Observed Universality of Phase Transitions in High-Dimensional Geometry, with Implications for Modern Data Analysis and Signal Processing

TL;DR

This paper explores the universal nature of phase transitions in high-dimensional geometry and data analysis, showing that these phenomena occur across various matrix ensembles and have critical implications for modern data processing methods.

Contribution

It provides empirical evidence supporting the universality of phase transitions across different matrix ensembles in high-dimensional problems.

Findings

Phase transitions occur at similar thresholds across different matrix ensembles.

Finite-sample universality can be rejected, indicating asymptotic behavior.

Thresholds define limits for model success, robustness, and compressed sensing.

Abstract

We review connections between phase transitions in high-dimensional combinatorial geometry and phase transitions occurring in modern high-dimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt breakdown of linear model selection, robust data fitting or compressed sensing reconstructions, when the complexity of the model or the number of outliers increases beyond a threshold. In combinatorial geometry these transitions appear as abrupt changes in the properties of face counts of convex polytopes when the dimensions are varied. The thresholds in these very different problems appear in the same critical locations after appropriate calibration of variables. These thresholds are important in each subject area: for linear modelling, they place hard limits on the degree to which the now-ubiquitous high-throughput data analysis can be successful; for robustness, they place hard limits on the degree to which standard robust fitting methods can tolerate outliers before breaking down; for compressed sensing, they define the sharp boundary of the undersampling/sparsity tradeoff in undersampling theorems. Existing derivations of phase transitions in combinatorial geometry assume the underlying matrices have independent and identically distributed (iid) Gaussian elements. In applications, however, it often seems that Gaussianity is not required. We conducted an extensive computational experiment and formal inferential analysis to test the hypothesis that these phase transitions are {\it universal} across a range of underlying matrix ensembles. The experimental results are consistent with an asymptotic large-$n$ universality across matrix ensembles; finite-sample universality can be rejected.