New cell structure on O ( 3 ) / O ( 1 ) 3 O(3)/O(1)^3 O ( 3 ) / O ( 1 ) 3 derived from injectivity radius computation.
problem Constructing equivariant cell structures on flag manifolds.
method Injectivity radius computation and Dirichlet-Voronoi domains.
result New S 3 \mathfrak{S}_3 S 3 -equivariant cell structure on O ( 3 ) / O ( 1 ) 3 O(3)/O(1)^3 O ( 3 ) / O ( 1 ) 3 . New method calculates cut locus on surfaces without boundary.
problem Computing the cut locus on compact submanifolds.
method Variational convex problem with conic constraints.
result Proven convergence of the approximation method.
Algorithm finds adversarial examples for k-NN classifiers using Voronoi diagrams.
problem Ensuring robustness of k-NN classifiers against adversarial attacks.
method Geometric approach expanding outwards from input points to find minimum-norm adversarial examples.
result Our method outperforms existing approaches on various datasets.
Proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.
problem Existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.
method Construct an isotopic map instead of edge-flipping algorithm, generalizing Dyer et al's method.
result Strict proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.
Estimates BV functions from noisy data using Voronoi diagrams.
problem Estimating multivariate BV functions from scattered noisy data.
method Form Voronoi diagram, solve optimization problem with discrete TV regularization.
result Voronoigram is minimax rate optimal for BV functions.
Algorithm constructs graphic matroid from graph's flow lattice.
problem Constructing graphic matroid from graph's flow lattice.
method Based on Amini's result linking Voronoi cell geometry to graph structure.
result Algorithmic construction of graphic matroid from lattice of integer flows.
Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadr…
Proves least Gaussian perimeter decomposition conjectures for 2-3 cells in n-dimensional space.
problem Finding least perimeter ways to divide space into cells of prescribed Gaussian measure.
method Analyzes stable clusters and uses Voronoi cells of equidistant points.
result Simplicial clusters are unique minimizers for 2-3 cells in n-dimensional space.
Adversarial training improves model robustness with Voronoi constraints.
problem Adversarial examples mislead machine learning models, leading to incorrect classifications.
method Geometric framework using Voronoi cells to constrain adversarial training.
result Adversarial training with Voronoi constraints produces robust models.
Proposes a new adversarial model to avoid accuracy vs. adversarial accuracy tradeoff.
problem Inherent tradeoff between accuracy and adversarial accuracy in existing adversarial robustness definitions.
method Introduces Voronoi-epsilon adversary that balances perturbation constraints.
result Voronoi-epsilon adversary avoids accuracy vs. adversarial accuracy tradeoff even with large ε ε ε . The paper proves actions of lattices in higher rank groups have cost one.
problem Fixed price question for higher rank semisimple Lie groups.
method Low intensity Poisson point processes and geometry of Voronoi tessellations.
result Proves all probability measure preserving actions of lattices in higher rank groups have cost one.
We prove an optimal systolic inequality for CAT(0) metrics on a genus~2 surface. We use a Voronoi cell technique, introduced by C.~Bavard in the hyperbolic context. The equality is saturated by a flat singular metric in the conformal class defined by the smooth completion of the curve y^2=x^5-x. Thus, among all CAT(0) …
In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi diagrams and compactifications of configuration spaces. We study moving and colliding p…
New insights into the top-K sparse softmax gating function for deep learning.
problem Understanding the theoretical effects of the top-K sparse softmax gating function on density and parameter estimations.
method Using a Gaussian mixture of experts, novel loss functions, and theoretical analysis.
result The convergence rates of density and parameter estimations are parametric under certain conditions, but slow under over-specified models.
Study higher rank inner products and their tilings to describe tori degenerations.
problem Understanding metric degenerations of tori.
method Introduce higher rank inner products and their tilings, use to describe degenerations.
result Describe metric degenerations of polarized tori and Hausdorff limits of tilings.
A hex sphere is a singular Euclidean sphere with four cones points whose cone angles are (integer) multiples of 2*pi/3 but less than 2*pi. Given a hex sphere M, we consider its Voronoi decomposition centered at the two cone points with greatest cone angles. In this paper we use elementary Euclidean geometry to describe…
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
New algorithm for computing Veech groups from translation surfaces.
problem Computing Veech groups for translation surfaces.
method Infinite translation surface containing copies of all surfaces in a stratum; associated affine automorphisms of the infinite surface map marked segments to other pairs of segments.
result Explicit hyperbolic ball condition for Fuchsian groups to agree with their Dirichlet domain.
