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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2406.03427 |
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| _version_ | 1866910473255911424 |
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| author | Klartag, Bo'az Ordentlich, Or |
| author_facet | Klartag, Bo'az Ordentlich, Or |
| contents | Let $ν$ and $μ$ be probability distributions on $\mathbb{R}^n$, and $ν_s,μ_s$ be their evolution under the heat flow, that is, the probability distributions resulting from convolving their density with the density of an isotropic Gaussian random vector with variance $s$ in each entry. This paper studies the rate of decay of $s\mapsto D(ν_s\|μ_s)$ for various divergences, including the $χ^2$ and Kullback-Leibler (KL) divergences. We prove upper and lower bounds on the strong data-processing inequality (SDPI) coefficients corresponding to the source $μ$ and the Gaussian channel. We also prove generalizations of de Brujin's identity, and Costa's result on the concavity in $s$ of the differential entropy of $ν_s$. As a byproduct of our analysis, we obtain new lower bounds on the mutual information between $X$ and $Y=X+\sqrt{s} Z$, where $Z$ is a standard Gaussian vector in $\mathbb{R}^n$, independent of $X$, and on the minimum mean-square error (MMSE) in estimating $X$ from $Y$, in terms of the Poincaré constant of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03427 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The strong data processing inequality under the heat flow Klartag, Bo'az Ordentlich, Or Information Theory Functional Analysis Let $ν$ and $μ$ be probability distributions on $\mathbb{R}^n$, and $ν_s,μ_s$ be their evolution under the heat flow, that is, the probability distributions resulting from convolving their density with the density of an isotropic Gaussian random vector with variance $s$ in each entry. This paper studies the rate of decay of $s\mapsto D(ν_s\|μ_s)$ for various divergences, including the $χ^2$ and Kullback-Leibler (KL) divergences. We prove upper and lower bounds on the strong data-processing inequality (SDPI) coefficients corresponding to the source $μ$ and the Gaussian channel. We also prove generalizations of de Brujin's identity, and Costa's result on the concavity in $s$ of the differential entropy of $ν_s$. As a byproduct of our analysis, we obtain new lower bounds on the mutual information between $X$ and $Y=X+\sqrt{s} Z$, where $Z$ is a standard Gaussian vector in $\mathbb{R}^n$, independent of $X$, and on the minimum mean-square error (MMSE) in estimating $X$ from $Y$, in terms of the Poincaré constant of $X$. |
| title | The strong data processing inequality under the heat flow |
| topic | Information Theory Functional Analysis |
| url | https://arxiv.org/abs/2406.03427 |