比较分布和使用核距离进行排序
Comparing Distributions and Shapes using the Kernel Distance
摘要：
从对象之间的相似性函数开始，可以定义对象对的距离度量，更一般地说，定义它们的概率分布。这些距离度量在泛函分析、测度理论和几何测度理论中有很深的基础，并且具有丰富的结构，包括等距嵌入到（可能是无限维的）希尔伯特空间中。它们最近被应用于机器学习和形状分析中的许多问题。在本文中，我们提供了这些距离度量的第一个算法分析。我们的主要贡献如下：（i）我们提出了快速的近似算法计算两个点集P和Q之间的核距离，运行在近线性时间的大小（P杯Q）（注意，显式计算将采取二次时间）。(ii)我们提出了多项式时间算法近似最小化的内核距离刚性变换下，他们运行时间O（n +聚（1/n，log n））。(iii)我们提供了几种通用的技术，用于减少复杂的对象，以方便的稀疏表示（特别是点集或点集的集合），近似保持内核距离。特别是，这使我们能够减少计算各种类型的对象（如曲线，曲面和分布）之间的核距离的问题，以计算点集之间的核距离。这些利用了再生核希尔伯特空间和一个新的关系连接二进制范围空间连续范围空间有界脂肪粉碎尺寸。
英文摘要：
Starting with a similarity function between objects, it is possible to define a distance metric on pairs of objects, and more generally on probability distributions over them. These distance metrics have a deep basis in functional analysis, measure theory and geometric measure theory, and have a rich structure that includes an isometric embedding into a (possibly infinite dimensional) Hilbert space. They have recently been applied to numerous problems in machine learning and shape analysis. In this paper, we provide the first algorithmic analysis of these distance metrics. Our main contributions are as follows: (i) We present fast approximation algorithms for computing the kernel distance between two point sets P and Q that runs in near-linear time in the size of (P cup Q) (note that an explicit calculation would take quadratic time). (ii) We present polynomial-time algorithms for approximately minimizing the kernel distance under rigid transformation; they run in time O(n + poly(1/epsilon, log n)). (iii) We provide several general techniques for reducing complex objects to convenient sparse representations (specifically to point sets or sets of points sets) which approximately preserve the kernel distance. In particular, this allows us to reduce problems of computing the kernel distance between various types of objects such as curves, surfaces, and distributions to computing the kernel distance between point sets. These take advantage of the reproducing kernel Hilbert space and a new relation linking binary range spaces to continuous range spaces with bounded fat-shattering dimension.
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