The sparsification theorem of Batson, Spielman and Srivastava
Let (Ω, μ) be a measure space and a = (a_k) a family of m square-integrable functions whose Gram matrix ∫ a a* dμ is the identity. Then for every n ≥ m there are points x₁, …, xₙ ∈ Ω and weights w₁, …, wₙ > 0 with (1 − √((m−1)/n))² I ≼ ∑ᵢ wᵢ a(xᵢ) a(xᵢ)* ≼ (1 + √((m−1)/n))² I in the Loewner order. For the counting measure on a finite set Ω this is the theorem for finitely many vectors: if v_k ∈ ℂᵐ, k ∈ Ω, satisfy ∑_k v_k v_k* = I, then for every n ≥ m there are weights s_k ≥ 0, at most n of them nonzero, with (1 − √((m−1)/n))² I ≼ ∑_k s_k v_k v_k* ≼ (1 + √((m−1)/n))² I, so a reweighted…
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