Universality laws for randomized dimension reduction, with applications

TL;DR

This paper establishes universality laws for randomized dimension reduction, showing a phase transition in success probability that applies broadly across data sets and maps, with implications for algorithms and statistical methods.

Contribution

It proves that the success of randomized dimension reduction exhibits a universal phase transition, independent of specific maps, across a wide class of data sets.

Findings

Success probability undergoes a phase transition as embedding dimension increases.

The phase transition point is consistent across different randomized maps.

Results apply to various applications in signal processing, statistics, and numerical algorithms.

Abstract

Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability. This paper studies a natural family of randomized dimension reduction maps and a large class of data sets. It proves that there is a phase transition in the success probability of the dimension reduction map as the embedding dimension increases. For a given data set, the location of the phase transition is the same for all maps in this family. Furthermore, each map has the same stability properties, as quantified through the restricted minimum singular value. These results can be viewed as new universality laws in high-dimensional stochastic geometry. Universality laws for randomized dimension reduction have many applications in applied mathematics, signal processing, and statistics. They yield design principles for numerical linear algebra algorithms, for compressed sensing measurement ensembles, and for random linear codes. Furthermore, these results have implications for the performance of statistical estimation methods under a large class of random experimental designs.