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.
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.
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.
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…
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 . 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 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.
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.
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.
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…
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.
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.
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.
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.
New chiral minimal surfaces derived from quartz network.
problem Finding new triply-periodic minimal surfaces.
method Using dual graphs of quartz and its dual, generating area-minimizing meshes, and identifying flat point structures.
result Identified a new family of chiral triply-periodic minimal surfaces.
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.
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.
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 . 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.
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.
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.
A new method learns quantization boundaries in continuous space using tessellation.
problem Mapping between discrete and continuous distributions is difficult.
method Constructs normalizing flows on convex polytopes with exact likelihood evaluations.
result Improves likelihood evaluation and quantization learning across various data modalities.
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…
Two methods using low-discrepancy points improve data compression for neural networks.
problem Efficiently compress large datasets for neural network training.
method Two methods based on low-discrepancy points: digital nets with averaging and clustering.
result Second method outperforms supercompress in compression error and neural network accuracy.
The Delaunay tessellation of a locally finite subset of hyperbolic space is constructed using convex hulls in Euclidean space of one higher dimension. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily modified from the Euclidean setting) of the empty circumsp…
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.
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.
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.
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.
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.
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.
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.
New method uses geometric properties for better density estimation.
problem Uncertainty quantification in ambiguous tasks.
method Winner-takes-all training with centroidal Voronoi tessellations.
result Improved quantization and density estimation.
For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain.
Estimates parameters in a deviated Gaussian mixture model.
problem Testing goodness-of-fit between a known function and a mixture of experts.
method Constructs novel Voronoi-based loss functions to estimate parameters.
result Characterizes local convergence rates of parameter estimation more accurately.
Study compares tessellation strategies for taxi demand-supply forecasting models.
problem Improving taxi demand-supply forecasting using neural networks.
method Compared Voronoi tessellation and Geohash tessellation for LSTM models.
result Variable-sized polygon tessellation yields superior performance in LSTM models.
Pycobra is a Python toolbox for ensemble learning and visualization.
problem Ensemble learning and visualization for machine learning tasks.
method Implementation of ensemble learning algorithms and a flexible interface for blending any machine learning algorithm.
result Visualisation tools like Voronoi tessellations for ensemble learning.
rMCL improves on MCL by preserving diversity in predictions for regression problems.
problem Multimodal density estimation in regression settings with multiple targets.
method rMCL uses a learned scoring scheme based on Voronoi tessellations to maintain diversity among predictions.
result rMCL outperforms existing MCL variants in sound source localization tasks.
Cluster analysis reveals diverse performance outcomes from innovation during crises.
problem How innovation strategies affect company performance during crises.
method Cluster analysis of innovation and performance data from Italian companies.
result Performance outcomes vary among companies that innovate during crises.
Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John's characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the 'hard part' of a generalization of John's theorem and shows the difficulties of…
Improved molecular property prediction using updated neural message passing.
problem Predicting properties of molecules and materials accurately.
method Extended neural message passing model with edge update network.
result Superior prediction of formation energies and other properties on multiple datasets.
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.
Model predicts DFT formation energies without atomic positions.
problem Fast prediction of material properties without atomic positions.
method Symmetry-labeled graphs and message passing neural network.
result Mean absolute error below 0.1 eV for selenides.
Analyzes convergence rates for Gaussian-gated MoE model.
problem Theoretical understanding of Gaussian-gated MoE model is incomplete.
method Maximum likelihood estimation with novel Voronoi loss functions.
result MLE has distinct behaviors under different settings of Gaussian gating function parameters.
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.