Ellipsoids approach Gaussian distribution in high dimensions.
problem Understanding convergence of high-dimensional ellipsoids to Gaussian spaces.
method Proof of convergence in Gromov's concentration topology.
result Solid ellipsoids converge to Gaussian space in high dimensions.
Unified framework for measuring concentration in weighted networks considering both weight distributions and network structure.
problem Traditional indices neglect the topology of relationships among network elements.
method Develops a family of topology-aware concentration indices that jointly account for weight distributions and network structure.
result The proposed indices preserve key properties and allow concentration to be evaluated across different dimensions of dependence.
Constructs flows on manifolds with small curvature, proving Euclidean topology.
problem Geometric structure of manifolds with unbounded curvature.
method Distance like functions with integral hessian bound, Ricci flows.
result Manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors n-regular and small curvature concentration are topologically Euclidean. This work explores the characteristics of financial contagion in networks whose links distributions approaches a power law, using a model that defines banks balance sheets from information of network connectivity. By varying the parameters for the creation of the network, several interbank networks are built, in which …
Topological constraints improve neural network generalization.
problem Improving generalization in neural networks with limited data.
method Imposing topological constraints on internal representations of neural networks.
result Topological constraints lead to better mass concentration around training instances, improving generalization.
A limit point p of a discrete group of Mobius transformations acting on S^n is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of U contains a local basis for the topology of S^n at p. For the case of Fuchsian groups (n = 1), every concentration point…
Sharp stability result for maps near infinitely concentrated minimisers.
problem Stability of maps near minimisers with infinite concentration.
method Dynamic approach to deform maps into harmonic maps, controlling topology changes.
result Sharp quantitative estimates on map distance to infinitely concentrated minimisers.
We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of ``bounded geometry''.
This is a sequel to our paper arXiv:1402.2546 to appear in the Journal of Geometric Analysis in which we concentrate on developing some of the topological properties of Sasaki-Einstein manifolds. In particular, we explicitly compute the cohomology rings for several cases not treated in arXiv:1402.2546 and give a formul…
In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This surprising classification of positively-curved Riemannian manifolds results from combin…
The manifold hypothesis states that many kinds of high-dimensional data are concentrated near a low-dimensional manifold. If the topology of this data manifold is non-trivial, a continuous encoder network cannot embed it in a one-to-one manner without creating holes of low density in the latent space. This is at odds w…
The paper studies bifurcations in discrete dynamical systems on manifolds.
problem Understanding bifurcations in discrete dynamical systems on manifolds.
method Topological techniques based on concentricity of manifolds.
result General result for attractors in n-dimensional manifolds.
Study deformations of compact Calabi-Yau conifolds with singularities.
problem Understanding deformations of compact Calabi-Yau conifolds with singularities.
method Analyzes the obstructions and local triviality of deformations under specific topological and geometric hypotheses.
result Obstruction to deformations concentrates at singularities, generalizing previous results.
Expands Bredon's trick for applications in geometry and topology.
problem Local-to-global extension principles in geometric and topological contexts.
method Novel applications and frameworks for stratified pseudomanifolds, Ricci flow, and persistent homology.
result Establishes Bredon's trick as a unifying framework.
This article introduces proximal planar vortex 1-cycles, resembling the structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867 and recent work on the proximity of sets that overlap either spatially or descriptively. Vortex cycles resemble Thomson's model of a vortex atom, inspired by P.G. Tait's …
A new definition of events of game-theoretic probability zero in continuous time is proposed and used to prove results suggesting that trading in financial markets results in the emergence of properties usually associated with randomness. This paper concentrates on "qualitative" results, stated in terms of order (or or…
A new method for graph-structured data improves transformer performance by incorporating topology.
problem Improving transformer performance on graph-structured data.
method Parameterizing topological masks as a learnable function of a weighted adjacency matrix, approximated with graph random features.
result Efficient masking algorithms provide strong performance gains for tasks on image and point cloud data.
Detect anomalies in complex networks using topological subspace detectors.
problem Detect anomalies in complex networks defined by simplicial complexes.
method Formulate a hypothesis testing framework using Neyman-Pearson matched topological subspace detectors.
result Effective detection of anomalies in foreign currency exchange networks and other real-world data.
Study geodesics on a special cylinder with arbitrary wind.
problem Global behavior of geodesics on a Randers metric cylinder.
method Solve Zermelo's navigation problem to define the Randers metric; analyze geodesics, conjugate, and cut loci.
result Characterize geodesics and their properties on the base manifold.
We describe a natural decomposition of a normal complex surface singularity (X,0) into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the…
Determines higher smooth surgery structure sets of complex projective spaces.
problem Understanding the higher smooth surgery structure sets of complex projective spaces.
method Analyzes the free subgroup and torsion in low dimensions.
result Obtains information in all dimensions for the free subgroup.
Theory for algebraic data on categories via concentration structures.
problem Defining algebraic structures on categories.
method Introducing concentration structures and concentration monoids.
result Every group can be represented as a concentration monoid of a trivial category.
The paper studies how norms of random vectors are preserved by random projections.
problem Understanding how random matrix affects norms of random vectors.
method Proved the distribution of the norm of random vector is preserved by random projection.
result Random matrix preserves the distribution of the norm of random vectors with i.i.d. entries.
Prototype selection improved using topological data analysis.
problem Improving prototype selection methods for data compression.
method Introducing two topological prototype selector variants: TPS and BoundaryTPS.
result BoundaryTPS achieves the lowest mean Friedman rank on H1 persistence-diagram preservation. Paper addresses concentration of distances for fractional quasi p-norms, identifying conditions for concentration and anti-concentration.
problem Understanding concentration of distances for fractional quasi p-norms in high dimensions.
method Analyzes conditions for concentration and anti-concentration of distances for fractional quasi p-norms.
result Identifies conditions for concentration and anti-concentration of fractional quasi p-norms, ruling out some approaches and specifying conditions for control.
