论文标题
解决矩阵Spencer的猜想最多可达多层级等级
Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank
论文作者
论文摘要
We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric $d \times d$ matrices $A_1,\ldots,A_n$ each with $\|A_i\|_{\mathsf{op}} \leq 1$ and rank at most $n/\log^3 n$, one can efficiently find $\pm 1$符号$ x_1，\ ldots，x_n $，使其签名和具有频谱norm $ \ | \ sum_ {i = 1}^n x_i a_i a_i \ | _ {\ MathSf {op}}} = o（\ sqrt {n}）$。此结果还意味着一个$ \ log n -ω（\ log \ log n）$ qubit lows bong of量子随机访问代码编码$ n $具有优势$ \ gg 1/\ sqrt {n} $的经典位。 我们的证明使用[Bandeira，Boedihardjo，Van Handel，2022]中的非交换性Khintchine不平等的改进，用于具有相关高斯条目的随机矩阵。
We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric $d \times d$ matrices $A_1,\ldots,A_n$ each with $\|A_i\|_{\mathsf{op}} \leq 1$ and rank at most $n/\log^3 n$, one can efficiently find $\pm 1$ signs $x_1,\ldots,x_n$ such that their signed sum has spectral norm $\|\sum_{i=1}^n x_i A_i\|_{\mathsf{op}} = O(\sqrt{n})$. This result also implies a $\log n - Ω( \log \log n)$ qubit lower bound for quantum random access codes encoding $n$ classical bits with advantage $\gg 1/\sqrt{n}$. 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.
