Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank

08/24/2022

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by   Nikhil Bansal, et al.

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We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric d × d matrices A_1,…,A_n each with A_i_𝗈𝗉≤ 1 and rank at most n/log^3 n, one can efficiently find ± 1 signs x_1,…,x_n such that their signed sum has spectral norm ∑_i=1^n x_i A_i_𝗈𝗉 = O(√(n)). This result also implies a log n - Ω( loglog n) qubit lower bound for quantum random access codes encoding n classical bits with advantage ≫ 1/√(n). Our proof uses the recent refinement of the non-commutative Khintchine inequality in [Bandeira, Boedihardjo, van Handel, 2022] for random matrices with correlated Gaussian entries.