We present a complete acyclic matching of the Hasse diagram associated with the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. We will then utilize this matching along with discrete Morse theory and some topological techniques to cl…
Study Finsler metric measure manifolds' concentration properties.
problem Understanding concentration properties in Finsler metric measure manifolds.
method Established relationships with observable diameter, isoperimetric inequalities, and first eigenvalue.
result Derived a Cheng type upper bound estimate for the first closed eigenvalue.
The main thrust of present note is a volume formula for hyperbolic surface bundle with the fundamental group G. The novelty consists in a purely algebraic approach to the above problem. Initially, we concentrate on the Baum-Connes morphism m(G): K(BG)--> K(C*G) for our class of manifolds, and then classify m(G) in term…
New method improves missing mass concentration bounds.
problem Missing mass concentration problem
method New method of estimating concentration of heterogenic sums
result Slightly improved state-of-the-art bounds
Theoretical study explains grokking in neural networks.
problem Understanding the abrupt transition from fitting to generalizing in neural networks.
method Characterized a shell-core topological configuration of the solution space induced by Adam's optimization dynamics.
result Derived grokking scaling laws for learning rate, batch size, and regularization coefficient.
Surfaces in 3-manifolds concentrate at curvature critical points.
problem Understanding concentration of surfaces in 3-manifolds.
method Proving surfaces concentrate at critical points of scalar curvature.
result Simply connected H-surfaces concentrate at curvature critical points.
TRA detects causal direction from bivariate data using geometric shapes.
problem Inferring causal direction from observational data is challenging and unreliable.
method TRA compares rank-based copula-standardized residual clouds to detect causal direction.
result TRA is robust and superior in detecting causal direction across various scenarios.
Sharp concentration bounds for i.i.d. variables.
problem Controlling the tail probabilities of independent variables.
method Extension of Sanov's theorem using large deviations and information theory.
result Matching concentration and anti-concentration bounds for i.i.d. samples of any size.
We consider principal component analysis (PCA) in decomposable Gaussian graphical models. We exploit the prior information in these models in order to distribute its computation. For this purpose, we reformulate the problem in the sparse inverse covariance (concentration) domain and solve the global eigenvalue problem …
We survey recent results related to the concentration of eigenfunctions. We also prove some new results concerning ball-concentration, as well as showing that eigenfunctions saturating lower bounds for L1-norms must also, in a measure theoretical sense, have extreme concentration near a geodesic.
A new model CDTM improves text classification by concentrating document topics.
problem Unsupervised text classification with diverse topic distributions.
method Imposes an exponential entropy penalty on document topic distribution to encourage concentration.
result More coherent topics and concentrated, sparse document-topic distributions.
Simplified proof of Gaussian concentration inequality using covariance.
problem Gaussian concentration inequality proof
method Covariance representation based on characteristic functions
result Elementary proof of Gaussian concentration inequality
Study on inequalities for multinomial variables.
problem Understanding concentration inequalities for multinomial variables.
method Investigation of Dirichlet and Multinomial random variables.
result Results on concentration inequalities for multinomial variables.
Study cohomology rings of 3D manifolds with round fold maps into the plane.
problem Understanding cohomology rings of 3D manifolds with round fold maps.
method Analyzing cohomology rings of 3D manifolds admitting round fold maps into the plane.
result Explicit new study showing relation between coefficient rings and topological types of round fold maps.
Study provides bounds for estimating intrinsic dimension using Gaussian kernels.
problem Estimating intrinsic dimension from data.
method Finite-sample concentration and anti-concentration bounds for Gaussian kernel sums.
result Explicit dependence on sample size, bandwidth, and geometric parameters.
Developed concentrated liquidity in n-dimensional AMM with polar coordinates in Rust.
problem Risk of stacking too many stablecoin pools.
method Building concentrated liquidity positions with ticks in polar coordinates in Rust.
result Hedging risk of stacking stablecoin pools.
Study reveals how dengue spread patterns vary across different years in Recife, Brazil.
problem Understanding spatial organization of dengue transmission in urban areas.
method Spatial analysis of dengue cases using topological data analysis and Vietoris-Rips filtrations.
result Critical percolation thresholds define distinct geometric regimes of dengue spread.
In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative 1-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concen…
Study on volume of tubes and concentration in Riemannian geometry.
problem Understanding concentration loci in Riemannian manifolds and their relation to tube volumes.
method Provided a general formula for tube volumes, specialized to totally geodesic submanifolds, and investigated concentration loci.
result Explicitly proved concentration for codimension one cases and explored characterizations in Wasserstein and Box distances.
Polluting fine dusts in South Korea which are mainly consisted of biomass burning and fugitive dust blown from dust belt is significant problem these days. Predicting concentrations of fine dust particles in Seoul is challenging because they are product of complicate chemical reactions among gaseous pollutants and also…
This study assesses risk concentration in MDB portfolios using Monte Carlo simulations.
problem Risk concentration in MDB portfolios of a few borrowers.
method Realistic MDB portfolio simulations and Monte Carlo analysis.
result Current risk adjustments may be overly conservative.
We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequa…
New approach to concentration inequalities for unbounded state space dynamical systems.
problem Concentration inequalities for unbounded state space dynamical systems.
method Functional analytic framework, transport-entropy inequality.
result Exponential concentration inequalities for sampling from stationary distribution.