Hilbert Space Embeddings and Metrics on Probability Measures
Hilbert Space Embeddings and Metrics on Probability Measures
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DOI：
10.5555/1756006.1859901
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发表时间：
2009-07
期刊：
影响因子：
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通讯作者：
Bharath K. Sriperumbudur;A. Gretton;K. Fukumizu;B. Scholkopf;Gert R. G. Lanckriet
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文献类型：
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作者：
Bharath K. Sriperumbudur;A. Gretton;K. Fukumizu;B. Scholkopf;Gert R. G. Lanckriet
最近提出了一种用于概率测度的Hilbert空间嵌入，其应用包括降维、同质性检验和独立性检验。这种嵌入表示任何概率测度作为再生核希尔伯特空间（RKHS）中的平均元素。概率测度空间上的伪度量可以定义为分布嵌入之间的距离：我们将其表示为γk，由定义RKHS中内积的核函数k索引。我们给出了γk的三个理论性质。首先，我们考虑确定γk是度量的核k上的条件的问题：这样的k被表示为特征核。与伪度量不同，只有当两个分布重合时度量才为零，从而确保RKHS嵌入唯一地映射所有分布（即，嵌入是单射的）。虽然以前公布的条件可能只适用于有限的情况下（例如，在紧域上），并且很难检查，我们的条件是直接和直观的：积分严格正定核是特征。另一方面，如果一个有界连续核是在全离散上的离散不变的，那么它是特征的当且仅当它的傅里叶变换的支集是全离散。其次，我们证明了γk下的分布之间的距离是由核的性质和分布之间的相互作用引起的，通过证明当它们的差异出现在更高的频率时，分布在嵌入空间中是接近的。第三，为了理解由γk诱导的拓扑的性质，我们将γk与概率测度上的其他流行度量联系起来，并给出γk度量弱拓扑的核k上的条件。
A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings: we denote this as γk, indexed by the kernel function k that defines the inner product in the RKHS. We present three theoretical properties of γk. First, we consider the question of determining the conditions on the kernel k for which γk is a metric: such k are denoted characteristic kernels. Unlike pseudometrics, a metric is zero only when two distributions coincide, thus ensuring the RKHS embedding maps all distributions uniquely (i.e., the embedding is injective). While previously published conditions may apply only in restricted circumstances (e.g., on compact domains), and are difficult to check, our conditions are straightforward and intuitive: integrally strictly positive definite kernels are characteristic. Alternatively, if a bounded continuous kernel is translation-invariant on ℜd, then it is characteristic if and only if the support of its Fourier transform is the entire ℜd. Second, we show that the distance between distributions under γk results from an interplay between the properties of the kernel and the distributions, by demonstrating that distributions are close in the embedding space when their differences occur at higher frequencies. Third, to understand the nature of the topology induced by γk, we relate γk to other popular metrics on probability measures, and present conditions on the kernel k under which γk metrizes the weak topology.