随机布尔张量幂在维度阈值处的线性独立性
Linear Independence of Random Boolean Tensor Powers at the Dimension Threshold
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- University of Colorado（科罗拉多大学）
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中文总结 AI 辅助
该论文证明在维度阈值 $D(n,d)$ 下，随机布尔张量幂以高概率线性独立，误差为 $O_d(\log^{C_d} n / n^{1/2})$，解决了 Baldi 和 Vershynin 的开放问题。
中文摘要 AI 辅助
设 $d \geq 1$ 固定，令 $D(n,d):= \sum_{j=0}^{d} \binom{n-1}{j}$。我们证明，若 $x^{(1)}, \dots, x^{(m)}$ 是 $\{\pm 1\}^n$ 中的独立均匀点，则对 $m \leq D(n,d)$ 一致地，存在常数 $C_d > 0$，使得 $\mathbb{P}((x^{(1)})^{\otimes d}, \dots, (x^{(m)})^{\otimes d}$ 线性独立$)= 1 - O_d\left(\frac{\log^{C_d} n}{n^{1/2}} \right)$。这达到了精确的维度阈值，并回答了 Baldi 和 Vershynin 提出的问题。
英文摘要
Let $d \geq 1$ be fixed and let \[ D(n,d) := \sum_{j=0}^{d} \binom{n-1}{j}. \] We show that if $x^{(1)}, \dots, x^{(m)}$ are independent uniform points of $\{\pm 1\}^n$ then uniformly for $m \leq D(n,d)$, there exists a constant $C_d > 0$ such that \[ \mathbb{P}((x^{(1)})^{\otimes d}, \dots, (x^{(m)})^{\otimes d} \text{ are linearly independent}) = 1 - O_d\left(\frac{\log^{C_d} n}{n^{1/2}} \right). \] This achieves the exact dimensional threshold and answers a question asked by Baldi and Vershynin. We discuss applications of the result to the semidefinite relaxation of the cut-polytope and to matrix factorization.
