Two-Sample Inference for Gaussian-Smoothed Wasserstein Costs with Finite Moments
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911724155699200 |
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| author | Yang, Jiaping Zhang, Yunxin |
| author_facet | Yang, Jiaping Zhang, Yunxin |
| contents | Gaussian smoothing has emerged as an effective technique for reducing the sample complexity of optimal transport. In this paper, we study the two-sample plug-in estimator of the Gaussian-smoothed Wasserstein cost \(T_p^{(σ)}(μ,ν)=W_p(μ*γ_σ,ν*γ_σ)^p\) on \(\R^d\). For fixed smoothing and finite polynomial moments \(M_{q_μ}(μ)<\infty\), \(M_{q_ν}(ν)<\infty\), with \(q_μ,q_ν>p\), we establish upper bounds in probability of order \(ρ_{q_μ,p,d}(m)+ρ_{q_ν,p,d}(n)\). Here \(ρ_{q,p,d}(N)=N^{-(q-p)/(q+d)}\) for \(p<q<d+2p\), \(N^{-1/2}\log N\) at \(q=d+2p\), and \(N^{-1/2}\) for \(q>d+2p\). This order also holds in expectation under \(q_μ,q_ν\ge2p\). When the smoothed population distance is positive, the cost bound yields this rate for the distance itself. For \(p>1\) and \(q_μ,q_ν>d+2p\), we also derive a first-order expansion, a separated two-sample central limit theorem, and a sample-splitting variance estimator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09084 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Two-Sample Inference for Gaussian-Smoothed Wasserstein Costs with Finite Moments Yang, Jiaping Zhang, Yunxin Statistics Theory Gaussian smoothing has emerged as an effective technique for reducing the sample complexity of optimal transport. In this paper, we study the two-sample plug-in estimator of the Gaussian-smoothed Wasserstein cost \(T_p^{(σ)}(μ,ν)=W_p(μ*γ_σ,ν*γ_σ)^p\) on \(\R^d\). For fixed smoothing and finite polynomial moments \(M_{q_μ}(μ)<\infty\), \(M_{q_ν}(ν)<\infty\), with \(q_μ,q_ν>p\), we establish upper bounds in probability of order \(ρ_{q_μ,p,d}(m)+ρ_{q_ν,p,d}(n)\). Here \(ρ_{q,p,d}(N)=N^{-(q-p)/(q+d)}\) for \(p<q<d+2p\), \(N^{-1/2}\log N\) at \(q=d+2p\), and \(N^{-1/2}\) for \(q>d+2p\). This order also holds in expectation under \(q_μ,q_ν\ge2p\). When the smoothed population distance is positive, the cost bound yields this rate for the distance itself. For \(p>1\) and \(q_μ,q_ν>d+2p\), we also derive a first-order expansion, a separated two-sample central limit theorem, and a sample-splitting variance estimator. |
| title | Two-Sample Inference for Gaussian-Smoothed Wasserstein Costs with Finite Moments |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2605.09084 |