Higher Lower Bounds for Sparse Oblivious Subspace Embeddings
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An oblivious subspace embedding (OSE), characterized by parameters m,n,d,ϵ,δ, is a random matrix Π∈ℝ^m× n such that for any d-dimensional subspace T⊆ℝ^n, _Π[∀ x∈ T, (1-ϵ)x_2 ≤Π x_2≤ (1+ϵ)x_2] ≥ 1-δ. When an OSE has 1/(9ϵ) nonzero entries in each column, we show it must hold that m = Ω(d^2/ϵ^1-O(δ)), which is the first lower bound with multiplicative factors of d^2 and 1/ϵ, improving on the previous Ω(ϵ^O(δ)d^2) lower bound due to Li and Liu (PODS 2022).
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