Related papers: Improved Topological Approximations by Digitizatio…
Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
We study the approximate range searching for three variants of the clustering problem with a set $P$ of $n$ points in $d$-dimensional Euclidean space and axis-parallel rectangular range queries: the $k$-median, $k$-means, and $k$-center…
In this paper, we consider a class of constrained clustering problems of points in $\mathbb{R}^{d}$, where $d$ could be rather high. A common feature of these problems is that their optimal clusterings no longer have the locality property…
We introduce \emph{local Urysohn width}, a complexity measure for classification problems on metric spaces. Unlike VC dimension, fat-shattering dimension, and Rademacher complexity, which characterize the richness of hypothesis…
The Euler characteristic (EC) is a powerful topological descriptor that can be used to quantify the shape of data objects that are represented as fields/manifolds. Fast methods for computing the EC are required to enable processing of…
Segmentation is often an essential intermediate step in image analysis. A volume segmentation characterizes the underlying volume image in terms of geometric information--segments, faces between segments, curves in which several faces…
Locality sensitive hashing (LSH) is a fundamental algorithmic toolkit used by data scientists for approximate nearest neighbour search problems that have been used extensively in many large scale data processing applications such as near…
This work develops a framework to create meshes with user-specified homology from potentially dirty geometry by coupling background grids, persistent homology, and a generalization of volume fractions. For a mesh with fixed grid size, the…
The indirect estimation of couplings in quantum dynamics relies on the measurement of the spectrum and the overlap of eigenvectors with some reference states. This data can be obtained by local measurements on some sites and eliminates the…
Classical unsupervised learning methods like clustering and linear dimensionality reduction parametrize large-scale geometry when it is discrete or linear, while more modern methods from manifold learning find low dimensional representation…
The Vapnik-Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and…
This paper proposes a hierarchical clustering approach for the segmentation of mobile LiDAR point clouds. We perform the hierarchical clustering on unorganized point clouds based on a proximity matrix. The dissimilarity measure in the…
In a geometric $k$-clustering problem the goal is to partition a set of points in $\mathbb{R}^d$ into $k$ subsets such that a certain cost function of the clustering is minimized. We present data structures for orthogonal range-clustering…
We study $k$-means clustering in a semi-supervised setting. Given an oracle that returns whether two given points belong to the same cluster in a fixed optimal clustering, we investigate the following question: how many oracle queries are…
$\newcommand{\eps}{\varepsilon}$ We observe that a $(1-\eps)$-approximation algorithm to Independent Set, that works for any induced subgraph of the input graph, can be used, via a polynomial time reduction, to provide a…
Galaxy morphologies and shape orientations are expected to correlate with their large-scale environment, since they grow by accreting matter from the cosmic web and are subject to interactions with other galaxies. Cosmic filaments are…
We investigate the fine-grained complexity of approximating the classical $k$-median / $k$-means clustering problems in general metric spaces. We show how to improve the approximation factors to $(1+2/e+\varepsilon)$ and…
In this paper we study the homology of a random Cech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for "homological connectivity" where the homology of…
Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to…