High-Dimensional Geometric Streaming for Nearly Low Rank Data

We study streaming algorithms for the $\ellp$ subspace approximation problem. Given points $a1, \ldots, an$ as an insertion-only stream and a rank parameter $k$, the $\ellp$ subspace approximation problem is to find a $k$-dimensional subspace $V$ such that $(\sum{i=1}ⁿ d(ai, V)^p){1/p}$ is minimized, where $d(a, V)$ denotes the Euclidean distance between $a$ and $V$ defined as $\min{v \in V}|{a - v}|{\infty}$. When $p = \infty$, we need to find a subspace $V$ that minimizes $\maxi d(ai, V)$. For $\ell{\infty}$ subspace approximation, we give a deterministic strong coreset construction algorithm and show that it can be used to compute a $\text{poly}(k, \log n)$ approximate solution. We show that the distortion obtained by our coreset is nearly tight for any sublinear space algorithm. For $\ellp$ subspace approximation, we show that suitably scaling the points and then using our $\ell_{\infty}$ coreset construction, we can compute a $\text{poly}(k, \log n)$ approximation. Our algorithms are easy to implement and run very fast on large datasets. We also use our strong coreset construction to improve the results in a recent work of Woodruff and Yasuda (FOCS 2022) which gives streaming algorithms for high-dimensional geometric problems such as width estimation, convex hull estimation, and volume estimation.