Relative Errors for Deterministic Low-Rank Matrix Approximations

Mina Ghashami, Jeff M. Phillips

Abstract: We consider processing an n × d matrix A in a stream with row-wise updates according to a recent algorithm called Frequent Directions (Liberty, KDD 2013). This algorithm maintains an ℓ × d matrix Q deterministically, processing each row in O(dℓ2) time; the processing time can be decreased to O(dℓ) with a slight modification in the algorithm and a constant increase in space. Then for any unit vector x, the matrix Q satisfies We show that if one sets ℓ = ⌈k + k/∊⌉ and returns Qk, a k × d matrix that is simply the top k rows of Q, then we achieve the following properties: and where πQk is the projection of A onto the rowspace of Qk then We also show that Frequent Directions cannot be adapted to a sparse version in an obvious way that retains ℓ original rows of the matrix, as opposed to a linear combination or sketch of the rows.