arXiv research

The study uses persistent homology to determine when Voronoi interpolation should stop.

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.

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…

Proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.

A new method for Bayesian optimization uses Voronoi tessellation candidates to reduce search time.

Topology-GS improves 3D GS for better structural and feature integrity.

Constructs an explicit cycle in arithmetic group cohomology.

Estimates BV functions from noisy data using Voronoi diagrams.

Deviance Voronoi residuals improve earthquake insurance risk assessment.

Proposes a new adversarial model to avoid accuracy vs. adversarial accuracy tradeoff.

Algorithm finds adversarial examples for k-NN classifiers using Voronoi diagrams.

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…

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…

Adversarial examples are a pervasive phenomenon of machine learning models where seemingly imperceptible perturbations to the input lead to misclassifications for otherwise statistically accurate models. We propose a geometric framework, drawing on tools from the manifold reconstruction literature, to analyze the high-…

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…

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…

New method calculates cut locus on surfaces without boundary.

Pólya's theorem extended to meromorphic functions on Riemann surfaces.

New upper bound for Cheeger constant of hyperbolic 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.

Quantized Variational Inference improves ELBO optimization with fast convergence.

The paper proves actions of lattices in higher rank groups have cost one.

New cell structure on $O(3)/O(1)^3$ derived from injectivity radius computation.

The paper analyzes how companies' investments before crises affect their performance after crises.

Study spider mechanism configuration spaces using squared distance function.

A new framework detects anomalies in structured data.

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.

New method uses geometric properties for better density estimation.

A new method learns quantization boundaries in continuous space using tessellation.

Separable Bregman divergences induce Riemannian metric spaces that are isometric to the Euclidean space after monotone embeddings. We investigate fixed rate quantization and its codebook Voronoi diagrams, and report on experimental performances of partition-based, hierarchical, and soft clustering algorithms with respe…

Two methods using low-discrepancy points improve data compression for neural networks.

rMCL improves on MCL by preserving diversity in predictions for regression problems.

When approximating a black-box function, sampling with active learning focussing on regions with non-linear responses tends to improve accuracy. We present the FLOLA-Voronoi method introduced previously for deterministic responses, and theoretically derive the impact of output uncertainty. The algorithm automatically p…

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…

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…

Study approximates probability measures using structured classes of functions.

Motivated by the prediction of cell loads in cellular networks, we formulate the following new, fundamental problem of statistical learning of geometric marks of point processes: An unknown marking function, depending on the geometry of point patterns, produces characteristics (marks) of the points. One aims at learnin…

New algorithm for computing Veech groups from translation surfaces.

Analyzes convergence rates for Gaussian-gated MoE model.

This paper shows how forward rate interpolations are equivalent to discount factor interpolations in yield curve construction.

New matching estimators correct bias in multivariate settings without smoothing parameters.

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) …

The paper proposes a method for generating uniform interpolations on data manifolds.

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…

s-RBFN integrates multiple hypotheses for efficient and diverse prediction.

New FX option interpolations impact implied volatilities.