Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank
Authors: 
Nikhil Bansal, Haotian Jiang, and Raghu MekaAuthors Info & Claims
Nikhil Bansal, Haotian Jiang, and Raghu MekaAuthors Info & ClaimsJune 2023
Pages 1814 - 1819
Abstract
We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric d × d matrices A1,…,An each with ||Ai||op ≤ 1 and rank at most n/log3 n, one can efficiently find ± 1 signs x1,…,xn such that their signed sum has spectral norm ||∑i=1n xi Ai||op = O(√n). This result also implies a logn − Ω( loglogn) 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, 2021] for random matrices with correlated Gaussian entries.
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Index Terms
- Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank
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Published: 02 June 2023
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