The Effect of Random Projection on Local Intrinsic Dimensionality

Much attention has been given in the research literature to the study of distance-preserving random projections of discrete data sets, the limitations of which are established by the classical Johnson-Lindenstrauss existence lemma In this theoretical paper, we analyze the effect of random projection on a natural measure of the local intrinsic dimensionality (LID) of smooth distance distributions in the Euclidean setting. The main contribution of the paper consists of upper and lower bounds on the LID in the vicinity of a reference point after random projection. The bounds depend only on the LID in the original data domain and the target dimension of the projection; as the difference between the target and intrinsic dimensionalities grows, these bounds converge to the LID of the original domain. The paper concludes with a brief discussion of the implications for applications in databases, machine learning and data mining.