arXiv research

New tools for constructing fixed point sets in digital topology.

We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the c…

A new model for point processes without intensity function trade-offs.

Study fixed-point sets of $S^{1}$-actions on quaternionic manifolds.

We show that the co-rays to a ray in a complete non-compact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the co-point set to a ray as the cut locus of such level sets. The structure theorem of the co-point set on a surface, namely that is a local tree, …

Study shows convergence speed for Fekete points on specific sets.

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most pi/2, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.

We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal…

Fixed point sets of certain group actions are contractible.

Study finds critical points in perimeter functional for fixed volume sets.

Paper analyzes algorithms for nonstationary saddle-point optimization problems.

Often noisy point clouds are given as an approximation of a particular compact set of interest. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets …

Study shows spectral gaps limit points on surfaces.

We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …

Rectifies flat singular points of area-minimizing currents with singularity degree > 1.

While Multiple Instance (MI) data are point patterns -- sets or multi-sets of unordered points -- appropriate statistical point pattern models have not been used in MI learning. This article proposes a framework for model-based MI learning using point process theory. Likelihood functions for point pattern data derived …

From computational geometry comes the notion of a Gabriel graph of a point set in the plane. The Gabriel graph consists of those edges connecting two points of the point set such that the circle whose diameter is the edge does not contain any point of the point set in its interior. We define a generalization of the Gab…

Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.

This paper generalizes the envelope of mid-lines to intermediate lines for a plane curve.

In this article, we classify all involutions on S^6 with 3-dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted 3-sphere and the classification of free involutions on homotopy CP^3's.

Algorithm classifies point clouds using deep set linearized optimal transport.

The level set of an elliptic function is a doubly periodic point set in C. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in C^2 and its sections (``cuts'') by C. We give S a crystallographic isometry in C^2 by defining a fundamental surface element as a conformal ma…

Study critical points of Laplace eigenfunctions in polygons.

We study the level sets of the distance function from a boundary point of a convex set in Euclidean space. We provide a lower bound for the range of connectivity of the level sets, in terms of the critical points of the distance function in the sense of Grove-Shiohama-Gromov-Cheeger.

We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We al…

The paper proves group actions on spheres with odd fixed points.

Deep learning within the context of point clouds has gained much research interest in recent years mostly due to the promising results that have been achieved on a number of challenging benchmarks, such as 3D shape recognition and scene semantic segmentation. In many realistic settings however, snapshots of the environ…

We study natural additional structures on real algebraic surfaces with trivial first homology mod 2 of the complexification. If the set of real points realizes the zero of the second homology mod 2 of the complexification, then the set of real points is equipped with a pair of opposite orientations and a Spin structure…

Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.

Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.

Sparse subspace clustering (SSC) is one of the current state-of-the-art methods for partitioning data points into the union of subspaces, with strong theoretical guarantees. However, it is not practical for large data sets as it requires solving a LASSO problem for each data point, where the number of variables in each…

MPMC generates low-discrepancy points using graph neural networks.

DALES offers a large annotated aerial LiDAR dataset for 3D deep learning.

A new method for non-rigid point set registration reduces computational complexity.

Proves robust transitivity for geodesic flows from metrics with conjugate points.

In this paper, we investigate the fixed-point set of an element of a CAT(0) group in its boundary. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$. Let $g\in G$ and let $\mathcal{F}_g$ be the fixed-point set of $g$ in the boundary $\partial X$. Then we show that $\mathcal{F}_g=L(Z_g)$, where $Z_g$ is …

The paper classifies involutions on S^4, proving linearities under certain conditions.

The classical $k$-means algorithm for partitioning $n$ points in $\mathbb{R}^d$ into $k$ clusters is one of the most popular and widely spread clustering methods. The need to respect prescribed lower bounds on the cluster sizes has been observed in many scientific and business applications. In this paper, we present an…

Study circle actions on unitary manifolds with discrete fixed points.

Formula for fixed points on noncompact spaces.

Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.

PointTriNet generates 3D triangulations from point clouds efficiently and scalably.

Study on cold and freezing sets in digital images.

Modeling 3D continua with singular points using Yin sets.

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kahler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the fixed-poin…

New clustering method using point-set kernel measures similarity.

Point patterns are sets or multi-sets of unordered elements that can be found in numerous data sources. However, in data analysis tasks such as classification and novelty detection, appropriate statistical models for point pattern data have not received much attention. This paper proposes the modelling of point pattern…

Distance function to a finite set is a topological Morse function.