基于凸性的等视野极限：采样调和-凹函数
The Limit of Convexity Based Isoperimetry: Sampling Harmonic-Concave Functions
摘要：
对数凹函数代表了R ^n中采样、优化和积分的有效算法的当前前沿。根据概率密度（其他两个问题可以减少）进行采样的有效采样算法依赖于良好的等周性，已知其适用于任意对数凹密度。在本文中，我们以两种方式扩展了这一前沿：首先，我们刻画了暗示良好等周性的类凸条件，即，沿每条线的函数值满足什么条件才能保证良好的等周性？答案是R ^n中的（1/（n-1））-调和凹函数的集合;我们还证明了这是具有良好等周性的函数沿着每条线的最佳可能的刻画。接下来，我们给出了第一个有效的算法，根据这样的函数的复杂性取决于一个光滑参数的采样。此外，注意到多变量柯西密度是这一类中的一个重要分布，我们利用柯西密度的某些性质，给出一个有效的采样算法的基础上随机游走的混合时间，匹配当前已知的最佳边界采样logconcave函数。
英文摘要：
Logconcave functions represent the current frontier of efficient algorithms for sampling, optimization and integration in R^n. Efficient sampling algorithms to sample according to a probability density (to which the other two problems can be reduced) relies on good isoperimetry which is known to hold for arbitrary logconcave densities. In this paper, we extend this frontier in two ways: first, we characterize convexity-like conditions that imply good isoperimetry, i.e., what condition on function values along every line guarantees good isoperimetry? The answer turns out to be the set of (1/(n-1))-harmonic concave functions in R^n; we also prove that this is the best possible characterization along every line, of functions having good isoperimetry. Next, we give the first efficient algorithm for sampling according to such functions with complexity depending on a smoothness parameter. Further, noting that the multivariate Cauchy density is an important distribution in this class, we exploit certain properties of the Cauchy density to give an efficient sampling algorithm based on random walks with a mixing time that matches the current best bounds known for sampling logconcave functions.
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