On Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems
Dennis Cheung, Felipe Cucker, Raphael Hauser
TL;DR
The paper analyzes the distribution tails and moments of the condition number $C(A)$ for Gaussian random matrices defining homogeneous linear conic systems. Employing a geometric approach on the sphere via extremal circular caps, it derives exact tail decay rates, refines moment bounds for $\log C(A)$, and proves limit theorems across regimes $n\gg m$ and $m\gg 1$. Key findings include exponential tails for $\log C(A)$ with rate $-1$, finiteness of all moments of $\log C(A)$ for $\gamma>0$ while $\mathbb{E}[C(A)^\gamma]$ diverges for $\gamma\ge1$, and limit results showing $C(A)$ converges to 1 in several scaling limits. These results reinforce the perspective that linear programming and interior-point methods are empirically strongly polynomial on average, by quantifying the rarity of large conditioning events under Gaussian and related distributions and linking tail behavior to algorithmic performance. The work provides refined probabilistic tools and limit theorems that illuminate the practical impact of conditioning on the complexity of feasibility algorithms, with potential extensions to broader input models beyond Gaussian data.
Abstract
In this paper we study the distribution tails and the moments of a condition number which arises in the study of homogeneous systems of linear inequalities. We consider the case where this system is defined by a Gaussian random matrix and characterise the exact decay rates of the distribution tails, improve the existing moment estimates, and prove various limit theorems for large scale systems. Our results are of complexity theoretic interest, because interior-point methods and relaxation methods for the solution of systems of linear inequalities have running times that are bounded in terms of the logarithm and the square of the condition number respectively.