Common Information Dimension
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909244774678528 |
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| author | Hanna, Osama Li, Xinlin Diggavi, Suhas Fragouli, Christina |
| author_facet | Hanna, Osama Li, Xinlin Diggavi, Suhas Fragouli, Christina |
| contents | The exact common information between a set of random variables $X_1,...,X_n$ is defined as the minimum entropy of a shared random variable that allows for the exact distributive simulation of $X_1,...,X_n$. It has been established that, in certain instances, infinite entropy is required to achieve distributive simulation, suggesting that continuous random variables may be needed in such scenarios. However, to date, there is no established metric to characterize such cases. In this paper, we propose the concept of Common Information Dimension (CID) with respect to a given class of functions $\mathcal{F}$, defined as the minimum dimension of a random variable $W$ required to distributively simulate a set of random variables $X_1,...,X_n$, such that $W$ can be expressed as a function of $X_1,\cdots,X_n$ using a member of $\mathcal{F}$. Our main contributions include the computation of the common information dimension for jointly Gaussian random vectors in a closed form, with $\mathcal{F}$ being the linear functions class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_06469 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Common Information Dimension Hanna, Osama Li, Xinlin Diggavi, Suhas Fragouli, Christina Information Theory The exact common information between a set of random variables $X_1,...,X_n$ is defined as the minimum entropy of a shared random variable that allows for the exact distributive simulation of $X_1,...,X_n$. It has been established that, in certain instances, infinite entropy is required to achieve distributive simulation, suggesting that continuous random variables may be needed in such scenarios. However, to date, there is no established metric to characterize such cases. In this paper, we propose the concept of Common Information Dimension (CID) with respect to a given class of functions $\mathcal{F}$, defined as the minimum dimension of a random variable $W$ required to distributively simulate a set of random variables $X_1,...,X_n$, such that $W$ can be expressed as a function of $X_1,\cdots,X_n$ using a member of $\mathcal{F}$. Our main contributions include the computation of the common information dimension for jointly Gaussian random vectors in a closed form, with $\mathcal{F}$ being the linear functions class. |
| title | Common Information Dimension |
| topic | Information Theory |
| url | https://arxiv.org/abs/2305.06469 |