论文标题
不同维度的概率分布之间的距离
Distances between probability distributions of different dimensions
论文作者
论文摘要
比较概率分布是机器学习和统计数据中必不可少且无处不在的任务。比较一对Borel概率度量的最常见方法是计算它们之间的度量，而到目前为止，使用最广泛的度量概念是Wasserstein Metric和总变异度量。下一个最常见的方法是计算它们之间的差异，在这种情况下，几乎所有已知的差异（例如Kullback） - Leibler，Jensen-Shannon，Rényi等，都是$ f $ divergence的特殊情况。然而，实际上，只有在相同维度的空间上，这些指标和差异才能实际上才能定义。例如，如何量化间隔$ [-1,1] $的统一分布与$ \ Mathbb {r}^3 $上的高斯分布之间的kl差异？我们表明，这些常见的指标和分歧概念在不同维度的空间上定义的Borel概率度量之间的自然距离，例如，一个on $ \ mathbb {r}^m $，另一个在$ \ mathbb {r}^n $ where $ m，n $ n $ n $ n $ where $ \ mathbb {r}^n $上是不同的，以便给出一个有意义的答案。
Comparing probability distributions is an indispensable and ubiquitous task in machine learning and statistics. The most common way to compare a pair of Borel probability measures is to compute a metric between them, and by far the most widely used notions of metric are the Wasserstein metric and the total variation metric. The next most common way is to compute a divergence between them, and in this case almost every known divergences such as those of Kullback--Leibler, Jensen--Shannon, Rényi, and many more, are special cases of the $f$-divergence. Nevertheless these metrics and divergences may only be computed, in fact, are only defined, when the pair of probability measures are on spaces of the same dimension. How would one quantify, say, a KL-divergence between the uniform distribution on the interval $[-1,1]$ and a Gaussian distribution on $\mathbb{R}^3$? We show that these common notions of metrics and divergences give rise to natural distances between Borel probability measures defined on spaces of different dimensions, e.g., one on $\mathbb{R}^m$ and another on $\mathbb{R}^n$ where $m, n$ are distinct, so as to give a meaningful answer to the previous question.
