Kernel Two-Sample Tests for Manifold Data
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arXiv
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| Format: | Preprint |
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2021
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| author | Cheng, Xiuyuan Xie, Yao |
| author_facet | Cheng, Xiuyuan Xie, Yao |
| contents | We present a study of a kernel-based two-sample test statistic related to the Maximum Mean Discrepancy (MMD) in the manifold data setting, assuming that high-dimensional observations are close to a low-dimensional manifold. We characterize the test level and power in relation to the kernel bandwidth, the number of samples, and the intrinsic dimensionality of the manifold. Specifically, when data densities $p$ and $q$ are supported on a $d$-dimensional sub-manifold ${M}$ embedded in an $m$-dimensional space and are Hölder with order $β$ (up to 2) on ${M}$, we prove a guarantee of the test power for finite sample size $n$ that exceeds a threshold depending on $d$, $β$, and $Δ_2$ the squared $L^2$-divergence between $p$ and $q$ on the manifold, and with a properly chosen kernel bandwidth $γ$. For small density departures, we show that with large $n$ they can be detected by the kernel test when $Δ_2$ is greater than $n^{- { 2 β/( d + 4 β) }}$ up to a certain constant and $γ$ scales as $n^{-1/(d+4β)}$. The analysis extends to cases where the manifold has a boundary and the data samples contain high-dimensional additive noise. Our results indicate that the kernel two-sample test has no curse-of-dimensionality when the data lie on or near a low-dimensional manifold. We validate our theory and the properties of the kernel test for manifold data through a series of numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_03425 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Kernel Two-Sample Tests for Manifold Data Cheng, Xiuyuan Xie, Yao Machine Learning Statistics Theory We present a study of a kernel-based two-sample test statistic related to the Maximum Mean Discrepancy (MMD) in the manifold data setting, assuming that high-dimensional observations are close to a low-dimensional manifold. We characterize the test level and power in relation to the kernel bandwidth, the number of samples, and the intrinsic dimensionality of the manifold. Specifically, when data densities $p$ and $q$ are supported on a $d$-dimensional sub-manifold ${M}$ embedded in an $m$-dimensional space and are Hölder with order $β$ (up to 2) on ${M}$, we prove a guarantee of the test power for finite sample size $n$ that exceeds a threshold depending on $d$, $β$, and $Δ_2$ the squared $L^2$-divergence between $p$ and $q$ on the manifold, and with a properly chosen kernel bandwidth $γ$. For small density departures, we show that with large $n$ they can be detected by the kernel test when $Δ_2$ is greater than $n^{- { 2 β/( d + 4 β) }}$ up to a certain constant and $γ$ scales as $n^{-1/(d+4β)}$. The analysis extends to cases where the manifold has a boundary and the data samples contain high-dimensional additive noise. Our results indicate that the kernel two-sample test has no curse-of-dimensionality when the data lie on or near a low-dimensional manifold. We validate our theory and the properties of the kernel test for manifold data through a series of numerical experiments. |
| title | Kernel Two-Sample Tests for Manifold Data |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2105.03425 |