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 lp 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 lp-regression, entrywise l1-low rank approximation, and approximate matrix multiplication.
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