We show that the guaranties of the Gram-Schmidt Walk Algorithm of Bansal, Dadush, Garg, and Lovett can be achieved by an Ising model. Given vectors $v_1, \ldots, v_n$ of norm less than 1, we give a closed form for a centrally symmetric distribution on a vector $σ\in \{\pm 1 \}^n$ so that the signed sum $\sum_j σ(j) v_j$ is $1$-subgaussian. We show that the Glauber dynamics for this model mix rapidly provided that the norms of the input vectors are bounded below 1 by an inverse-polynomial. The proofs employ the theory of Lorentzian polynomials, and we use these techniques to show fast mixing bounds for other spin systems.
Large Growth Happens: Gaussian Elimination with Partial Pivoting on Random Matrices
Daniel A. Spielman, Xifan Yu
AI 速览：作者证明对高斯随机矩阵做带部分主元的高斯消元时，出现大幅增长的概率至少是逆拟多项式级别，从而挑战了平滑分析与平均情形分析中的标准猜想。
We prove that the probability that Gaussian elimination with partial pivoting on $n \times n$ Gaussian random matrices has growth $ρ$ is at least inverse quasi-polynomial: $Ω\left(\exp\left(-c \log^2 \left(\fracρ{\sqrt{n}}\right)\log(n)\right)\right)$ for some constant $c > 0$. In particular, the probability of growth $ρ\ge n^{1/2 + ε}$ is at least $\exp(- O (ε^2 \log^3 (n) ))$. This lower bound breaks standard conjectures in the smoothed and average-case analysis of Gaussian elimination. To the best of our knowledge, it is the first non-trivial lower bound on the probability of large growth for Gaussian random matrices.