler information is introduced and shown to satisfy all proposed properties.
Algorithm reconstructs vertex positions in random geometric graphs with improved accuracy.
problem Reconstructing vertex positions in random geometric graphs with high accuracy.
method Hybrid of graph distances and short-range estimates based on common neighbors.
result Algorithm reconstructs vertex positions with error of O(nβ), improving over previous results. New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.
problem Lack of explicit representations and symmetry in heat kernels in sub-Riemannian geometry.
method Establishes a new heat semigroup characterisation using integral decoupling property.
result Characterizes Sobolev and BV spaces in Carnot groups.
Study on symmetries in geometric structures, finding unique symmetries imply affine symmetry.
problem Investigating symmetries in geometric structures.
method Analyzing (local) automorphisms of parabolic geometries.
result Many parabolic geometries have at most one generalized geodesic symmetry at points with non-zero harmonic curvature.
Sym-NET detects human symmetries in photos, outperforming existing models.
problem Capturing human symmetry perception in real-world images.
method Deep-learning neural network (Sym-NET) trained on MS-COCO dataset with human labels.
result Sym-NET significantly outperforms existing algorithms on unseen MS-COCO photos.
Study shows nonextendibility of warped spacelike singularities in specific spacetimes.
problem Nonextendibility of warped spacelike singularities in specific spacetimes.
method Establishes a local obstruction through integrability conditions and radial compression.
result Imply C0-inextendibility for the one-horizon Birmingham-Kottler family.