Related papers: Improved Topological Approximations by Digitizatio…
Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously.…
Storing information in DNA molecules is of great interest because of its advantages in longevity, high storage density, and low maintenance cost. A key step in the DNA storage pipeline is to efficiently cluster the retrieved DNA sequences…
Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from,…
Clustering aims to form groups of similar data points in an unsupervised regime. Yet, clustering complex datasets containing critically intertwined shapes poses significant challenges. The prevailing clustering algorithms widely depend on…
The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this…
With the emergence of graph databases, the task of frequent subgraph discovery has been extensively addressed. Although the proposed approaches in the literature have made this task feasible, the number of discovered frequent subgraphs is…
We introduce the notion of t-restricted doubling dimension of a point set in Euclidean space as the local intrinsic dimension up to scale t. In many applications information is only relevant for a fixed range of scales. We present an…
We study the design of efficient approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in high dimensional Euclidean and Hamming spaces. Our main tool is randomized dimension reduction.…
We extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to…
In this note we analyze the use of Pad\'e approximants for downward continuation beyond the radius of convergence of spherical harmonic expansions (SHEs), and for identifying the complex singularities of the gravitational potential. SHEs…
Simplicial complexes are higher-order combinatorial structures which have been used to represent real-world complex systems. In this paper, we concentrate on the local patterns in simplicial complexes called simplets, a generalization of…
We present data streaming algorithms for the $k$-median problem in high-dimensional dynamic geometric data streams, i.e. streams allowing both insertions and deletions of points from a discrete Euclidean space $\{1, 2, \ldots \Delta\}^d$.…
The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality. In the first part of the paper we concentrate…
Given a data set of size $n$ in $d'$-dimensional Euclidean space, the $k$-means problem asks for a set of $k$ points (called centers) so that the sum of the $\ell_2^2$-distances between points of a given data set of size $n$ and the set of…
Given a set of $n$ points in the plane, the Unit Disk Cover (UDC) problem asks to compute the minimum number of unit disks required to cover the points, along with a placement of the disks. The problem is NP-hard and several approximation…
A topology preserving skeleton is a synthetic representation of an object that retains its topology and many of its significant morphological properties. The process of obtaining the skeleton, referred to as skeletonization or thinning, is…
This paper focuses on minimizing a smooth function combined with a nonsmooth regularization term on a compact Riemannian submanifold embedded in the Euclidean space under a decentralized setting. Typically, there are two types of approaches…
We study the topological signature of euclidean isometries in pair separations histograms (PSH) and elucidate some unsettled issues regarding distance correlations between cosmic sources in cosmic crystallography. Reducing the noise of…
We investigate convergence of point processes associated with critical faces for a \v{C}ech filtration built over a homogeneous Poisson point process in the $d$-dimensional flat torus. The convergence of our point process is established in…
In this paper, we introduce a distributed algorithm to compute th\v{e} Cech complex. This algorithm is aimed at solving coverage problems in self organized wireless networks. Two applications based on the distributed computation of th\v{e}…