The study uses persistent homology to determine when Voronoi interpolation should stop.
problem Interpolating complex topological data sets accurately.
method Persistent homology is applied to the Voronoi tessellation to detect changes in the data's topology.
result The method effectively identifies when the interpolation has captured the data's topology changes.
Study homogenizes equations on parallelizable manifolds using tensor localization and periodicity.
problem Homogenizing oscillating linear elliptic equations on parallelizable manifolds.
method Two-scale convergence through localization and periodicity induced by geometry.
result Explicit cell formulae for the homogenization limit and a theory of two-scale convergence of tensors.
Predict cell loads in cellular networks using statistical learning of geometric marks.
problem Predicting cell loads in cellular networks using geometric marks.
method Statistical regression model and scattering moments of random measures.
result Scattering moments can capture similar geometry information as baseline approach and improve performance.
A new method for Bayesian optimization uses Voronoi tessellation candidates to reduce search time.
problem Efficiently optimizing black-box functions with minimal overhead.
method Using Voronoi tessellation candidates for continuous optimization of acquisition functions.
result Significantly improved execution time with no loss in accuracy.
New matching estimators correct bias in multivariate settings without smoothing parameters.
problem Bias in nearest-neighbor and matching estimators in multiple dimensions.
method Polynomial least squares fits on Voronoi tessellations.
result Novel estimators converge at n \sqrt{n} n rate under mild smoothness assumptions. Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.
problem Creating space-filling shapes without sharp corners.
method Edge bending algorithm to deform polyhedral tilings into soft tilings.
result Soft tilings derived from minimal surfaces can be continuously transformed into one another.
Constructs an explicit cycle in arithmetic group cohomology.
problem Cohomology of SL n ( Z ) _n(\mathbb{Z}) n ( Z ) at virtual cohomological dimension. method Geometric rigidity of Voronoi tessellations and abstract framework for polyhedral tessellations.
result Explicit canonical cycle in top-dimensional homology of Voronoi complex.
Deviance Voronoi residuals improve earthquake insurance risk assessment.
problem Assessing earthquake insurance risk using spatio-temporal point process models.
method Extended Voronoi residuals and created simulation-based approach.
result Proposed formula for country-wide minimum capital test.
Improved taxi demand-supply forecasts using graph-based LSTM.
problem Accurate taxi demand-supply forecasting with complex spatial and temporal patterns.
method Investigated impact of spatial partitioning techniques (Voronoi vs. Geohash) on LSTM network performance.
result GraphLSTM offers competitive performance against ConvLSTM, at lower complexity, across real-world data sets.
Distributional lattices on Riemannian symmetric spaces are studied, leading to new insights on random walks.
problem Understanding distributional lattices on Riemannian symmetric spaces.
method Introduced distributional lattices, used amenability equivalence, and developed graph speed for Poisson-Voronoi tessellations.
result Simple random walk on distributional lattices in nonamenable spaces has positive embedded speed.
The paper provides convergence bounds for approximating a distribution using point clouds.
problem Approximating a distribution using discrete points with minimal Wasserstein distance.
method Lloyd's algorithm with Power cells, analyzed using gradient descent.
result Explicit upper bounds for the convergence speed of the Lloyd-type algorithm.
Generates infinite-depth hierarchical clusters from few examples.
problem Inadequate finite-sample clustering methods for fine-scale hierarchical structures.
method Classification fields generated by a local refinement rule, approximated by predictors.
result Learned predictors can approximate infinite-depth hierarchical structures.
Consider a set represented by an inequality. An interesting phenomenon which occurs in various settings in mathematics is that the interior of this set is the subset where strict inequality holds, the boundary is the subset where equality holds, and the closure of the set is the closure of its interior. This paper disc…
Confirms isoperimetric conjectures on R^n and S^n for q ≤ min(5, n+1).
problem Minimizing total perimeter among bubbles enclosing prescribed volume.
method Tandem consideration of R^n and S^n, Möbius geometry, conformal Killing fields.
result Spherical interfaces and connected cells in minimizers, resolving Heppes conjecture.
The paper explores clustering methods using Bregman divergences.
problem Developing efficient clustering algorithms for complex data.
method Investigates fixed rate quantization and Voronoi diagrams in Riemannian metric spaces induced by separable Bregman divergences.
result Experimental results show improved performance of clustering algorithms using these metrics.
