Measuring Dependency Via Intrinsic Dimensionality

Measuring the amount of dependency among multiple variables is an important task in pattern recognition. In the last few years, many new dependency measures have been developed for the exploration of functional relationships. In this paper, we develop a dependency measure between variables based on an extreme-value theoretic treatment of intrinsic dimensionality. Our measure identifies variables with low intrinsic dimension that is, those that support embeddings of the data within low-dimensional manifolds. To build a dependency measure on strong foundations, we theoretically prove a connection between information theory and intrinsic dimensionality theory. This allows us also to propose novel estimators of intrinsic dimensionality. Finally, we show that our dependency measure enables to find patterns that cannot be found by other state-of-the-art measures on real and synthetic data.