Efficient l_{alpha} Distance Approximation for High Dimensional Data Using alpha-Stable Projection

TL;DR

This paper introduces an efficient method for approximating l_{alpha} distances in high-dimensional data using alpha-stable projections, enabling fast and statistically efficient dimension reduction.

Contribution

It presents a novel projection-based algorithm leveraging alpha-stable distributions for accurate l_{alpha} distance approximation with low computational cost.

Findings

Achieves full statistical efficiency in distance recovery.

Runs in O(k) time after initial setup.

Applicable to large high-dimensional datasets.

Abstract

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 alpha-stable distributions. We show that l_{alpha} 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.