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.
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48 results for “centroidal Voronoi tessellations”
Develops a new non-adversarial framework for better generative models.
problem Inaccurate approximation of target distribution in latent space.
method Tessellated Wasserstein Auto-Encoders (TWAE) using centroidal Voronoi tessellation (CVT) to tessellate latent space.
result Significantly enhances generative performance in terms of FID compared to existing models.
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.
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.
Accurate taxi demand-supply forecasting is a challenging application of ITS (Intelligent Transportation Systems), due to the complex spatial and temporal patterns. We investigate the impact of different spatial partitioning techniques on the prediction performance of an LSTM (Long Short-Term Memory) network, in the con…
Structured Radial Basis Function Network: Modelling Diversity for Multiple Hypotheses Predictioncs.LG
s-RBFN integrates multiple hypotheses for efficient and diverse prediction.
problem Integrating multiple hypotheses into learning models for regression.
method Structured Radial Basis Function Network (s-RBFN) using Voronoi tessellations and least-squares training.
result s-RBFN achieves superior generalization and efficiency compared to other models.
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.
Constructs an explicit cycle in arithmetic group cohomology.
problem Cohomology of SL 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.
A Riemannian symmetric space is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. On such spaces, we introduce the notion of a distributional lattice, generalizing the notion of lattice. Distributional lattices exist in any Riemannian symmetric space: th…
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.
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.
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…
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.
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.
This paper analyzes the connection between innovation activities of companies -- implemented before crisis -- and their performance -- measured at time of crisis. The companies listed in the STAR Market Segment of the Italian Stock Exchange are analyzed. Innovation is measured through the level of investments in total …
Taxi Demand-Supply Forecasting: Impact of Spatial Partitioning on the Performance of Neural Networkscs.LG
In this paper, we investigate the significance of choosing an appropriate tessellation strategy for a spatio-temporal taxi demand-supply modeling framework. Our study compares (i) the variable-sized polygon based Voronoi tessellation, and (ii) the fixed-sized grid based Geohash tessellation, using taxi demand-supply GP…
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.
The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.
problem Understanding the geometric structure of decorated hyperbolic surfaces.
method Developing a characterisation of canonical tessellations and dual decompositions using hyperbolic geometry.
result Decorations on hyperbolic surfaces induce unique canonical tessellations and dual decompositions.
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.
A key problem in location-based modeling and forecasting lies in identifying suitable spatial and temporal resolutions. In particular, judicious spatial partitioning can play a significant role in enhancing the performance of location-based forecasting models. In this work, we investigate two widely used tessellation s…
Dually flat structure with escort probability and its application to alpha-Voronoi diagramscond-mat.stat-mech
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…
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 rate under mild smoothness assumptions.
We propose Dirichlet Simplex Nest, a class of probabilistic models suitable for a variety of data types, and develop fast and provably accurate inference algorithms by accounting for the model's convex geometry and low dimensional simplicial structure. By exploiting the connection to Voronoi tessellation and properties…
In our physically inspired in-tree (IT) based clustering algorithm and the series after it, there is only one free parameter involved in computing the potential value of each point. In this work, based on the Delaunay Triangulation or its dual Voronoi tessellation, we propose a nonparametric process to compute potentia…
We consider a scenario where the aim of a group of agents is to perform the optimal coverage of a region according to a sensory function. In particular, centroidal Voronoi partitions have to be computed. The difficulty of the task is that the sensory function is unknown and has to be reconstructed on line from noisy me…
We introduce \texttt{pycobra}, a Python library devoted to ensemble learning (regression and classification) and visualisation. Its main assets are the implementation of several ensemble learning algorithms, a flexible and generic interface to compare and blend any existing machine learning algorithm available in Pytho…
Neural Message Passing with Edge Updates for Predicting Properties of Molecules and Materialsstat.ML
Neural message passing on molecular graphs is one of the most promising methods for predicting formation energy and other properties of molecules and materials. In this work we extend the neural message passing model with an edge update network which allows the information exchanged between atoms to depend on the hidde…
FiberNet integrates geometry into machine learning for clearer classification.
problem Lack of interpretability in traditional deep learning.
method Reformulates classification as geometric optimization on fiber bundles, introducing learnable Riemannian metrics and variational prototype optimization.
result Clear geometric interpretability and efficiency in classification.
K-Means and RBF networks are shown to be equivalent under certain conditions.
problem Discrete clustering vs. continuous optimization in machine learning.
method Established variational and gradient-based equivalence between K-Means and RBF networks.
result Gradient-based updates of RBF centers recover K-Means centroid update rule.
Generative adversarial networks (GANs) are the state of the art in generative modeling. Unfortunately, most GAN methods are susceptible to mode collapse, meaning that they tend to capture only a subset of the modes of the true distribution. A possible way of dealing with this problem is to use an ensemble of GANs, wher…
Limits of Voronoi Diagramsmath.MG
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…
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.
Tessellations of surfacesmath.HO
Tessellations cover planes without gaps or overlaps.
problem None explicitly stated, focuses on beauty and majesty of tessellations.
method Explains tessellations as tile coverings of a plane.
result Explains the beauty and majesty of tessellations and triangulation of surfaces.
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…
Up to isomorphism there are six fixed-point free crystallographic groups in Euclidean Space generated by twists (screw motions). In each case, an orientable 3-manifold is obtained as the quotient of E3 by such a group. The cubic tessellation of E3 induces tessellations on each such manifold. These tessellations of the …
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.
By regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we pr…
This paper is the second part of a two-part study of an elementary functorial construction of tesselated surfaces from finite groups. This elementary construction was discussed in the first part and generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to…
We present an explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F(n) embedded in CP3 defined by the equation z0^n + z1^n + z2^n + z3^n = 0 in the standard homogeneous coordinates [z0, z1, z2, z3], where n is any positive integer. Note that F(4) in particular is a K3 surface. Our tessellation…
Gradient-based clustering method for various cost functions.
problem Distance-based clustering for various cost functions.
method Iterative alternating update procedure for cluster assignments and centers.
result Converges to fixed points under mild assumptions.
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.
This paper is the first part in a 2 part study of an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this construction results in a collection of compact, connected, oriented tesselated smooth su…
Farey Bryophyllamath.GT
The paper generalizes Farey tessellation to 3D hyperbolic space.
problem Generalizing Farey tessellation to higher dimensions.
method Introducing conformal bryophylla and classifying them.
result Properties of conformal bryophylla's limiting sets studied.
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 .
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.
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…
Centroid Decision Foreststat.ML
CDF uses centroids to split features for high-dimensional classification.
problem High-dimensional classification problems with complex class structures.
method CDF introduces a centroid-driven splitting strategy in decision trees.
result CDF outperforms conventional methods in high-dimensional classification.