Lower Bounds for Sparse Oblivious Subspace Embeddings

An oblivious subspace embedding (OSE), characterized by parameters m,n, d, , δ, is a random matrix Π ∈ Rm×n such that for any d-dimensional subspace T ⊆ R, PrΠ[∀x ∈ T, (1 − )‖x‖2 ≤ ‖Πx‖2 ≤ (1 + )‖x‖2] ≥ 1− δ. For and δ at most a small constant, we show that any OSE with one nonzero entry in each column must satisfy that m = Ω(d/( δ)), establishing the optimality of the classical Count-Sketch matrix. When an OSE has 1/(9 ) nonzero entries in each column, we show it must hold that m = Ω( d), improving on the previous Ω( d) lower bound due to Nelson and Nguyẽ̂n (ICALP 2014).