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 ℓ_2. 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 = ∞, and (3) can be implemented in both distributed and streaming environments. We apply our results to ℓ_p-regression, entrywise ℓ_1-low rank approximation, and approximate matrix multiplication.
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