@online{Marx_arXiv2110.13883,
TITLE = {{Estimating Mutual Information via Geodesic $k$NN}},
AUTHOR = {Marx, Alexander and Fischer, Jonas},
LANGUAGE = {eng},
URL = {https://arxiv.org/abs/2110.13883},
EPRINT = {2110.13883},
EPRINTTYPE = {arXiv},
YEAR = {2021},
ABSTRACT = {Estimating mutual information (MI) between two continuous random variables<br>$X$ and $Y$ allows to capture non-linear dependencies between them,<br>non-parametrically. As such, MI estimation lies at the core of many data<br>science applications. Yet, robustly estimating MI for high-dimensional $X$ and<br>$Y$ is still an open research question.<br> In this paper, we formulate this problem through the lens of manifold<br>learning. That is, we leverage the common assumption that the information of<br>$X$ and $Y$ is captured by a low-dimensional manifold embedded in the observed<br>high-dimensional space and transfer it to MI estimation. As an extension to<br>state-of-the-art $k$NN estimators, we propose to determine the $k$-nearest<br>neighbours via geodesic distances on this manifold rather than form the ambient<br>space, which allows us to estimate MI even in the high-dimensional setting. An<br>empirical evaluation of our method, G-KSG, against the state-of-the-art shows<br>that it yields good estimations of the MI in classical benchmark, and manifold<br>tasks, even for high dimensional datasets, which none of the existing methods<br>can provide.<br>},
}