On the k-Means/Median Cost Function

In this work, we study the $k$-means cost function. Given a dataset $X \subseteq \mathbb{R}^d$ and an integer $k$, the goal of the Euclidean $k$-means problem is to find a set of $k$ centers $C \subseteq \mathbb{R}^d$ such that $Φ(C, X) \equiv \sum_{x \in X} \min_{c \in C} ||x - c||^2$ is minimized. Let $Δ(X,k) \equiv \min_{C \subseteq \mathbb{R}^d} Φ(C, X)$ denote the cost of the optimal $k$-means solution. For any dataset $X$, $Δ(X,k)$ decreases as $k$ increases. In this work, we try to understand this behaviour more precisely. For any dataset $X \subseteq \mathbb{R}^d$, integer $k \geq 1$, and a precision parameter $\varepsilon > 0$, let $L(X, k, \varepsilon)$ denote the smallest integer such that $Δ(X, L(X, k, \varepsilon)) \leq \varepsilon \cdot Δ(X,k)$. We show upper and lower bounds on this quantity. Our techniques generalize for the metric $k$-median problem in arbitrary metric spaces and we give bounds in terms of the doubling dimension of the metric. Finally, we observe that for any dataset $X$, we can compute a set $S$ of size $O \left(L(X, k, \varepsilon/c) \right)$ using $D^2$-sampling such that $Φ(S,X) \leq \varepsilon \cdot Δ(X,k)$ for some fixed constant $c$. We also discuss some applications of our bounds.