Four Deviations Suffice for Rank 1 Matrices
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We prove a matrix discrepancy bound that strengthens the famous Kadison-Singer result of Marcus, Spielman, and Srivastava. Consider any independent scalar random variables ξ_1, ..., ξ_n with finite support, e.g. {± 1 } or { 0,1 }-valued random variables, or some combination thereof. Let u_1, ..., u_n ∈C^m and σ^2 = ∑_i=1^n Var[ ξ_i ] (u_i u_i^*)^2 . Then there exists a choice of outcomes ε_1,...,ε_n in the support of ξ_1, ..., ξ_n s.t. ∑_i=1^n E [ ξ_i] u_i u_i^* - ∑_i=1^n ε_i u_i u_i^* ≤ 4 σ. A simple consequence of our result is an improvement of a Lyapunov-type theorem of Akemann and Weaver.
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