Deterministic coreset for Lp subspace

We introduce the first iterative algorithm for constructing a ε-coreset that guarantees deterministic ℓp subspace embedding for any p ∈ [1, ∞) and any ε > 0. For a given full rank matrix X ∈ R n×d where n ≫ d, X′ ∈ R m×d is an (ε, ℓp)-subspace embedding of X, if for every q ∈ R d , (1−ε)∥Xq∥ p p ≤ ∥X′q∥ p p ≤ (1+ε)∥Xq∥ p p . Specifically, in this paper, X′ is a weighted subset of rows of X which is commonly known in the literature as a coreset. In every iteration, the algorithm ensures that the loss on the maintained set is upper and lower bounded by the loss on the original dataset with appropriate scalings. So, unlike typical coreset guarantees, due to bounded loss, our coreset gives a deterministic guarantee for the ℓp subspace embedding. For an error parameter ε, our algorithm takes O(poly(n, d, ε−1 )) time and returns a deterministic ε-coreset, for ℓp subspace embedding whose size is O d max{1,p/2} ε 2 . Here, we remove the log factors in the coreset size, which had been a long-standing open problem [6]. Our coresets are optimal as they are tight with the lower bound. As an application, our coreset can also be used for approximately solving the ℓp regression problem in a deterministic manner.