We develop a novel connection between discrepancy minimization and (quantum) communication complexity. As an application, we resolve a substantial special case of the Matrix Spencer conjecture. In particular, we show that for every collection of symmetric $n \times n$ matrices $A_1,\ldots,A_n$ with $\|A_i\| \leq 1$ and $\|A_i\|_F \leq n^{1/4}$ there exist signs $x \in \{ \pm 1\}^n$ such that the maximum eigenvalue of $\sum_{i \leq n} x_i A_i$ is at most $O(\sqrt n)$. We give a polynomial-time algorithm based on partial coloring and semidefinite programming to find such $x$. Our techniques open a new avenue to use tools from communication complexity and information theory to study discrepancy. The proof of our main result combines a simple compression scheme for transcripts of repeated (quantum) communication protocols with quantum state purification, the Holevo bound from quantum information, and tools from sketching and dimensionality reduction. Our approach also offers a promising avenue to resolve the Matrix Spencer conjecture completely -- we show it is implied by a natural conjecture in quantum communication complexity.
翻译:我们开发了差异最小化和(量)通信复杂性之间的新联系。 作为一种应用, 我们解决了MITSpencer预测中一个非常特殊的例子。 特别是, 我们展示了每套对称美元和美元总基数( $A_ 1,\ldots, A_ n$, A_ 美元与$A_ i ⁇ \\ leq 1美元和 $A_ iq\ i1/4美元 美元之间的新联系。 我们的技术为使用通信复杂程度和信息理论的工具来研究差异开辟了新的途径。 我们的主要结果证据是, 一个简单的压缩计划, 用于重复( Quantum) 通信协议的记录, 与量子状态净化, 最大A_ i A 美元最高为O(\\ sqrt n) 美元。 我们给出了基于部分颜色和半确定值程序来找到这种美元。 我们给出的多元时间算法算算算法, 我们从一个有希望的直观的直观的直观的直观信息, 和直观的直观的直观的直观矩阵工具, 展示了我们从直观的直观的直观的直观的直观的直观路径。