Given a collection of $n$ points in $\mathbb{R}^d$, the goal of the $(k,z)$-clustering problem is to find a subset of $k$ "centers" that minimizes the sum of the $z$-th powers of the Euclidean distance of each point to the closest center. Special cases of the $(k,z)$-clustering problem include the $k$-median and $k$-means problems. Our main result is a unified two-stage importance sampling framework that constructs an $\varepsilon$-coreset for the $(k,z)$-clustering problem. Compared to the results for $(k,z)$-clustering in [Feldman and Langberg, STOC 2011], our framework saves a $\varepsilon^2 d$ factor in the coreset size. Compared to the results for $(k,z)$-clustering in [Sohler and Woodruff, FOCS 2018], our framework saves a $\operatorname{poly}(k)$ factor in the coreset size and avoids the $\exp(k/\varepsilon)$ term in the construction time. Specifically, our coreset for $k$-median ($z=1$) has size $\tilde{O}(\varepsilon^{-4} k)$ which, when compared to the result in [Sohler and Woodruff, STOC 2018], saves a $k$ factor in the coreset size. Our algorithmic results rely on a new dimensionality reduction technique that connects two well-known shape fitting problems: subspace approximation and clustering, and may be of independent interest. We also provide a size lower bound of $Ω\left(k\cdot \min \left\{2^{z/20},d \right\}\right)$ for a $0.01$-coreset for $(k,z)$-clustering, which has a linear dependence of size on $k$ and an exponential dependence on $z$ that matches our algorithmic results.
Given a set of $n$ points in $d$ dimensions, the Euclidean $k$-means problem (resp. the Euclidean $k$-median problem) consists of finding $k$ centers such that the sum of squared distances (resp. sum of distances) from every point to its closest center is minimized. The arguably most popular way of dealing with this problem in the big data setting is to first compress the data by computing a weighted subset known as a coreset and then run any algorithm on this subset. The guarantee of the coreset is that for any candidate solution, the ratio between coreset cost and the cost of the original in…
Designing coresets--small-space sketches of the data preserving cost of the solutions within $(1\pm ε)$-approximate factor--is an important research direction in the study of center-based $k$-clustering problems, such as $k$-means or $k$-median. Feldman and Langberg [STOC'11] have shown that for $k$-clustering of $n$ points in general metrics, it is possible to obtain coresets whose size depends logarithmically in $n$. Moreover, such a dependency in $n$ is inevitable in general metrics. A significant amount of recent work in the area is devoted to obtaining coresests whose sizes are independen…
Vertical federated learning (VFL), where data features are stored in multiple parties distributively, is an important area in machine learning. However, the communication complexity for VFL is typically very high. In this paper, we propose a unified framework by constructing coresets in a distributed fashion for communication-efficient VFL. We study two important learning tasks in the VFL setting: regularized linear regression and $k$-means clustering, and apply our coreset framework to both problems. We theoretically show that using coresets can drastically alleviate the communication complex…
A set of points $P$ in a metric space and a constant integer $k$ are given. The $k$-center problem finds $k$ points as centers among $P$, such that the maximum distance of any point of $P$ to their closest centers $(r)$ is minimized. Doubling metrics are metric spaces in which for any $r$, a ball of radius $r$ can be covered using a constant number of balls of radius $r/2$. Fixed dimensional Euclidean spaces are doubling metrics. The lower bound on the approximation factor of $k$-center is $1.822$ in Euclidean spaces, however, $(1+ε)$-approximation algorithms with exponential dependency on $\f…
In the Euclidean $k$-center problem in sliding window model, input points are given in a data stream and the goal is to find the $k$ smallest congruent balls whose union covers the $N$ most recent points of the stream. In this model, input points are allowed to be examined only once and the amount of space that can be used to store relative information is limited. Cohen-Addad et al.~\cite{cohen-2016} gave a $(6+ε)$-approximation for the metric $k$-center problem using O($k/ε\log α$) points, where $α$ is the ratio of the largest and smallest distance and is assumed to be known in advance. In th…
Google Research（谷歌研究院）；Aarhus University（奥胡斯大学）；Sorbonne Université（索邦大学）
