Fast, Space-Optimal Streaming Algorithms for Clustering and Subspace Embeddings

Abstract: We show that both clustering and subspace embeddings can be performed in the streaming model with the same asymptotic efficiency as in the central/offline setting. For $(k, z)$-clustering in the streaming model, we achieve a number of words of memory which is independent of the number $n$ of input points and the aspect ratio $\Delta$, yielding an optimal bound of $\tilde{\mathcal{O}}\left(\frac{dk}{\min(\varepsilon^4,\varepsilon^{z+2})}\right)$ words for accuracy parameter $\varepsilon$ on $d$-dimensional points. Additionally, we obtain amortized update time of $d\,\log(k)\cdot\text{polylog}(\log(n\Delta))$, which is an exponential improvement over the previous $d\,\text{poly}(k,\log(n\Delta))$. Our method also gives the fastest runtime for $(k,z)$-clustering even in the offline setting. For subspace embeddings in the streaming model, we achieve $\mathcal{O}(d)$ update time and space-optimal constructions, using $\tilde{\mathcal{O}}\left(\frac{d^2}{\varepsilon^2}\right)$ words for $p\le 2$ and $\tilde{\mathcal{O}}\left(\frac{d^{p/2+1}}{\varepsilon^2}\right)$ words for $p>2$, showing that streaming algorithms can match offline algorithms in both space and time complexity.