Given a set S of n points in general position, we consider all k-th order Voronoi diagrams on S, for k=1,...,n, simultaneously. We deduce symmetry relations for the number of faces, number of vertices and number of circles of certain orders. These symmetry relations are independent of the position of the sites in S. As…
FLOLA-Voronoi improves function approximation with uncertain outputs.
problem Approximating expensive, uncertain functions with active learning.
method Active learning focusing on uncertain regions, derived from theoretical analysis of output uncertainty.
result Algorithm provides more information to models by emphasizing exploration.
Study connects spectral geometry with Coulomb interactions in perforated manifolds.
problem Understanding spectral properties of perforated manifolds and their interactions.
method Optimal convergence rates for Steklov eigenvalues and expansions, derived from Green function and Coulomb-type energy.
result Identified two correction scales for Steklov eigenvalues in dimensions two and three.
Pólya's theorem extended to meromorphic functions on Riemann surfaces.
problem Distribution of zeros of iterated derivatives of meromorphic functions.
method Recasting local arguments into translation surfaces and using flat metrics.
result Asymptotic distribution of zeros on compact Riemann surfaces.
New upper bound for Cheeger constant of hyperbolic surfaces.
problem Bounding the Cheeger constant of hyperbolic surfaces.
method Random construction based on Poisson--Voronoi tessellation.
result The Cheeger constant of closed hyperbolic surfaces is less than that of the hyperbolic plane.
This paper studies geometrical structure of the manifold of escort probability distributions and shows its new applicability to information science. In order to realize escort probabilities we use a conformal transformation that flattens so-called alpha-geometry of the space of discrete probability distributions, which…
Bayesian model captures mean and variance of response variables.
problem Complex, predictor-dependent relationships and heteroscedastic patterns in data.
method Sum-of-tessellations for mean, product-of-tessellations for variance.
result Model captures nuanced variance structures and provides reliable predictive uncertainty.
Quantized Variational Inference improves ELBO optimization with fast convergence.
problem Maximizing Evidence Lower Bound (ELBO) for variational inference.
method Optimal Voronoi Tesselation for variance-free gradients, Richardson extrapolation for asymptotic improvement.
result Quantized Variational Inference leads to fast convergence with comparable computational cost.
Standard bubbles and partitions are stable in various model spaces.
problem Stability of standard bubbles and partitions in different model spaces.
method New conjugated Brascamp-Lieb inequality and conformally flattening boundary potential.
result Stability of standard bubbles and partitions in R n \mathbb{R}^n R n , S n \mathbb{S}^n S n , and H n \mathbb{H}^n H n . Proposes Dirichlet Simplex Nest for probabilistic modeling of various data types.
problem Modeling and inference for diverse data types.
method Probabilistic models based on Dirichlet distribution and Voronoi tessellation, with fast and accurate inference algorithms exploiting convex geometry and simplicial structure.
result Inference algorithms achieve consistency and strong error bounds across various settings and data distributions.
Study on soap bubble clusters, showing manifold properties.
problem Understanding the space of planar soap bubble clusters.
method Analysis of soap bubble clusters as generalized Voronoi partitions.
result The space of planar clusters with positive second variation is an n-dimensional manifold.
The paper analyzes how companies' investments before crises affect their performance after crises.
problem Understanding how companies' investments before financial crises impact their performance afterward.
method Cluster analysis using Voronoi tessellation with statistical outliers identified.
result Positive investments before crises are associated with better performance after crises.
Study spider mechanism configuration spaces using squared distance function.
problem Understand configuration spaces of spider mechanisms.
method Use Morse theory of squared distance function from body to fixed point.
result List and describe critical manifolds of squared distance function as products of polygon spaces.
New method reduces uncertainty in high-dimensional circuits by automatically determining tensor rank and adaptive sampling.
problem Uncertainty quantification in high-dimensional circuits due to fabrication process variations.
method Tensor regression with ℓ q / ℓ 2 \ell_{q}/ \ell_{2} ℓ q / ℓ 2 group-sparsity regularization for rank determination and adaptive sampling. result Captures uncertainty with only 100-600 simulation samples for 19-100 random variables.
A new framework detects anomalies in structured data.
problem Detecting anomalies in samples not conforming to low-dimensional manifolds.
method Preference Isolation Forest (PIF) framework combining adaptive isolation methods and preference embedding.
result Anomalies identified as isolated points in a high-dimensional preference space.