Given a set of points in a metric space, the $(k,z)$-clustering problem consists of finding a set of $k$ points called centers, such that the sum of distances raised to the power of $z$ of every data point to its closest center is minimized. Special cases include the famous k-median problem ($z = 1$) and k-means problem ($z = 2$). The $k$-median and $k$-means problems are at the heart of modern data analysis and massive data applications have given raise to the notion of coreset: a small (weighted) subset of the input point set preserving the cost of any solution to the problem up to a multipl…
We present an algorithm for computing $ε$-coresets for $(k, \ell)$-median clustering of polygonal curves in $\mathbb{R}^d$ under the Fréchet distance. This type of clustering is an adaption of Euclidean $k$-median clustering: we are given a set of $n$ polygonal curves in $\mathbb{R}^d$, each of complexity (number of vertices) at most $m$, and want to compute $k$ median curves such that the sum of distances from the given curves to their closest median curve is minimal. Additionally, we restrict the complexity of the median curves to be at most $\ell$ each, to suppress overfitting, a problem sp…
University of Science and Technology of China（中国科学技术大学）
We study the problem of $k$-center clustering with outliers in arbitrary metrics and Euclidean space. Though a number of methods have been developed in the past decades, it is still quite challenging to design quality guaranteed algorithm with low complexity for this problem. Our idea is inspired by the greedy method, Gonzalez's algorithm, for solving the problem of ordinary $k$-center clustering. Based on some novel observations, we show that this greedy strategy actually can handle $k$-center clustering with outliers efficiently, in terms of clustering quality and time complexity. We further…
Clustering is the task of partitioning a given set of geometric objects. This is thoroughly studied when the objects are points in the euclidean space. There are also several approaches for points in general metric spaces. In this thesis we consider clustering polygonal curves, i.e., curves composed of line segments, under the Fréchet distance. We obtain clusterings by minimizing an objective function, which yields a set of centers that induces a partition of the input. The objective functions we consider is the so called $(k,l)$-\textsc{center}, where we are to find the $k$ center-curves that…
Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Aalto University（阿尔托大学）；Tel Aviv University（特拉维夫大学）；California State University Northridge（北岭加州州立大学）；Tufts University（塔夫茨大学）；Max Planck Institute for Informatics（马克斯·普朗克计算机科学研究所）；University of Copenhagen（哥本哈根大学）
Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We also show some interesting connections of gap ratio to Delaunay triangulation and discrepancy in the Euclidean plane. The major focus of this work is on solv…
Point location problems for $n$ points in $d$-dimensional Euclidean space (and $\ell_p$ spaces more generally) have typically had two kinds of running-time solutions: * (Nearly-Linear) less than $d^{poly(d)} \cdot n \log^{O(d)} n$ time, or * (Barely-Subquadratic) $f(d) \cdot n^{2-1/Θ(d)}$ time, for various $f$. For small $d$ and large $n$, "nearly-linear" running times are generally feasible, while "barely-subquadratic" times are generally infeasible. For example, in the Euclidean metric, finding a Closest Pair among $n$ points in ${\mathbb R}^d$ is nearly-linear, solvable in $2^{O(d)} \cdot n…
We study the k nearest neighbors problem in the plane for general, convex, pairwise disjoint sites of constant description complexity such as line segments, disks, and quadrilaterals and with respect to a general family of distance functions including the L_p-norms and additively weighted Euclidean distances. For point sites in the Euclidean metric, after four decades of effort, an optimal data structure has recently been developed with O( n ) space, O( log n + k ) query time, and O( n log n ) preprocessing time. We develop a static data structure for the general setting with nearly optimal O(…
Let $P$ be a set of $n$ points in $\mathbb{R}^d$. In the projective clustering problem, given $k, q$ and norm $ρ\in [1,\infty]$, we have to compute a set $\mathcal{F}$ of $k$ $q$-dimensional flats such that $(\sum_{p\in P}d(p, \mathcal{F})^ρ)^{1/ρ}$ is minimized; here $d(p, \mathcal{F})$ represents the (Euclidean) distance of $p$ to the closest flat in $\mathcal{F}$. We let $f_k^q(P,ρ)$ denote the minimal value and interpret $f_k^q(P,\infty)$ to be $\max_{r\in P}d(r, \mathcal{F})$. When $ρ=1,2$ and $\infty$ and $q=0$, the problem corresponds to the $k$-median, $k$-mean and the $k$-center clust…
