G. Cormode, C. Dickens, and D. P. Woodruff. Leveraging well-conditioned bases: Streaming and distributed summaries in minkowski p-norms. In International Conference on Machine Learning, (ICML), 2018.

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 l₂. We study other l_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 >=1, including p infinite, and (3) can be implemented in both distributed and streaming environments. We apply our results to l_p-regression, entrywise l₁-low rank approximation, and approximate matrix multiplication.

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