熵与组合维数
Entropy and the Combinatorial Dimension
摘要：
我们解决了Talagrand的熵问题：每个一致有界函数类的L_2-覆盖数在其破碎维数上是指数的.这扩展了Dudley定理关于{0，1}-值函数类，其中破碎维数是Vapnik-Chervonenkis维数。在凸几何中，该解意味着凸体K的熵由K的坐标投影中包含的固定边的立方体的最大维数控制。这有一些后果，包括最佳埃尔顿定理和估计的一致中心极限定理在实值的情况下。
英文摘要：
We solve Talagrand's entropy problem: the L_2-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley's theorem on classes of {0,1}-valued functions, for which the shattering dimension is the Vapnik-Chervonenkis dimension. In convex geometry, the solution means that the entropy of a convex body K is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of K. This has a number of consequences, including the optimal Elton's Theorem and estimates on the uniform central limit theorem in the real valued case.
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