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

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.Comment: 47 pages, 24 figures, 10 table

Orthonormal Expansion l1-Minimization Algorithms for Compressed Sensing

Compressed sensing aims at reconstructing sparse signals from significantly reduced number of samples, and a popular reconstruction approach is $\ell_1$-norm minimization. In this correspondence, a method called orthonormal expansion is presented to reformulate the basis pursuit problem for noiseless compressed sensing. Two algorithms are proposed based on convex optimization: one exactly solves the problem and the other is a relaxed version of the first one. The latter can be considered as a modified iterative soft thresholding algorithm and is easy to implement. Numerical simulation shows that, in dealing with noise-free measurements of sparse signals, the relaxed version is accurate, fast and competitive to the recent state-of-the-art algorithms. Its practical application is demonstrated in a more general case where signals of interest are approximately sparse and measurements are contaminated with noise.Comment: 7 pages, 2 figures, 1 tabl

On Phase Transition of Compressed Sensing in the Complex Domain

The phase transition is a performance measure of the sparsity-undersampling tradeoff in compressed sensing (CS). This letter reports our first observation and evaluation of an empirical phase transition of the $\ell_1$ minimization approach to the complex valued CS (CVCS), which is positioned well above the known phase transition of the real valued CS in the phase plane. This result can be considered as an extension of the existing phase transition theory of the block-sparse CS (BSCS) based on the universality argument, since the CVCS problem does not meet the condition required by the phase transition theory of BSCS but its observed phase transition coincides with that of BSCS. Our result is obtained by applying the recently developed ONE-L1 algorithms to the empirical evaluation of the phase transition of CVCS.Comment: 4 pages, 3 figure