Mutual information of high-dimensional random variables: estimation by frontier mutual information

Many natural sources of information, including genes, yield intricate datasets characterized by high-dimensional random variables. However, the computational constraints and information loss have often limited the accuracy of mutual information (MI) computations in such datasets. To address these limitations, we introduce a novel metric, micromutual information, which measures the information exchange at each cell level within high-dimensional contingency tables. This methodology represents an extension of our previous techniques and employs a linear index approach. The method simplifies complex, high-dimensional genetic data into a one-dimensional format, thereby improving computational efficiency while preserving the intricate structure of gene interactions. Theorems are developed which demonstrate how the sum of micromutual information asymptotically converges to the total MI for multidimensional variables. Our findings indicate that the maximum value of micromutual information, termed MIfront, adheres to an extreme value distribution. The observation of MIfront provides a streamlined approach to estimating the total MI, due to the simplicity of measuring the micromutual information of just one cell. This approach has the potential to improve data analysis in genomics and other fields that deal with multidimensional information.