Distance Estimation for High-Dimensional Discrete Distributions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908521458565120 |
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| author | Kumar, Gunjan Meel, Kuldeep S. Pote, Yash |
| author_facet | Kumar, Gunjan Meel, Kuldeep S. Pote, Yash |
| contents | Given two distributions $\mathcal{P}$ and $\mathcal{Q}$ over a high-dimensional domain $\{0,1\}^n$, and a parameter $\varepsilon$, the goal of distance estimation is to determine the statistical distance between $\mathcal{P}$ and $\mathcal{Q}$, up to an additive tolerance $\pm \varepsilon$. Since exponential lower bounds (in $n$) are known for the problem in the standard sampling model, research has focused on richer query models where one can draw conditional samples. This paper presents the first polynomial query distance estimator in the conditional sampling model ($\mathsf{COND}$).
We base our algorithm on the relatively weaker \textit{subcube conditional} sampling ($\mathsf{SUBCOND}$) oracle, which draws samples from the distribution conditioned on some of the dimensions. $\mathsf{SUBCOND}$ is a promising model for widespread practical use because it captures the natural behavior of discrete samplers. Our algorithm makes $\tilde{\mathcal{O}}(n^3/\varepsilon^5)$ queries to $\mathsf{SUBCOND}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04264 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Distance Estimation for High-Dimensional Discrete Distributions Kumar, Gunjan Meel, Kuldeep S. Pote, Yash Data Structures and Algorithms Probability Given two distributions $\mathcal{P}$ and $\mathcal{Q}$ over a high-dimensional domain $\{0,1\}^n$, and a parameter $\varepsilon$, the goal of distance estimation is to determine the statistical distance between $\mathcal{P}$ and $\mathcal{Q}$, up to an additive tolerance $\pm \varepsilon$. Since exponential lower bounds (in $n$) are known for the problem in the standard sampling model, research has focused on richer query models where one can draw conditional samples. This paper presents the first polynomial query distance estimator in the conditional sampling model ($\mathsf{COND}$). We base our algorithm on the relatively weaker \textit{subcube conditional} sampling ($\mathsf{SUBCOND}$) oracle, which draws samples from the distribution conditioned on some of the dimensions. $\mathsf{SUBCOND}$ is a promising model for widespread practical use because it captures the natural behavior of discrete samplers. Our algorithm makes $\tilde{\mathcal{O}}(n^3/\varepsilon^5)$ queries to $\mathsf{SUBCOND}$. |
| title | Distance Estimation for High-Dimensional Discrete Distributions |
| topic | Data Structures and Algorithms Probability |
| url | https://arxiv.org/abs/2308.04264 |