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We review connections between phase transitions in high-dimensionalcombinatorial geometry and phase transitions occurring in modernhigh-dimensional data analysis and signal processing. In data analysis, suchtransitions arise as abrupt breakdown of linear model selection, robust datafitting or compressed sensing reconstructions, when the complexity of the modelor the number of outliers increases beyond a threshold. In combinatorialgeometry these transitions appear as abrupt changes in the properties of facecounts of convex polytopes when the dimensions are varied. The thresholds inthese very different problems appear in the same critical locations afterappropriate 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-ubiquitoushigh-throughput data analysis can be successful; for robustness, they placehard limits on the degree to which standard robust fitting methods can tolerateoutliers before breaking down; for compressed sensing, they define the sharpboundary of the undersampling/sparsity tradeoff in undersampling theorems.  Existing derivations of phase transitions in combinatorial geometry assumethe underlying matrices have independent and identically distributed (iid)Gaussian elements. In applications, however, it often seems that Gaussianity isnot required. We conducted an extensive computational experiment and formalinferential analysis to test the hypothesis that these phase transitions are{\it universal} across a range of underlying matrix ensembles. The experimentalresults are consistent with an asymptotic large-$n$ universality across matrixensembles; finite-sample universality can be rejected.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

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