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.
The sum-of-squares complexity of linear-algebraic principles
Dmitriy Kunisky, Xifan Yu
AI 速览：本文研究了实线性代数中若干基本原理的平方和（SoS）证明所需度数，给出了弱秩原理等问题的精确度数下界与上界。
We study the degree needed for sum-of-squares (SoS) proofs of several basic principles of real linear algebra. For any $n\times n$ matrix $A$ of rank greater than $r$, the Weak Rank Principle states that the equations $XY^\top=A$ with $n\times r$ matrix variables $X, Y$ have no solution. We show that this principle requires degree exactly $2r+2$ to prove. On the other hand, for $A=I_n$, setting $X = Y$ gives the Tight Frame Principle, which has a degree-$4$ proof. For $n\times n$ matrices, proving that a right inverse is also a left inverse requires degree exactly $2n+2$, while proving that orthonormal rows imply orthonormal columns again has a degree-4 proof. Using our lower bounds, we also derive related SoS degree lower bounds for polynomial systems encoding sign rank, matrix rigidity, small-width homogeneous non-commutative algebraic branching programs, and tensor CP rank. Lastly, we show that quantitative bounds on the Frobenius norms of the factors in the Weak Rank Principle can lower the degree required to refute $XY^\top=I_n$, but sufficiently large bounds still preserve the exact degree $2r+2$ from the unbounded case, giving a quantitative version of the Archimedean property arising in classical results on SoS proofs.