Relative errors for deterministic low-rank matrix approximations

We consider processing an n x 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 x 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 [EQUATION] We show that if one sets ℓ = ⌈k + k/ε⌉ and returns Qk, a k x d matrix that is simply the top k rows of Q, then we achieve the following properties: [EQUATION] and where πQk (A) is the projection of A onto the rowspace of Qk then [EQUATION] 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.