论文标题
大偏差，中度偏差和KLS猜想
Large deviations, moderate deviations, and the KLS conjecture
论文作者
论文摘要
Kannan-Lovász-Simonovits（KLS）的猜想起源于理论计算机科学，是当今渐近凸几何形状和高维概率理论的主要开放问题之一。在这项工作中，我们建立了这种猜想与对各向同性对数凸线随机向量的大偏差的研究之间的新联系，从而提供了解决猜想的新颖可能性。然后，我们研究了$ \ ell_p^n $ -ball中随机正交的随机向量的欧几里得规范的中等偏差。这导致了许多有趣的观察：（a）$ \ ell_1^n $ - ball对于新方法至关重要； （b）对于$ p \ geq 2 $，中等偏差原理中的速率函数会经历相变，具体取决于缩放率是低于子空间维度的平方根还是可比较； （c）$ 1 \ leq p <2 $和可比的子空间维度，速率函数再次显示相变的相对于$ n^{p/2} $的增长。
Having its origin in theoretical computer science, the Kannan-Lovász-Simonovits (KLS) conjecture is one of the major open problems in asymptotic convex geometry and high-dimensional probability theory today. In this work, we establish a new connection between this conjecture and the study of large and moderate deviations for isotropic log-concave random vectors, thereby providing a novel possibility to tackle the conjecture. We then study the moderate deviations for the Euclidean norm of random orthogonally projected random vectors in an $\ell_p^n$-ball. This leads to a number of interesting observations: (A) the $\ell_1^n$-ball is critical for the new approach; (B) for $p\geq 2$ the rate function in the moderate deviations principle undergoes a phase transition, depending on whether the scaling is below the square-root of the subspace dimensions or comparable; (C) for $1\leq p<2$ and comparable subspace dimensions, the rate function again displays a phase transition depending on its growth relative to $n^{p/2}$.
