Leveraging Well-Conditioned Bases: Streaming \& Distributed Summaries in Minkowski $p$-Norms

Work on approximate linear algebra has led to efficient distributed and streaming algorithms for problems such as approximate matrix multiplication, low rank approximation, and regression, primarily for the Euclidean norm $\ell_2$. We study other $\ell_p$ norms, which are more robust for $p < 2$, and can be used to find outliers for $p > 2$. Unlike previous algorithms for such norms, we give algorithms that are (1) deterministic, (2) work simultaneously for every $p \geq 1$, including $p = \infty$, and (3) can be implemented in both distributed and streaming environments. We apply our results to $\ell_p$-regression, entrywise $\ell_1$-low rank approximation, and approximate matrix multiplication.