# Sketching Matrix Least Squares via Leverage Scores Estimates

We consider the matrix least squares problem of the form ππ-π_F^2 where the design matrix πββ^N Γ r is tall and skinny with N β« r. We propose to create a sketched version πΜπ-πΜ_F^2 where the sketched matrices πΜ and πΜ contain weighted subsets of the rows of π and π, respectively. The subset of rows is determined via random sampling based on leverage score estimates for each row. We say that the sketched problem is Ο΅-accurate if its solution πΜ_opt = argmin πΜπ-πΜ_F^2 satisfies ππΜ_opt-π_F^2 β€ (1+Ο΅) minππ-π_F^2 with high probability. We prove that the number of samples required for an Ο΅-accurate solution is O(r/(Ξ²Ο΅)) where Ξ²β (0,1] is a measure of the quality of the leverage score estimates.

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