链式法、插值与凸性
Chaining, Interpolation, and Convexity
AI总结：
本文证明在凸集条件下，利用实插值作为自然链式机制，仅需计算“薄”子集的熵数即可系统性改进随机过程上确界的经典链式界，且在经典熵界非最优时仍保持精确。
摘要：
我们证明了，当底层集合为凸集时，用熵数表示随机过程上确界的经典链式界可以被系统性地改进：无需对整个集合计算熵数，只需对某些“薄”子集计算即可。这一现象源于如下观察：实插值可作为一种自然的链式机制。与 Talagrand 的通用链式方法的一般形式（该方法精确但往往难以使用）不同，所得的界仅涉及熵数，但在经典熵界非最优的许多情形下依然是精确的。这些界在具体例子中易于进行显式计算，并且我们发现了一些用于控制链式泛函的旧有与全新几何原理作为其特例。
英文摘要：
We show that classical chaining bounds on the suprema of random processes in terms of entropy numbers can be systematically improved when the underlying set is convex: the entropy numbers need not be computed for the entire set, but only for certain "thin" subsets. This phenomenon arises from the observation that real interpolation can be used as a natural chaining mechanism. Unlike the general form of Talagrand's generic chaining method, which is sharp but often difficult to use, the resulting bounds involve only entropy numbers but are nonetheless sharp in many situations in which classical entropy bounds are suboptimal. Such bounds are readily amenable to explicit computations in specific examples, and we discover some old and new geometric principles for the control of chaining functionals as special cases.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
