Structure preservation via the Wasserstein distance
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917888372244480 |
|---|---|
| author | Bartl, Daniel Mendelson, Shahar |
| author_facet | Bartl, Daniel Mendelson, Shahar |
| contents | We show that under minimal assumptions on a random vector $X\in\mathbb{R}^d$ and with high probability, given $m$ independent copies of $X$, the coordinate distribution of each vector $(\langle X_i,θ\rangle)_{i=1}^m$ is dictated by the distribution of the true marginal $\langle X,θ\rangle$. Specifically, we show that with high probability, \[\sup_{θ\in S^{d-1}} \left( \frac{1}{m}\sum_{i=1}^m \left|\langle X_i,θ\rangle^\sharp - λ^θ_i \right|^2 \right)^{1/2} \leq c \left( \frac{d}{m} \right)^{1/4},\] where $λ^θ_i = m\int_{(\frac{i-1}{m}, \frac{i}{m}]} F_{ \langle X,θ\rangle }^{-1}(u)\,du$ and $a^\sharp$ denotes the monotone non-decreasing rearrangement of $a$. Moreover, this estimate is optimal.
The proof follows from a sharp estimate on the worst Wasserstein distance between a marginal of $X$ and its empirical counterpart, $\frac{1}{m} \sum_{i=1}^m δ_{\langle X_i, θ\rangle}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07058 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Structure preservation via the Wasserstein distance Bartl, Daniel Mendelson, Shahar Statistics Theory Functional Analysis Probability We show that under minimal assumptions on a random vector $X\in\mathbb{R}^d$ and with high probability, given $m$ independent copies of $X$, the coordinate distribution of each vector $(\langle X_i,θ\rangle)_{i=1}^m$ is dictated by the distribution of the true marginal $\langle X,θ\rangle$. Specifically, we show that with high probability, \[\sup_{θ\in S^{d-1}} \left( \frac{1}{m}\sum_{i=1}^m \left|\langle X_i,θ\rangle^\sharp - λ^θ_i \right|^2 \right)^{1/2} \leq c \left( \frac{d}{m} \right)^{1/4},\] where $λ^θ_i = m\int_{(\frac{i-1}{m}, \frac{i}{m}]} F_{ \langle X,θ\rangle }^{-1}(u)\,du$ and $a^\sharp$ denotes the monotone non-decreasing rearrangement of $a$. Moreover, this estimate is optimal. The proof follows from a sharp estimate on the worst Wasserstein distance between a marginal of $X$ and its empirical counterpart, $\frac{1}{m} \sum_{i=1}^m δ_{\langle X_i, θ\rangle}$. |
| title | Structure preservation via the Wasserstein distance |
| topic | Statistics Theory Functional Analysis Probability |
| url | https://arxiv.org/abs/2209.07058 |