Rate of convergence of the smoothed empirical Wasserstein distance
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| Format: | Preprint |
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2022
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| _version_ | 1866909484233785344 |
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| author | Block, Adam Jia, Zeyu Polyanskiy, Yury Rakhlin, Alexander |
| author_facet | Block, Adam Jia, Zeyu Polyanskiy, Yury Rakhlin, Alexander |
| contents | Consider an empirical measure $\mathbb{P}_n$ induced by $n$ iid samples from a $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$ and let $γ= N(0,σ^2 I_d)$ be the isotropic Gaussian measure. We study the speed of convergence of the smoothed Wasserstein distance $W_2(\mathbb{P}_n * γ, \mathbb{P}*γ) = n^{-α+ o(1)}$ with $*$ being the convolution of measures. For $K<σ$ and in any dimension $d\ge 1$ we show that $α= {1\over2}$. For $K>σ$ in dimension $d=1$ we show that the rate is slower and is given by $α= {(σ^2 + K^2)^2\over 4 (σ^4 + K^4)} < 1/2$. This resolves several open problems in [GGNWP20], and in particular precisely identifies the amount of smoothing $σ$ needed to obtain a parametric rate. In addition, for any $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$, we also establish that $D_{KL}(\mathbb{P}_n * γ\|\mathbb{P}*γ)$ has rate $O(1/n)$ for $K<σ$ but only slows down to $O({(\log n)^{d+1}\over n})$ for $K>σ$. The surprising difference of the behavior of $W_2^2$ and KL implies the failure of $T_{2}$-transportation inequality when $σ< K$. Consequently, it follows that for $K>σ$ the log-Sobolev inequality (LSI) for the Gaussian mixture $\mathbb{P} * N(0, σ^{2})$ cannot hold. This closes an open problem in [WW+16], who established the LSI under the condition $K<σ$ and asked if their bound can be improved. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_02128 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rate of convergence of the smoothed empirical Wasserstein distance Block, Adam Jia, Zeyu Polyanskiy, Yury Rakhlin, Alexander Probability Information Theory Statistics Theory Consider an empirical measure $\mathbb{P}_n$ induced by $n$ iid samples from a $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$ and let $γ= N(0,σ^2 I_d)$ be the isotropic Gaussian measure. We study the speed of convergence of the smoothed Wasserstein distance $W_2(\mathbb{P}_n * γ, \mathbb{P}*γ) = n^{-α+ o(1)}$ with $*$ being the convolution of measures. For $K<σ$ and in any dimension $d\ge 1$ we show that $α= {1\over2}$. For $K>σ$ in dimension $d=1$ we show that the rate is slower and is given by $α= {(σ^2 + K^2)^2\over 4 (σ^4 + K^4)} < 1/2$. This resolves several open problems in [GGNWP20], and in particular precisely identifies the amount of smoothing $σ$ needed to obtain a parametric rate. In addition, for any $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$, we also establish that $D_{KL}(\mathbb{P}_n * γ\|\mathbb{P}*γ)$ has rate $O(1/n)$ for $K<σ$ but only slows down to $O({(\log n)^{d+1}\over n})$ for $K>σ$. The surprising difference of the behavior of $W_2^2$ and KL implies the failure of $T_{2}$-transportation inequality when $σ< K$. Consequently, it follows that for $K>σ$ the log-Sobolev inequality (LSI) for the Gaussian mixture $\mathbb{P} * N(0, σ^{2})$ cannot hold. This closes an open problem in [WW+16], who established the LSI under the condition $K<σ$ and asked if their bound can be improved. |
| title | Rate of convergence of the smoothed empirical Wasserstein distance |
| topic | Probability Information Theory Statistics Theory |
| url | https://arxiv.org/abs/2205.02128 |