The Bane of Low-Dimensionality Clustering

TL;DR

This paper establishes strong computational lower bounds for k-median and k-means clustering problems in low-dimensional Euclidean spaces, showing they are computationally hard even in dimensions as low as three or four.

Contribution

It provides the first conditional lower bounds for these clustering problems in low dimensions, contrasting with other geometric problems that are easier in higher dimensions.

Findings

Lower bound of $n^{ ilde{ ext{Omega}}(k)}$ for k-median and k-means in 4D.

Lower bound of $n^{o( oot{ ext{sqrt}}{k})}$ in 2D for penalized clustering.

Matching upper bounds in 2D for penalized clustering.

Abstract

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.