  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.
