arXiv research

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

New method calculates cut locus on surfaces without boundary.

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

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

Estimates BV functions from noisy data 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…

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

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-…

The paper proves 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) …

We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a "simplicial cluster", obtained from the Voronoi cells of $q$ equidistant points. Moreover, we prove that…

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.

New insights into the top-K sparse softmax gating function for deep learning.

The lattice of integer flows of a graph is known to determine the graph up to 2-isomorphism (work of Su--Wagner and Caporaso--Viviani). In this paper we give an algorithmic construction of the graphic matroid $\calM(G)$ of a graph $G$, given its lattice of integer flows $\calF(G)$. The algorithm can then be applied to …

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.

New algorithm for computing Veech groups from translation surfaces.

Study homogenizes equations on parallelizable manifolds using tensor localization and periodicity.

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

Constructs an explicit cycle in arithmetic group cohomology.

Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.

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

Deviance Voronoi residuals improve earthquake insurance risk assessment.

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…

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…

The paper provides convergence bounds for approximating a distribution using point clouds.

In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the p…

Generates infinite-depth hierarchical clusters from few examples.

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).

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…

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.

Standard bubbles and partitions are stable in various model spaces.

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

Study spider mechanism configuration spaces using squared distance function.

New method reduces uncertainty in high-dimensional circuits by automatically determining tensor rank and adaptive sampling.

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.