Hadamard流形上优化测度的几何方法
Geometric Approach For Majorizing Measures on Hadamard Manifolds
摘要：
高斯过程可以看作是标准希尔伯特空间的子集，然而随机过程的指数空间与其凸包之间的体积大小关系并不清楚.对这种体积大小关系的理解可以帮助我们从几何上建立一个优化测度定理。本文假定随机过程的基本指标空间是截面曲率小于负的单连通流形（Hadamard流形）。我们推导出了指数空间的体积与其凸包的体积之比的上界。然后，我们应用这个体积比几何证明优测度定理。
英文摘要：
Gaussian processes can be treated as subsets of a standard Hilbert space, however, the volume size relation between the underlying index space of random processes and its convex hull is not clear. The understanding of such volume size relations can help us to establish a majorizing measure theorem geometrically. In this paper, we assume that the underlying index space of random processes is a simply connected manifold with sectional curvature less than negative one (Hadamard manifold). We derive the upper bound for the ratio between the volume of the underlying index space and the volume of its convex hull. We then apply this volume ratio to prove the majorizing measure theorem geometrically.
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