Asymmetry index for data and its verification in dimensionality reduction and data visualization

We propose an asymmetry index as a measure of degree of asymmetry of a given dataset. It provides an additional information on a dataset allowing to guide and improve any further analysis. The index reflects the intensity of the asymmetric relationships among data resulting from hierarchical data structure. Using the information retrieved by our asymmetry index, one obtains a justification and explanation of the effectiveness of the subsequent asymmetric data analysis methods, as well as helpful preparation to asymmetrizing the tools for the further analysis. The asymmetry index is based on the k-nearest neighbors graph representing the considered data. Therefore, it uses the intrinsic geometry-based information on the data, in this way, providing an insight into the data structure. Our experiments on real data are designed to verify the usefulness of the asymmetry index and the correctness of its theoretical fundamentals. In our empirical validation, we employ the symmetric and asymmetric dimensionality reduction algorithms and evaluate their results on the basis of clustering in the 2-dimensional visualization space. We test, whether our index indeed predicts the level of superiority of the asymmetric methods over their symmetric counterparts.