EPTAS for $k$-means Clustering of Affine Subspaces

Abstract: We consider a generalization of the fundamental $k$-means clustering for data with incomplete or corrupted entries. When data objects are represented by points in $\mathbb{R}<sup>d$, a data point is said to be incomplete when some of its entries are missing or unspecified. An incomplete data point with at most $\Delta$ unspecified entries corresponds to an axis-parallel affine subspace of dimension at most $\Delta$, called a $\Delta$-point. Thus we seek a partition of $n$ input $\Delta$-points into $k$ clusters minimizing the $k$-means objective. For $\Delta=0$, when all coordinates of each point are specified, this is the usual $k$-means clustering. We give an algorithm that finds an $(1+ \epsilon)$-approximate solution in time $f(k,\epsilon, \Delta) \cdot n² \cdot d$ for some function $f$ of $k,\epsilon$, and $\Delta$ only.