High-Dimensional Geometric Streaming for Nearly Low Rank Data

TL;DR

This paper introduces efficient streaming algorithms with strong coresets for high-dimensional $\, ext{l}_p$ subspace approximation, enabling near-optimal solutions for low-rank data in large datasets.

Contribution

It presents a deterministic coreset construction for $\, ext{l}_ ext{infinity}$ subspace approximation and extends it to general $\, ext{l}_p$ cases, improving streaming algorithms for high-dimensional geometric problems.

Findings

Deterministic coreset for $\, ext{l}_ ext{infinity}$ subspace approximation with near-tight distortion.

Poly$(k, \, ext{log}\, n)$ approximation algorithms for $\, ext{l}_p$ subspace approximation.

Enhanced streaming algorithms for geometric problems like width, convex hull, and volume estimation.

Abstract

We study streaming algorithms for the $\ell_p$ subspace approximation problem. Given points $a_1, \ldots, a_n$ as an insertion-only stream and a rank parameter $k$, the $\ell_p$ subspace approximation problem is to find a $k$-dimensional subspace $V$ such that $(\sum_{i=1}^n d(a_i, V)^p)^{1/p}$ 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 $\max_i d(a_i, V)$. 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 $\ell_p$ 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.