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In this paper, we give a conditional lower bound of $n^{Omega(k)}$ on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in low-dimensional Euclidean space of dimension four, assuming the Exponential Time Hypothesis (ETH). We also consider k-median (and k-means) with penalties where each point need not be assigned to a center, in which case it must pay a penalty, and extend our lower bound to at least three-dimensional Euclidean space. This stands in stark contrast to many other geometric problems such as the traveling salesman problem, or computing an independent set of unit spheres. While these problems benefit from the so-called (limited) blessing of dimensionality, as they can be solved in time $n^{O(k^{1-1/d})}$ or $2^{n^{1-1/d}}$ in d dimensions, our work shows that widely-used clustering objectives have a lower bound of $n^{Omega(k)}$, even in dimension four. We complete the picture by considering the two-dimensional case: we show that there is no algorithm that solves the penalized version in time less than $n^{o(sqrt{k})}$, and provide a matching upper bound of $n^{O(sqrt{k})}$. The main tool we use to establish these lower bounds is the placement of points on the moment curve, which takes its inspiration from constructions of point sets yielding Delaunay complexes of high complexity.
We show how to approximate a data matrix $mathbf{A}$ with a much smaller sketch $mathbf{tilde A}$ that can be used to solve a general class of constrained k-rank approximation problems to within $(1+epsilon)$ error. Importantly, this class of problem
We study the classic $k$-median and $k$-means clustering objectives in the beyond-worst-case scenario. We consider three well-studied notions of structured data that aim at characterizing real-world inputs: Distribution Stability (introduced by Awast
We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is $alpha$-stable if the underlying optimal clustering continues to remain optimal even when all pairwise
In this paper we present novel algorithms for several multidimensional data processing problems. We consider problems related to the computation of restricted clusters and of the diameter of a set of points using a new distance function. We also cons
Given points $p_1, dots, p_n$ in $mathbb{R}^d$, how do we find a point $x$ which maximizes $frac{1}{n} sum_{i=1}^n e^{-|p_i - x|^2}$? In other words, how do we find the maximizing point, or mode of a Gaussian kernel density estimation (KDE) centered