Efficient lα Distance Approximation for High Dimensional Data Using α-Stable Projection

In recent years, large high-dimensional data sets have become commonplace in a wide range of applications in science and commerce. Techniques for dimension reduction are of primary concern in statistical analysis. Projection methods play an important role. We investigate the use of projection algorithms that exploit properties of the α-stable distributions. We show that lα distances and quasi-distances can be recovered from random projections with full statistical efficiency by L-estimation. The computational requirements of our algorithm are modest; after a once-and-for-all calculation to determine an array of length k, the algorithm runs in O(k) time for each distance, where k is the reduced dimension of the projection.